Research Article Approximate Preservers on Banach Algebras...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 757646 12 pageshttpdxdoiorg1011552013757646

Research ArticleApproximate Preservers on Banach Algebras and Clowast-Algebras

M Burgos1 A C Maacuterquez-Garciacutea2 and A Morales-Campoy2

1 Campus de Jerez Facultad de Ciencias Sociales y de la Comunicacion Avenida de la Universidad sn Jerez 11405 Cadiz Spain2Departamento de Matematicas Universidad de Almerıa 04120 Almerıa Spain

Correspondence should be addressed to M Burgos mariaburgosucaes

Received 3 October 2013 Accepted 6 November 2013

Academic Editor Antonio M Peralta

Copyright copy 2013 M Burgos et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The aim of the present paper is to give approximate versions of Huarsquos theorem and other related results for Banach algebras andClowast-algebras We also study linear maps approximately preserving the conorm between unital Clowast-algebras

1 Preliminaries

Awell known formulation of the celebratedHuarsquos theorem [1]asserts that every bijective additive map 119879 119870 rarr 119870 on adivision ring 119870 such that 119879(1) = 1 and 119879(119909

minus1

) = 119879(119909)minus1

for every invertible element 119909 is either an automorphism oran antiautomorphism This result was later moved to matrixalgebras in [2] and finally extended to Banach algebras in [3](see also [4]) In [3] the author called the previous relationstrongly preserving invertibility

Let 119860 and 119861 be unital Banach algebras Recall thatan additive map 119879 119860 rarr 119861 is a Jordan homomorphismif 119879(119886119887 + 119887119886) = 119879(119886)119879(119887) + 119879(119887)119879(119886) for every 119886 119887 isin 119860or equivalently 119879(119886

2

) = 119879(119886)2 for all 119886 isin 119860 Obvious

examples of Jordan homomorphisms are homomorphismsand anti-homomorphisms It is well known that every unital(ie 119879(1) = 1) Jordan homomorphism strongly preservesinvertibility Reciprocally one of the results in [4] proves thatevery additive map between Banach algebras strongly pre-serving invertibility is amultiple of a Jordan homomorphismIn particular if the map is unital the map itself is a Jordanhomomorphism

There exist also other versions of Huarsquos theorem involvingsome important kinds of generalized invertibility Given aring 119860 and an element 119886 isin 119860 119886 is said to be Drazininvertible if there exist 119887 isin 119860 and a nonnegative integer 119896

such that

119887119886119887 = 119887 119886119896

119887119886 = 119886119896

119886119887 = 119887119886 (1)

Such 119887 is unique whenever it exists In this case it is called theDrazin inverse of 119886 and it is denoted by 119887 = 119886

119863This notionwas introduced by Drazin in [5] and it has proved useful inmany fields of pure and appliedmathematics (see for instance[6 7]) If the previous identities are satisfied for some 119887 isin 119860

and 119896 = 1 119887 is called the group inverse of 119886 and we willdenote it by 119886

119866 Let 119860minus1 119860

119863 and 119860119866 denote the sets of

all invertibleDrazin invertible and group invertible elementsin 119860 respectively Clearly

119860minus1

sub 119860119866

sub 119860119863

(2)

Linear or additive maps between unital Banach algebrasstrongly preserving group andDrazin invertibilitywere intro-duced in [3 4] (see also [8] where the author described addi-tive maps between operator algebras of infinite-dimensionalHilbert spaces strongly preserving Drazin invertibility) If119879 119860 rarr 119861 is a Jordan homomorphism between Banachalgebras it was shown in [3 Theorem 21] that

(i) 119879 strongly preserves group invertibility that is119879(119886119866

) = 119879(119886)119866 for every 119886 isin 119860

119866(ii) 119879 strongly preserves Drazin invertibility that is

119879(119886119863

) = 119879(119886)119863 for every 119886 isin 119860

119863

Conversely if 119879 119860 rarr 119861 is an additive map stronglypreserving invertibility group invertibility or Drazin invert-ibility and 119879(1) = 1 (resp 119879(1) is invertible or 1 isin 119879(119860))then 119879 (resp 119879(1)119879) is a unital Jordan homomorphism

2 Abstract and Applied Analysis

and 119879(1) commutes with the image of 119879 [4 Theorem42] In [9] the authors showed that the same holds withoutany hypothesis on 119879(1) and even when 119861 is not necessarilyunital

Recall that an element 119886 isin 119860 is called regular if thereexists 119887 isin 119860 (not necessarily unique) such that 119886 = 119886119887119886

and 119887 = 119887119886119887 Notice that the first equality 119886 = 119886119887119886 isa necessary and sufficient condition for 119886 to be regular andthat 119901 = 119886119887 and 119902 = 119887119886 are idempotents in 119860 fulfilling119886119860 = 119901119860 and 119860119886 = 119860119902 For an element 119886 isin 119860 let usconsider the left and right multiplication operators 119871

119886 119909 997891rarr

119886119909 and 119877119886 119909 997891rarr 119909119886 respectively If 119886 is regular then

so are 119871119886

and 119877119886 and thus its ranges 119886119860 = 119871

119886(119860) and

119860119886 = 119877119886(119860) are both closed

Regular elements in a unital 119862lowast-algebra 119860 were studiedby Harte and Mbekhta in [10 11]

An element 119886 isin 119860 has Moore-Penrose inverse 119887 when119886 = 119886119887119886 119887 = 119887119886119887 and the associated idempotents 119886119887 and119887119886 are self-adjoint In the aforementioned papers by Harteand Mbekhta it is shown that every regular element in a 119862lowast-algebra has a Moore-Penrose inverse and that it is uniqueFor a regular element 119886 in a 119862lowast-algebra 119860 119886dagger will denoteits Moore-Penrose inverse The set of all Moore-Penroseinvertible elements in a 119862lowast-algebra 119860 will be denoted by119860daggerLet 119860 and 119861 be unital 119862lowast-algebras A linear map

119879 119860 rarr 119861 strongly preserves Moore-Penrose invert-ibility if 119879(119886

dagger

) = 119879(119886)dagger for every 119886 isin 119860

daggerEvery Jordan lowast-homomorphism strongly preserves Moore-Penrose invertibility and the question is whether or notthe converse holds Some partial positive answers are givenby Mbekhta in [3] and more recently by the authors ofthe present paper in [12] when 119860 has a rich structure ofprojections The problem for general 119862

lowast-algebras remainsopen However we can consider an alternative approach

Recall that the class of119862lowast-algebras is contained in a widerclass of Banach spaces the so called 119869119861lowast-triples in which theconcept of regularity extends the one given for 119862lowast-algebrasA 119869119861lowast-triple is a complex Banach space 119864 together with a

continuous triple product sdot sdot sdot 119864 times 119864 times 119864 rarr 119864 whichis conjugate linear in the middle variable and symmetric andbilinear in the outer variables satisfying that

(a) 119871(119886 119887)119871(119909 119910) = 119871(119909 119910)119871(119886 119887) + 119871(119871(119886 119887)119909 119910) minus

119871(119909 119871(119887 119886)119910) where 119871(119886 119887) is the operatoron 119864 given by 119871(119886 119887)119909 = 119886 119887 119909

(b) 119871(119886 119886) is an hermitian operator with nonnegativespectrum

(c) 119871(119886 119886) = 1198862

For each 119909 in a 119869119861lowast-triple 119864 119876(119909) will stand for the

conjugate linear operator on 119864 defined by 119876(119909)(119910) =

119909 119910 119909An element 119886 in a 119869119861lowast-triple 119864 is called von Neumann

regular if there exists (a unique) 119887 isin 119864 such that 119876(119886)(119887) =

119886 119876(119887)(119886) = 119887 and 119876(119886)119876(119887) = 119876(119887)119876(119886) Theelement 119887 is called the generalized inverse of 119886 and it willbe denoted by 119886

and We will also note by 119864and the set of all

elements in 119864 with generalized inverse We refer to [13ndash17]for basics results on von Neumann regularity in 119869119861lowast-triples

Every 119862lowast-algebra is a 119869119861lowast-triple via the triple product

given by

119909 119910 119911 =

1

2

(119909119910lowast

119911 + 119911119910lowast

119909) (3)

For a 119862lowast-algebra 119860 it is well known that 119860dagger = 119860and and for

every regular element 119886 in 119860 we have 119886and = (119886dagger

)lowast

A linear map 119879 119864 rarr 119865 between 119869119861lowast-triples is a triplehomomorphism if

119879 (119909 119910 119911) = 119879 (119909) 119879 (119910) 119879 (119911) (4)

for every 119909 119910 119911 isin 119864 Every triple homomorphism 119879

119864 rarr 119865 between 119869119861lowast-triples strongly preserves generalizedinvertibility that is 119879(119909

and

) = 119879(119909)and for every 119909 isin 119864

andIn [9] the authors characterized the triple homomorphismbetween 119862

lowast-algebras as the linear maps strongly preservinggeneralized invertibility As a consequence it is proved that aself-adjoint linearmap from a unital119862lowast-algebra 119860 into a119862lowast-algebra 119861 is a triple homomorphism if and only if it stronglypreserves Moore-Penrose invertibility [9 Theorem 35]

The Hua-type descriptions belong to the framework oflinear preserver problemsThis has become an active researcharea in many topics of matrix theory operator theory andBanach algebras theory Some of the most popular preserverproblems are those dealing with determining the linear mapspreserving properties related to invertibility Every Jordanisomorphism 119879 119860 rarr 119861 between unital Banach algebrasis unital and preserves invertibility in both directions [18Proposition 13] or equivalently preserves the spectrum thatis 120590(119879(119886)) = 120590(119886) for every 119886 isin 119860 The celebratedKaplanskyrsquos conjecture [19] reformulated by Aupetit in [20]states that every unital surjective linear map 119879 betweenunital semisimple Banach algebras preserving invertibilityin both directions is a Jordan isomorphism Many partialpositive results are known so far [18 20ndash24] but the generalproblem is still open even in the class of 119862lowast-algebras In thecommutative setting the classical Gleason-Kahane-Zelazkotheorem (see [23]) states that a linear functional 120593 on aunital complex Banach algebra 119860 is multiplicative if andonly if 120593(119886) isin 120590(119886) for all 119886 isin 119860 On the other handJafarian and Sourour proved in [22] that every spectrumpreserving surjective linear map 119879 B(119883) rarr B(119884)

is either an isomorphism or an anti-isomorphism whereB(119883) denotes the Banach algebra of all bounded linearoperators on a Banach space 119883

The authors of [25 26] consider the problem of charac-terizing the approximately multiplicative linear functionalsamong all linear functionals on a commutative Banachalgebra in terms of spectra More recently in [27] (see also[28]) Alaminos Extremera and Villena investigate approxi-mate versions of Kaplanskyrsquos problem by providing approx-imate formulations of [18 22] They considered linear mapsthat approximately preserve spectrum or spectral radius onoperator algebras and established the relationship betweenapproximately preserving spectrum (resp spectral radius)

Abstract and Applied Analysis 3

and approximately being a Jordan homomorphism (respweighted Jordan homomorphism)

Let 119860 and 119861 be Banach algebras and 119879 119860 rarr 119861 be abounded linear map Following [27 29] the multiplicativityantimultiplicativity and Jordan-multiplicativity of 119879 can bemeasured by considering the following values

mult (119879)

= sup 119879 (119886119887) minus 119879 (119886) 119879 (119887) 119886 119887 isin 119860 119886 = 119887 = 1

amult (119879)

= sup 119879 (119886119887) minus 119879 (119887) 119879 (119886) 119886 119887 isin 119860 119886 = 119887 = 1

jmult (119879) = sup 10038171003817100381710038171003817119879 (1198862

) minus 119879(119886)210038171003817100381710038171003817 119886 isin 119860 119886 = 1

(5)

respectively Obviously 119879 is a homomorphism (anti-homomorphism Jordan homomorphism) if and only ifmult(119879) = 0 (resp amult(119879) = 0 jmult(119879) = 0)

For a bounded linear map 119879 119860 rarr 119861 between 119862lowast-

algebras we define the triple multiplicativity and the self-adjointness of 119879 respectively as the following quantities

tmult (119879)

= sup 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119886 = 119887 = 119888 = 1

sa (119879) = sup 10038171003817100381710038171003817119879(119886lowast

)

lowast

minus 119879 (119886)

10038171003817100381710038171003817 119886 = 1

(6)

Clearly 119879 119860 rarr 119861 is a triple homomorphism if andonly if tmult(119879) = 0 and 119879 is self-adjoint if and only ifsa(119879) = 0

The aim of the present paper is to bring Hua type theo-rems into this framework In order to make this possible wehave adapted some techniques from [27] involving ultraprod-ucts of Banach algebras Section 2 contains all the technicalresults about invertibility and coset representatives in ultra-products of Banach algebras that we will need throughoutthe paper Section 3 provides approximate versions of Huarsquostheorem for invertibility and group invertibility in Banachalgebras We translate the strongly invertibility preservingcondition 119879(119886

minus1

) = 119879(119886)minus1 into

sup119886=1119886isin119860

minus1

10038171003817100381710038171003817119879 (119886minus1

) minus 119879(119886)minus110038171003817100381710038171003817lt 120576 (7)

and the condition 119879(119886119866

) = 119879(119886)119866 into

sup119886=1119886isin119860

119866

10038171003817100381710038171003817119879 (119886119866

) minus 119879(119886)11986610038171003817100381710038171003817lt 120576 (8)

for some 120576 gt 0 We prove that for every unital Banachalgebras 119860 and 119861 if 120576 rarr 0 in (7) or (8) thenjmult(119879(1)119879) rarr 0 uniformly on any set of linear maps119879 119860 rarr 119861 with norms bounded above

Section 4 includes an approximate formulation of [9Theorem 35] The condition 119879(119886

dagger

) = 119879(119886)dagger for every 119886 isin

119860dagger is replaced by

sup119886=1119886isin119860

dagger

10038171003817100381710038171003817119879 (119886dagger

) minus 119879(119886)dagger10038171003817100381710038171003817lt 120576 (9)

We show that for every unital119862lowast-algebras 119860 and 119861 if 120576 rarr

0 in (9) then tmult(119879) rarr 0 uniformly on any set of linearmaps whose norms are bounded above

In this section we also study linear maps that approxi-mately preserve the conorm Recall that the conorm 119888(119886) ofan element 119886 in a unital Banach algebra is defined asthe reduced minimum modulus of the left multiplicationoperator by 119886 119888(119886) = 120574(119871

119886) [11] For a bounded linear

operator 119879 on a complex Banach space 119883 its reducedminimum modulus is given by

120574 (119879) = inf 119879119909 dist (119909 ker (119879)) ge 1 (10)

It is well known that 120574(119879) gt 0 if and only if 119879 has closedrange In [10] it is shown that an element 119886 in a unital 119862lowast-algebra 119860 is regular if and only if 119888(119886) gt 0 In this case

119888 (119886) =

10038171003817100381710038171003817119886dagger10038171003817100381710038171003817

minus1 (11)

(see [11 Theorem 2]) In [30] the authors characterizedthe linear maps between unital 119862lowast-algebras preserving theconorm By [30 Theorem 31] if 119879 119860 rarr 119861 is a unitallinear map such that 119888(119879(119886)) = 119888(119886) for every 119886 isin 119860 then119879 is an isometric Jordan lowast-homomorphism Also from [30Theorem 32] if 119879 119860 rarr 119861 is a surjective linear mapsuch that 119888(119879(119886)) = 119888(119886) for every 119886 isin 119860 then 119879 is anisometric Jordan lowast-homomorphism multiplied by a unitaryelement Hence we replace the condition 119888(119879(119886)) = 119888(119886) by

sup119886=1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120576 (12)

in order to get approximate versions of [30Theorems 31 and32]

2 Ultraproducts of Banach AlgebrasBasic Tools

Given a free ultrafilter U on N and a sequence of Banachspaces 119883

119899119899isinN the so called ultraproduct of the sequence is

defined as follows

(119883119899)

U=

ℓinfin

(N 119883119899)

NU

(13)

where ℓinfin

(N 119883119899) is the Banach space of all bounded

sequences 119909119899119899isinN with 119909

119899isin 119883119899

for all 119899 isin N equippedwith the ℓ

infinnorm and

NU = 119909119899119899isinN

isin ℓinfin

(N 119883119899) lim

U

1003817100381710038171003817119909119899

1003817100381710038171003817= 0 (14)

If the sequence 119883119899119899isinN = 119883 is constant 119883

U=

ℓinfin

(N 119883)NU is called the ultrapower of 119883 with respectto the ultrafilter U We will denote by x = [119909

119899] the

equivalence class of the sequence 119909119899119899isinN The ultrapower

of a Banach space is also a Banach space provided with thefollowing norm

x = limU

1003817100381710038171003817119909119899

1003817100381710038171003817 (15)


Of course the ultrapower 119860U of a Banach algebra (resp

119862lowast-algebra) is also a Banach algebra (resp 119862lowast-algebra) with

respect to the pointwise operationsFinally for every Banach spaces 119883 and 119884 the canonical

linear isometry B(119883 119884)Urarr B(119883

U 119884

U) given by

T (x) = [119879119899(119909119899)] (16)

for every T = [119879119899] isin B(119883 119884)

U and x = [119909119899] isin

119883U allows us to consider B(119883 119884)

U as a closed subspaceof B(119883

U 119884

U) For 119883 = 119884 the canonical map gives an

isometric unital homomorphism from B(119883)U to B(119883

U)

The reader can see [31] in order to find basic results onultraproducts

Let 119860 be a unital Banach algebra and U a freeultrafilter on N The following proposition is devoted to thedescription of invertible elements in 119860

U through certaincoset representatives The result is probably well known butthe lack of an adequate reference moves us to include it here

Proposition 1 Let a isin 119860U The following assertions are

equivalent

(1) a is invertible(2) a has a coset representative [119906

119899] such that 119906

119899isin 119860minus1

for all 119899 isin N and 119906minus1

119899119899isinN is bounded

Proof For (2) rArr (1) just note that [119906minus1119899] isin 119860

U is an inversefor [119906

119899]

Reciprocally assume that a = [119886119899] is invertible Then

there exists b = [119887119899] isin 119860

U such that ab = ba = 1 That is

limU

1003817100381710038171003817119886119899119887119899minus 1100381710038171003817

1003817= 0

limU

1003817100381710038171003817119887119899119886119899minus 1100381710038171003817

1003817= 0

(17)

Fix 0 lt 120575 lt 1 The above identities imply that

119877 = 119899 isin N

1003817100381710038171003817119886119899119887119899minus 1100381710038171003817

1003817lt 120575 isin U

119871 = 119899 isin N 1003817100381710038171003817119887119899119886119899minus 1100381710038171003817

1003817lt 120575 isin U

(18)

In particular 119886119899is right invertible for every 119899 isin 119877 and 119886

119899

is left invertible for every 119899 isin 119871 Thus 119886119899

is invertible forevery 119899 isin 119868 = 119877 cap 119871 isin U Moreover

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817119887119899(119886119899119887119899)

minus110038171003817100381710038171003817le1003817100381710038171003817119887119899

1003817100381710038171003817

10038171003817100381710038171003817(119886119899119887119899)

minus110038171003817100381710038171003817

le

1003817100381710038171003817119887119899

1003817100381710038171003817

1 minus

10038171003817100381710038171 minus 119886119899119887119899

1003817100381710038171003817

le

1003817100381710038171003817119887119899

1003817100381710038171003817

1 minus 120575

(19)

which shows that 119886minus1

119899 119899 isin 119868 is bounded Therefore we

can assume without loss of generality that 119886119899119899isinN consists

of invertible elements and 119886minus1

119899119899isinN is bounded (Otherwise

we choose

1198861015840

119899=

119886119899 if 119899 isin 119868

1 if 119899 notin 119868(20)

Clearly [119886119899] = [119886

1015840

119899])

Remark 2 It is clear that for a = [119886119899] isin 119860

U with a = 1 wecan choose a coset representative a = [119887

119899] such that 119887

119899 =

1 for all 119899 isin N as follows

119887119899=

119886119899

1003817100381710038171003817119886119899

1003817100381710038171003817

if 119886119899

= 0

1 if 119886119899= 0

(21)

Hence for every invertible element a in 119860U we can find

a coset representative a = [119886119899] fulfilling the conditions in

Proposition 1 and satisfying 119886119899 = a for all 119899 isin N We

will name this one a normalized representative for a

3 Approximate Preservers in Banach Algebras

Let 119860 and 119861 be unital Banach algebras Recall that Boudiand Mbekhta proved in [4 Theorem 22] that an additivemap 119879 119860 rarr 119861 strongly preserves invertibility if andonly if 119879(1)119879 is a unital Jordan homomorphism and 119879(1)

commutes with the range of 119879 Hence for a bounded linearmap 119879 119860 rarr 119861 between unital Banach algebras weconsider the unit-commutativity of 119879 defined as

ucomm (119879) = sup119886=1

119879 (119886) 119879 (1) minus 119879 (1) 119879 (119886) (22)

in order to measure how close is our ldquoapproximately preserv-ing invertibilityrdquo map to fulfilling that property

Obviously every bounded linear map satisfies ucomm(119879) le 2119879

2 The next lemma shows the good behaviour ofthis concept with the ultraproduct of operators

Some arguments in this section are inspired in [27]

Lemma 3 Let 119879119899119899isinN be a bounded sequence of linear maps

between Banach algebras 119860 and 119861 where 119860 is supposed tobe unital Consider T = [119879

119899] 119860

Urarr 119861

U Then

limU

ucomm (119879119899) = ucomm (T) (23)

Proof Given a isin 119860U with a = 1 we can choose a =

[119886119899] with 119886

119899 = 1 for every 119899 isin N Therefore

T (a)T (1) minus T (1)T (a)

= limU

1003817100381710038171003817119879119899(119886119899) 119879119899(1) minus 119879

119899(1) 119879119899(119886119899)

1003817100381710038171003817le lim

Uucomm (119879

119899)

(24)

and hence

ucomm (T) le limU

ucomm (119879119899) (25)

Reciprocally for each 119899 isin N there exists 119886119899isin 119860 with

119886119899 = 1 such that

ucomm (119879119899) minus

1

119899

lt

1003817100381710038171003817119879119899(119886119899) 119879119899(1) minus 119879

119899(1) 119879119899(119886119899)

1003817100381710038171003817 (26)


Taking limit along U we obtain

limU

ucomm (119879119899)

le limU

1003817100381710038171003817119879119899(119886119899) 119879119899(1) minus 119879

119899(1) 119879119899(119886119899)

1003817100381710038171003817

= T (a)T (1) minus T (1)T (a) le ucomm (T)

(27)

Our first main result provides an approximate version ofHuarsquos theorem for Banach algebras [4 Theorem 22] in theabove mentioned

Theorem 4 Let 119860 and 119861 be unital Banach algebras and119870 120576 gt 0 Then there exists 120575 gt 0 such that for every linearmap 119879 119860 rarr 119861 with 119879 lt 119870 the condition

sup119886=1119886isin119860

minus1

10038171003817100381710038171003817119879 (119886minus1

) minus 119879(119886)minus110038171003817100381710038171003817lt 120575 (28)

implies that

jmult (119879 (1) 119879) lt 120576 ucomm (119879) lt 120576 (29)

Proof Suppose that the assertion of the theorem is falseThen we can find 119870

0 1205760gt 0 and a sequence 119879

119899119899isinN of

linear maps from 119860 to 119861 such that for every 119899 isin N

(i) 119879119899 lt 119870

0

(ii) sup119886=1

119879119899(119886minus1

) minus 119879119899(119886)minus1

lt 1119899(iii) jmult(119879

119899(1)119879119899) ge 1205760or ucomm(119879

119899) ge 1205760

Consider that T = [119879119899] 119860

Urarr 119861

U We claim that Tstrongly preserves invertibility Indeed let a isin 119860

U be aninvertible elementWe can suppose without loss of generalitythat a = 1 Let [119886

119899] be its normalized representative with

119886minus1

119899 lt 120572 for some 120572 gt 0 (see Proposition 1 andRemark 2)

As10038171003817100381710038171003817119879119899(119886minus1

119899)

10038171003817100381710038171003817le

1003817100381710038171003817119879119899

1003817100381710038171003817

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817lt 1198700120572

10038171003817100381710038171003817119879119899(119886minus1

119899) minus 119879119899(119886119899)

minus110038171003817100381710038171003817lt 1

(30)

we get

10038171003817100381710038171003817119879119899(119886119899)

minus110038171003817100381710038171003817lt 1198700120572 + 1 (31)

for all 119899 isin N Hence T(a) is invertible and [119879119899(119886119899)minus1

] isits inverse This yields

10038171003817100381710038171003817T (aminus1) minus T(a)minus11003817100381710038171003817

1003817

= limU

10038171003817100381710038171003817119879119899(119886minus1

119899) minus 119879119899(119886119899)

minus110038171003817100381710038171003817le lim

U

1

119899

= 0

(32)

Thus T(aminus1) = T(a)minus1 for every invertible element a isin

119860U By [4 Theorem 22] T(1)T(a2) = (T(1)T(a))2 and

T(1)T(a) = T(a)T(1) for every a isin 119860U We apply

[27 Lemmas 34] and Lemma 3 to obtain respectively thefollowing

0 = jmult (T (1)T) = limU

jmult (119879119899(1) 119879119899)

0 = ucomm (T) = limU

ucomm (119879119899)

(33)

Consequently

119868 = 119899 isin N jmult (119879119899(1) 119879119899) lt 1205760 isin U

119869 = 119899 isin N ucomm (119879119899) lt 1205760 isin U

(34)

Finally 119868 cap 119869 isin U gives us the desired contradiction

Our goal now is to achieve a group invertibility versionfor the previous theorem Recall that given an additive map119879 119860 rarr 119861 from a unital Banach algebra 119860 into a Banachalgebra 119861 by [9 Theorem 24] if 119879 strongly preservesgroup invertibility then 119879(1)119879 is a Jordan homomorphismand 119879(1) commutes with the range of 119879 In order to takeadvantage of Proposition 1 our first step is to improve [9Theorem 24] by showing that all the information requiredis located in 119860

minus1Recall that the so-called Huarsquos identity asserts that if

119886 119887 and 119886 minus 119887minus1 are invertible elements in a ring then

(119886minus1

minus (119886 minus 119887minus1

)

minus1

)

minus1

= 119886 minus 119886119887119886 (35)

Theorem 5 Let 119860 and 119861 be Banach algebras 119860 beingunital and 119879 119860 rarr 119861 be an additive map suchthat 119879(119886minus1) = 119879(119886)

119866 for all 119886 isin 119860minus1Then 119879(1)119879 is a Jordanhomomorphism and 119879(1) commutes with 119879(119860)

Proof A look to the arguments employed in [9 Lemma 21]allows us to show that 119879 preserves the cubes of the invertibleelements Indeed given 119906 isin 119860

minus1 and 120582 isin Q with 0 lt |120582| lt

119906minus1

minus2 as 120582minus1

119906 and 119906 minus 120582119906minus1 are invertible elements we

can apply Huarsquo s identity to obtain

(119906minus1


)

minus1

)

minus1

= 119906 minus 119906 (120582minus1

119906) 119906 = 119906 minus 120582minus1

1199063

(36)

Let us assume that 119879(119906) = 0 Since 119879(119906) isin 119861119866 it follows that

119879(119906) is invertible in the unital Banach algebra 119901119861119901 for 119901 =

119879(119906)119879(119906)119866 with inverse 119879(119906)119866 Identity (36) applied for 119879(119906)

and 0 lt |120582| lt 119879(119906)119866

minus2 gives

119879 (119906) minus 120582minus1

119879(119906)3

= (119879(119906)119866

minus (119879 (119906) minus 120582119879(119906)119866

)

119866

)

119866

(37)


Hence for every 120582 isin Q such that 0 lt |120582| lt

min119906minus1minus2 119879(119906)119866minus2

we get

119879 (119906) minus 120582minus1

119879(119906)3

= (119879 (119906minus1

) minus 119879(119906 minus 120582119906minus1

)

119866

)

119866

= 119879(119906minus1


)

minus1

)

119866

= 119879((119906minus1


)

minus1

)

minus1

)

= 119879 (119906 minus 120582minus1

1199063

) = 119879 (119906) minus 120582minus1

119879 (1199063

)

(38)

Hence 119879(1199063

) = 119879(119906)3 as desired From this last identity

reasoning as in [9 Proposition 23] we deduce that thefollowing equalities hold for every 119909 isin 119860

3119879 (119909) = 119879(1)2119879 (119909) + 119879 (119909) 119879(1)2 + 119879 (1) 119879 (119909) 119879 (1)

3119879 (1199092

) = 119879(119909)2

119879 (1) + 119879 (1) 119879(119909)2 + 119879 (119909) 119879 (1) 119879 (119909) (39)

Finally it only remains to repeat the arguments in (2) rArr

(3) in [9 Theorem 24] to conclude the proof

Now we can state the following result

Theorem 6 Let 119860 and 119861 be Banach algebras where 119860 isunital and 119870 120576 gt 0 Then there exists 120575 gt 0 such that forevery linear map 119879 119860 rarr 119861 with 119879 lt 119870 the condition

sup119886=1119886isin119860

119866

10038171003817100381710038171003817119879 (119886119866

) minus 119879(119886)11986610038171003817100381710038171003817lt 120575 (40)

implies that

jmult (119879 (1) 119879) lt 120576 ucomm (119879) lt 120576 (41)

Proof First notice that if b isin 119860U has a coset representativeb = [119887

119899] where 119887

119899is group invertible for every 119899 isin

N and 119887119866

119899119899isinN is bounded then b is group invertible

and b119866 = [119887119866

119899] Hence the same arguments used in

Theorem 4 produce an operator T = [119879119899] 119860

Urarr

119861U satisfying T(aminus1) = T(a)119866 for every invertible element

a isin 119860U Now Theorem 5 proves that T(1)T is a Jordan

homomorphism and T(1) commutes with T(119860U) Again

the final argument inTheorem 4 completes the proof

In [32 Proposition 25] the authors proved in particularthat if an additive map 119879 119860 rarr 119861 between unital Banachalgebras satisfies

119879 (119886) 119879 (119886minus1

) = 119879(1)2

for every 119886 isin 119860minus1 (42)

and 119879(1) is invertible then 119879(1)minus1

119879 is a Jordan homomor-phism and 119879(1) commutes with 119879(119860) It is clear now thatfor a sequence of linear operators 119879

119899 119860 rarr 119861 satisfying

that 119879119899 119879119899(1)minus1

lt 119870 for all 119899 isin N and

10038171003817100381710038171003817119879119899(119886) 119879119899(119886minus1

) minus 119879119899(1)210038171003817100381710038171003817lt

1

119899

forall119899 isin N (43)

its ultrapoduct T 119860U

rarr 119861U fulfills T(a)T(aminus1) =

T(1)2 for every invertible a isin 119860U Therefore T(1)minus1T is

a Jordan homomorphism and T(1) commutes with T(119860U)

This leads us to the following approximate formulation of [32Proposition 25]

Theorem 7 Let 119860 and 119861 be unital Banach algebras and119870 120576 gt 0 Then there exists 120575 gt 0 such that for every linearmap 119879 119860 rarr 119861 with 119879 119879(1)

minus1

lt 119870 the followingcondition

sup119886isin119860minus1

10038171003817100381710038171003817119879 (119886) 119879 (119886

minus1

) minus 119879(1)210038171003817100381710038171003817lt 120575 (44)

implies that

jmult (119879(1)minus1119879) lt 120576 ucomm (119879) lt 120576 (45)

4 Approximate Preservers in 119862lowast-Algebras

The aim of this section is twofold On the one hand weprove that linear maps approximately preserving generalizedinvertibility in 119862

lowast-algebras are close to be triple homomor-phisms On the other hand we study linear maps approxi-mately preserving the conorm

41 Approximate Preservers of the Moore-Penrose Inverse andthe Generalized Inverse Given a 119869119861lowast-triple 119864 the triple cubeof an element 119909 isin 119864 is defined as 119909[3] = 119909 119909 119909 Anelement satisfying 119890[3] = 119890 is called a tripotent The followingpolarization identity allows us to write the triple product aslinear combination of triple cubes

119909 119910 119911 = sum

1205722

=1

sum

1205734

=1

120572120573(119909 + 120572119910 + 120573119911)

[3]

for119909 119910 119911 isin 119864

(46)

Hence a linearmap between 119869119861lowast-triples is a triple homomor-phism if and only if it preserves triple cubes

Each tripotent 119890 in 119864 gives rise to the so-called Peircedecomposition of 119864 associated to 119890 that is

119864 = 1198642(119890) oplus 119864

1(119890) oplus 119864

0(119890) (47)

where for 119894 = 0 1 2 119864119894(119890) is the 1198942 eigenspace of 119871(119890 119890)

The peirce space 1198642(119890) is a 119869119861lowast-algebra with product 119909 sdot119910 =

119909 119890 119910 and involution 119909♯

= 119890 119909 119890 Moreover the tripleproduct induced on 119864

2(119890) by this Jordan lowast-algebra structure

coincides with its original triple productIt is proved in [16 Lemma 32] (compare with [13

Theorem 34]) that for every regular element 119886 in a 119869119861lowast-

triple 119864 there exists a tripotent 119890 isin 119864 such that 119886 is a self-adjoint invertible element in the 119869119861lowast-algebra 119864

2(119890) If 119886 is

invertible its inverse is denoted as usual by 119886minus1 Moreoverif 119886 and 119887 are invertible elements in the Jordan algebra 119869 =

(119869 ∘) such that 119886minus119887minus1 is also invertible then 119886minus1

+(119887minus1

minus119886)minus1

is invertible and the Hua identity

(119886minus1

+ (119887minus1

minus 119886)

minus1

)

minus1

= 119886 minus 119880119886(119887) (48)

holds where 119880119886(119909) = 2119886 ∘ (119886 ∘ 119909) minus 119886

2

∘ 119909 (see [33] (11))


Let 119860 be a unital 119862lowast-algebra and 119906 isin 119860and

0 Thenthere exists a unique partial isometry 119890 such that 119906 is self-adjoint and invertible in the Jordan algebra (119864

119860)2(119890) =

119890119890lowast

119860119890lowast

119890 with inverse 119906and Hence for every 120582 isin C with 0 lt

|120582| lt 119906and

minus2 the element 119906 minus 120582119906and is invertible in 119890119890

lowast

119860119890lowast

119890Reciprocally the inverses of 119906 minus 120582119906

and and 119906and

minus (119906 minus

120582119906and

)and in 119890119890

lowast

119860119890lowast

119890 are their generalized inverses in 119860 By theHua identity (48) we obtain

119906 minus 120582minus1

119906[3]

= (119906and

minus (119906 minus 120582119906and

)

and

)

and

(49)

Theorem 8 Let 119860 and 119861 be 119862lowast-algebras 119860 being unital

and 119879 119860 rarr 119861 a bounded linear map satisfying 119879(119909minus1) =119879(119909)and for every self-adjoint invertible element 119909 isin 119860 Then

119879 is a triple homomorphism

Proof Arguing as in [9 Lemma 31] pick a self-adjointinvertible element 119906 isin 119860 We may assume that 119879(119906) = 0Then given 120582 isin C with 0 lt |120582| lt min119906minus1minus2 119879(119906)andminus2from identities (36) and (49) we get

119879 (119906) minus 120582minus1

119879(119906)[3]

= (119879(119906)and

minus (119879 (119906) minus 120582119879(119906)and

)

and

)

and

= 119879 (119906) minus 120582minus1

119879 (1199063

)

(50)

which shows that 119879(1199063) = 119879(119906)[3] Once we have proved

that 119879 preserves cubes of self-adjoint invertible elementsgiven a self-adjoint element 119886 isin 119860 and 120572 isin R with |120572| gt

119886 as the element 119886 + 120572 is self-adjoint and invertible weget 119879((119886 + 120572)

3

) = 119879(119886 + 120572)[3] Expanding this las equation

we obtain

119879 (1198863

) + 3120572119879 (1198862

) + 31205722

119879 (119886) + 1205723

119879 (1)

= 119879(119886)[3]

+ 1205723

119879 (1) + 2120572 119879 (119886) 119879 (119886) 119879 (1)

+ 120572 119879 (119886) 119879 (1) 119879 (119886) + 21205722

119879 (1) 119879 (1) 119879 (119886)

+ 1205722

119879 (1) 119879 (119886) 119879 (1)

(51)

for every 119886 isin 119860 and |120572| gt 119886 From this we deducethat 119879(1198863) = 119879(119886)

[3] for every 119886 isin 119860 That is 119879 preservestriple cubes of self-adjoint elements By [34 Theorem20] 119879 is a triple homomorphism

Recall that wemeasure how close is a linearmap 119879 119860 rarr

119861 between 119862lowast-algebras to being a triple homomorphism

or self-adjoint by the triple multiplicativity and the self-adjointness of 119879 respectively as follows

tmult (119879) = sup 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119886 = 119887 = 119888 = 1

sa (119879) = sup119886=1

10038171003817100381710038171003817119879(119886lowast

)

lowast

minus 119879 (119886)

10038171003817100381710038171003817

(52)

Remark 9 Let 119860 and 119861 be 119862lowast-algebras It is clear that everyJordan lowast-homomorphism 119879 119860 rarr 119861 is a triple homo-morphismWe ask whether jmult(119879) and sa(119879) being smallimply tmult(119879) being small

Let 119879 119860 rarr 119861 be a bounded linear map Define

119879119869(119886 119887) = 119879 (119886 ∘ 119887) minus 119879 (119886) ∘ 119879 (119887)

119879119879(119886 119887 119888) = 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119879lowast(119886) = 119879(119886

lowast

)

lowast

minus 119879 (119886)

(53)

for every 119886 119887 119888 isin 119860 Then

119879 (119886 119887 119888)

= 119879 ((119886 ∘ 119887lowast

) ∘ 119888) + 119879 (119886 ∘ (119887lowast

∘ 119888)) minus 119879 ((119886 ∘ 119888) ∘ 119887lowast

)

= 119879 (119886 ∘ 119887lowast

) ∘ 119879 (119888) + 119879119869(119886 ∘ 119887lowast

119888)

+ 119879 (119886) ∘ 119879 (119887lowast

∘ 119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879 (119886 ∘ 119888) ∘ 119879 (119887lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

= (119879 (119886) ∘ 119879 (119887lowast

)) ∘ 119879 (119888) + 119879119869(119886 119887lowast

) ∘ 119879 (119888)

+ 119879119869(119886 ∘ 119887lowast

119888) + 119879 (119886) ∘ (119879 (119887lowast

) ∘ 119879 (119888))

+ 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus (119879 (119886) ∘ 119879 (119888)) ∘ 119879 (119887lowast

) minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

)

minus 119879119869(119886 ∘ 119888 119887

lowast

)

= 119879 (119886) 119879(119887lowast

)

lowast

119879 (119888) + 119879119869(119886 119887lowast

) ∘ 119879 (119888)

+ 119879119869(119886 ∘ 119887lowast

119888) + 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

(54)

Therefore

119879119879(119886 119887 119888)

= 119879 (119886) 119879lowast(119887) 119879 (119888) + 119879

119869(119886 119887lowast

) ∘ 119879 (119888) + 119879119869(119886 ∘ 119887lowast

119888)

+ 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

(55)

This implies that

tmult (119879) le 1198792sa (119879) + 3 (119879 + 1) jmult (119879) (56)

As in Lemma 3 it can be shown that for an operator T =

[119879119899] the following holds

limU

tmult (119879119899) = tmult (T)

limU

sa (119879119899) = sa (T)

(57)


We will omit the proof of the next result in order toavoid repetition The argument is analogous to the one usedin the proofs of Theorems 4 and 6 assuming the contrarywe can construct a map T = [119879

119899] 119860

Urarr 119861

U betweenthe ultrapowers fulfilling T(xminus1) = T(x)and for every self-adjoint invertible x isin 119860

U By Theorem 8 T is a triplehomomorphism

Theorem 10 Let 119860 and 119861 be 119862lowast-algebras where 119860 is unitaland 119870 120576 gt 0 Then there exists 120575 gt 0 such that for everylinear map 119879 119860 rarr 119861 with 119879 lt 119870 the condition

sup119886=1119886=119886

lowast

10038171003817100381710038171003817119879 (119886and

) minus 119879(119886)and10038171003817100381710038171003817lt 120575 (58)

implies that

tmult (119879) lt 120576 (59)

Remark 11 Notice that in Theorem 8 it can be also obtainedthat 119879(1)

lowast

119879 is a Jordan lowast-homomorphism Hence thehypothesis of Theorem 10 also yields to jmult(119879(1)lowast119879) lt 120576

and sa(119879(1)lowast119879) lt 120576

Corollary 12 Let 119860 and 119861 be 119862lowast-algebras where 119860 is unitaland 119870 120576 gt 0 Then there exists 120575 gt 0 such that for everylinear map 119879 119860 rarr 119861 with 119879 lt 119870 the conditions

sup119886=1119886isin119860

dagger

10038171003817100381710038171003817119879 (119886dagger

) minus 119879(119886)dagger10038171003817100381710038171003817lt 120575 sa (119879) lt 120575 (60)

imply that

tmult (119879) lt 120576 (61)

Proof Let us briefly sketch the proof assuming the contrarythere exist 119870


119899119899isinN of linear maps

from 119860 to 119861 such that for every 119899 isin N

1003817100381710038171003817119879119899

1003817100381710038171003817lt 1198700 sup

119886=1

10038171003817100381710038171003817119879119899(119886dagger

) minus 119879119899(119886)dagger10038171003817100381710038171003817lt

1

119899

sa (119879119899) lt

1

119899

tmult (119879119899) ge 1205760

(62)

Then T = [119879119899] 119860

Urarr 119861

U is a self-adjoint mapsuch that T(aminus1) = T(a)dagger for every invertible element a isin

119860U In particular T(aminus1) = T(a)and for every invertible self-

adjoint element a isin 119860U From Theorem 8 T is a triple

homomorphism (Contradiction)

42 Maps Approximately Preserving the Conorm Let 119860 and119861 be unital 119862lowast-algebras Kadison proved in [35] that asurjective linear map 119879 119860 rarr 119861 is an isometry if andonly if 119879 is a Jordanlowast-isomorphism multiplied by a unitaryelement in 119861 In [30] the authors address the question ofcharacterizing surjective linear maps preserving some spec-tral quantities Given an element 119886 of a Banach algebra 119860

the minimum modulus and the surjectivity modulus of 119886 aredefined respectively by

119898(119886) = inf 119886119909 119909 isin 119860 119909 = 1

119902 (119886) = inf 119909119886 119909 isin 119860 119909 = 1

(63)

Obviously 119898(119886) = 0 (resp 119902(119886) = 0) if and only if 119886 is a left(resp right) topological divisor of zero It is well known thatfor any invertible element 119886 isin 119860

119898(119886) = 119902 (119886) =

10038171003817100381710038171003817119886minus110038171003817100381710038171003817

minus1

(64)

Moreover

119898(119886)119898 (119887) le 119898 (119886119887) le 119886119898 (119887)

|119898 (119886) minus 119898 (119887)| le 119886 minus 119887

(65)

for every 119886 119887 isin 119860 If 119860 is a 119862lowast-algebra then 119898(119886) = 119902(119886lowast

)In particular 119898(119886) gt 0 (resp 119902(119886) gt 0) if and only if 119886 is left(resp right) invertible

Recall also from the introduction that the conorm of anelement 119886 in a Banach algebra 119860 is defined as

119888 (119886) =

inf 119886119909 dist (119909 ker (119871119886)) ge 1 if 119886 = 0

infin if 119886 = 0

(66)

For a regular element 119886 in a 119862lowast-algebra 119860

119888 (119886) =

10038171003817100381710038171003817119886dagger10038171003817100381710038171003817

minus1

(67)

Let 119860 and 119861 be unital 119862lowast-algebras By Theorems 31 and32 in [30] if 119879 119860 rarr 119861 is a linear map preserving anyof these spectral quantities then 119879 is an isometric Jordan lowast-homomorphism whenever 119879 is unital and 119879 is an isometricJordan lowast-homomorphism multiplied by a unitary elementwhenever 119879 is surjective In the next results we show thatthe same holds if we just impose the preserving conditionfor invertible elements Notice that we focus our attention onthe conorm but identical results can be established for theminimum and surjective modulus

Theorem 13 Let 119860 and 119861 be unital 119862lowast-algebras and 119879

119860 rarr 119861 a unital linear map satisfying 119888(119879(119909)) = 119888(119909) forall 119909 isin 119860minus1 Then 119879 is a Jordan lowast-homomorphism

Proof First let us prove that 119879 is injective Take 1198860isin 119860 such

that 119879(1198860) = 0 and let 120572 isin C be sufficiently small so that 1 +

1205721198860is invertible Then

1 = 119888 (119879 (1)) = 119888 (119879 (1 + 1205721198860)) = 119888 (1 + 120572119886

0) (68)

In particular we get

1 le 119888 (1 + 1198941199051198860) 1 le 119888 (1 minus 119905119886

0) (69)

as 119905 rarr 0 Hence by [30 Lemma 41] both 1198941198860and 119886

0are

self-adjoint and consequently 1198860= 0


We claim that 119879 is positive Indeed given a self-adjointelement 119886 isin 119860 we know that

1 + 119900 (119905) le 119888 (1 + 119894119905119886) (as 119905 rarr 0) (70)

Since 1 + 119894119905119886 is invertible for 119905 isin R small enough it followsthat

1 + 119900 (119905) le 119888 (1 + 119894119905119886) = 119888 (1 + 119894119905119879 (119886)) (as 119905 rarr 0) (71)

This implies that 119879(119886) is self-adjointMoreover given 119909 isin 119860 and 120582 notin 120590(119909) there exists a

neighborhood 119880120582of 120582 such that 119880

120582cap 120590(119909) = 0 If 120582 notin 120590(119909)

then 120582 notin 120590119870(119909) where 120590

119870(119909) denotes the Kato spectrum

of 119909 as follows

120590119870(119886) = 120582 isin C lim

120583rarr120582

119888 (119886 minus 120583) = 0 (72)

As for 120583 isin 119880120582 the element 119909 minus 120583 is invertible then we

have 119888(119879(119909) minus 120583) = 119888(119909 minus 120583) for every 120583 isin 119880120582 Consequently

lim120583rarr120582

119888 (119879 (119909) minus 120583) gt 0 (73)

and 120582 notin 120590119870(119879(119909)) Since 120597120590(119886) sube 120590

119870(119886) sube 120590(119886) for every 119886 isin

119860 (see [36 Sections 12 13]) we have just proved that

120597120590 (119879 (119909)) sube 120590119870(119879 (119909)) sube 120590 (119909) (74)

for every 119909 isin 119860 Being 119879 self-adjoint this implies that 119879 ispositive and hence 119879 = 1

Arguing as in [30 Theorem 51] given a self-adjointelement 119886 isin 119860 and 119905 sufficiently small so that 119906 = 119890

119894119905119886 is aunitary element with spectrum strictly contained in the unitcircle T since

120597120590 (119879 (119906)) ⫋ T 119879 (119906) le 1

10038171003817100381710038171003817119879(119906)minus110038171003817100381710038171003817= 119888(119879 (119906))

minus1

= 119888(119906)minus1

= 1

(75)

the element 119879(119906) is unitary From

1 = 119879 (119906) 119879(119906)lowast

= 119879 (119906) 119879 (119906lowast

)

= (1 + 119894119905119879 (119886) minus 1

2

1199052

119879 (1198862

) + sdot sdot sdot )

times (1 minus 119894119905119879 (119886) minus 1

2

1199052

119879 (1198862


(76)

we deduce that 119879(1198862) = 119879(119886)2 as desired


119860 rarr 119861 a surjective linear map satisfying 119888(119879(119909)) = 119888(119909) forall 119909 isin 119860minus1 Then 119879 is a Jordan lowast-homomorphismmultipliedby a unitary element in 119861

Proof First let us prove that 119887 = 119879(1) is invertibleSince 119888(119879(1)) = 119888(1) = 1 119887 is regular Let 119910 = 1 minus 119887119887

daggerand 119909 isin 119860 such that 119910 = 119879(119909) Notice that 119887lowast119910 = 119910

lowast

119887 = 0For 120572 isin C sufficiently small such that 1 + 120572119909 isin 119860minus1

119888(1 + 120572119909)2

= 119888(119887 + 120572119910)

2

= 119888 (119887119887lowast

+ |120572|2

119910119910lowast

) (77)

Hence

lim|120572|rarr0

119888 (119887119887lowast

+ |120572|2

119910119910lowast

) = lim|120572|rarr0

119888(1 + 120572119909)2

= 1 = 119888 (119887119887lowast

)

(78)

Reasoning in a similar way to [30 Theorem 62] we get

119888(1 + 120572119909)2

= 119888 (119887119887lowast

+ |120572| 119910119910lowast

) ge 1 minus |120572|210038171003817100381710038171199101003817100381710038171003817

2

(79)

and therefore

119888 (1 + 119894119905119909) ge (1 minus 119905210038171003817100381710038171199101003817100381710038171003817

2

)

12

119888 (1 minus 119905119909) ge (1 minus 11990521003817100381710038171003817119910

1003817100381710038171003817

2

)

12

(80)

for small enough 119905 isin R From these inequalities weget respectively that 119909 and 119894119909 are self-adjoint This showsthat 119909 = 0 and thus 119910 = 0 Consequently 1 = 119887119887

dagger thatis 119887 is right invertible Similarly it can be proved that 119887 is leftinvertible

Note that as in the previous theorem 119879 is injectiveTherefore 119878 = 119887

minus1

119879 is a unital and bijective linear mapsatisfying

119898(119878 (119909)) = 119898 (119887minus1

119879 (119909)) le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119898 (119879 (119909))

= 119898 (119879 (119909)) le 119888 (119879 (119909)) = 119888 (119909) forall119909 isin 119860minus1

(81)

Let 119910 be a self-adjoint element in 119861 and 119905 isin R smallsuch that 1+119894119905119878minus1(119910) is invertible Taking 119909 = 1+119894119905119878

minus1

(119910) inthe previous identity we have

119898(1 + 119894119905119910) le 119888 (1 + 119894119905119878minus1

(119910)) (82)

It follows that 119878minus1 is self-adjoint and so is 119878We claim that 119878 is positive Note that for every 119909 isin

119860dagger and 119906 isin 119860

minus1 it is clear that 119906119909 isin 119860dagger with

(119906119909) (119909dagger

119906minus1

) (119906119909) = 119906119909

(119909dagger

119906minus1

) (119906119909) (119909dagger

119906minus1

) = 119909dagger

119906minus1

(83)

This implies by [11 Theorem 2] the following1

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le 119888 (119906119909)

le

10038171003817100381710038171003817119909dagger

119906minus1

119906119909

10038171003817100381710038171003817

10038171003817100381710038171003817119906119909119909dagger

119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le

119906

10038171003817100381710038171003817119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

(84)

Hence for every 119909 isin 119860minus1 we have

119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909)) ge

1

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

ge

1

119887

10038171003817100381710038171003817119879(119909)dagger10038171003817100381710038171003817

= 119887minus1

119888 (119879 (119909))

119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909))

le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119879 (1)

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

le 119887 |

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2

119888 (119879 (119909))

(85)


So we have shown so far the following

119887minus1

119888 (119909) le 119888 (119878 (119909)) le 119887

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2

119888 (119909) forall119909 isin 119860minus1

(86)

The first inequality can be used to show the following

120597120590 (119878 (119909)) sub 120590119870(119878 (119909)) sube 120590 (119909) (87)

in a similar way as in the previous theorem As aconsequence 119878 is positive In order to conclude that 119878 isan isometric Jordan lowast-isomorphism it suffices to provethat 119878minus1 is also positive (see for instance [35 Corollary 5])So let ℎ = 119878(119886) be a positive element As 119878minus1 is self-adjoint 119886 is self-adjoint We can therefore write 119886 = 119909 minus 119910where 119909 and 119910 are positive elements and 119909119910 = 119910119909 = 0 Forevery 120583 isin C we have

120597120590 (119878 (119909) + 120583119878 (119910)) sub 120590119870(119878 (119909) + 120583119878 (119910))

sub 120590 (119909 + 120583119910) sub R cup 120583R(88)

(Recall that if 119908119911 = 119911119908 = 0 then 120590(119908 + 119911) 0 = (120590(119908)

0) cup (120590(119911) 0)) The previous spectral inclusion gives

120590 (119878 (119909) + 120583119878 (119910)) sub R cup 120583R (89)

By Lemmas B and C in [37] we get 119878(119910) = 0 and so 119910 = 0Consequently 119886 is positive as desiredWe conclude the proofby showing that 119887 is unitary Indeed since 119878 is a Jordan lowast-isomorphism and 119879(119909) = 119887119878(119909) it is clear that for every 119909 isin119860minus1 119879(119909) is invertible with inverse 119879(119909)minus1 = 119878(119909

minus1

)119887minus1

Moreover 119888(119879(119909)) = 119888(119909) that is10038171003817100381710038171003817119879(119909)minus110038171003817100381710038171003817=

10038171003817100381710038171003817119909minus110038171003817100381710038171003817 (90)

for every 119909 isin 119860minus1 Since 119878 is an isometry10038171003817100381710038171003817119878 (119909minus1

)

10038171003817100381710038171003817=

10038171003817100381710038171003817119909minus110038171003817100381710038171003817=

10038171003817100381710038171003817119878 (119909minus1

) 119887minus110038171003817100381710038171003817 (91)

or equivalently1003817100381710038171003817119910

1003817100381710038171003817=

10038171003817100381710038171003817119910119887minus110038171003817100381710038171003817 (92)

for every 119910 isin 119861minus1 This yields that 119887 is unitary

Note that for every invertible element a isin 119860U where a =[119886119899] is a normalized representative

119888 (a) = 10038171003817100381710038171003817aminus11003817100381710038171003817

1003817

minus1

= limU

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817

minus1

= limU

119888 (119886119899) (93)

We are now in position to prove the main results in thissection

Theorem 15 Let 119860 and 119861 be unital 119862lowast-algebras and 119870 120576 gt

0 Then there exists 120575 gt 0 such that for every linear map 119879

119860 rarr 119861 with 119879 lt 119870 the following conditions

sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 119879 (1) minus 1 lt 120575 (94)

imply that

jmult (119879) lt 120576 sa (119879) lt 120576 (95)

Proof Note that if b isin 119861U has a coset representative b =

[119887119899] where 119887

119899has Moore-Penrose inverse for every 119899 isin

N and 119887dagger

119899119899isinN is bounded then b is Moore-Penrose invert-

ible and bdagger = [119887dagger

119899]

As we did above suppose that the assertion is false thatis we can find 119870


119899119899isinN of linear

maps from 119860 to 119861 satisfying

(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119879

119899(1) minus 1 lt

1119899(iii) jmult(119879

119899) ge 1205760or sa(119879

119899) ge 1205760

for every 119899 isin N Consider T = [119879119899] 119860

Urarr 119861

U Weclaim that T is unital and preserves the conorm of invertibleelements On the one hand

T (1) minus 1 = limU

1003817100381710038171003817119879119899(1) minus 1

1003817100381710038171003817le lim

U

1

119899

= 0 (96)

so T(1) = 1 On the other hand for an invertibleelement a isin 119860

U with norm 1 let [119886119899] be its normalized

representative 119886119899 = 1 and 119886

minus1

119899 lt 120572 for some positive 120572

We know that

119888 (119886119899) minus

1

119899

lt 119888 (119879119899(119886119899)) for every 119899 isin N (97)

Hence 119879119899(119886119899) isin 119860

dagger with 119879119899(119886119899)dagger

lt 2120572 for 119899 gt 2120572 Thatis 119879119899(119886119899) is Moore-Penrose invertible almost everywhere

with their respective Moore-Penrose inverses uniformlybounded in norm This ensures that T(a) is Moore-Penroseinvertible and 119888(T(a)) = limU119888(119879119899(119886119899)) Finally Consider thefollowing

|119888 (T (a)) minus 119888 (a)| = limU

1003816100381610038161003816119888 (119879119899(119886119899)) minus 119888 (119886

119899)

1003816100381610038161003816le lim

U

1

119899

= 0

(98)

ByTheorem 13 T is a Jordan lowast-homomorphism which givesthe contradiction

Let 119883 and 119884 be complex Banach spaces andlet B(119883 119884) be the algebra of all bounded linear operatorsfrom 119883 to 119884 Recall that the surjectivity modulus of 119879 isgiven by 119902(119879) = sup120576 ge 0 120576119861

119884sube 119879(119861

119883) whereas

usual 119861119883denotes the closed unit ball of 119883 Note that 119902(119879) gt

0 if and only if 119879 is surjective and 119902(119879) = inf119878119879 119878 isinB(119883) 119878 = 1 (see [36 Theorem II911])



119860 rarr 119861 with 119879 lt 119870 and 119902(119879) gt 119870minus1 the following

condition

sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 (99)

implies that

jmult (119879(1)lowast119879) lt 120576 sa (119879(1)lowast119879) lt 120576 (100)


Proof If we assume the contrary hence as above wefind 119870

0 1205760

gt 0 and a sequence 119879119899119899isinN of linear maps

from 119860 to 119861 satisfying

(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119902(119879

119899) ge 119870

minus1

0

(iii) jmult(119879119899(1)lowast

119879119899) ge 1205760or sa(119879

119899(1)lowast

119879119899) ge 1205760

for every 119899 isin NAs in the previous theorem the map T = [119879

119899] preserves

the conorm of all invertible elements Moreover

119902 (T) = limU119902 (119879119899) ge 119870

minus1

0gt 0 (101)

and thus T is surjective By Theorem 14 T(1) is unitaryand T(1)lowastT is a unital Jordan lowast-isomorphism

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors were partially supported by I+D MEC projectsno MTM 2011-23843 MTM 2010-17687 and Junta deAndalucıa Grants FQM 375 and FQM 3737 The secondauthor is also supported by a Plan Propio de Investigaciongrant from University of Almerıa The authors would like tothank the referees for their comments and suggestions

References

[1] L-K Hua ldquoOn the automorphisms of a sfieldrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 35 pp 386ndash389 1949

[2] H Essannouni and A Kaidi ldquoLe theoreme de Hua pour lesalgebres artiniennes simplesrdquo Linear Algebra and Its Applica-tions vol 297 no 1ndash3 pp 9ndash22 1999

[3] M Mbekhta ldquoA Hua type theorem and linear preserver prob-lemsrdquoMathematical Proceedings of the Royal Irish Academy vol109 no 2 pp 109ndash121 2009

[4] N Boudi and M Mbekhta ldquoAdditive maps preserving stronglygeneralized inversesrdquo Journal of Operator Theory vol 64 no 1pp 117ndash130 2010

[5] M P Drazin ldquoPseudo-inverses in associative rings and semi-groupsrdquoThe American Mathematical Monthly vol 65 pp 506ndash514 1958

[6] A Ben-Israel and T N E Greville Generalized Inverses TheoryandApplications Springer NewYork NYUSA Second edition2003

[7] S L Campbell Recent Applications of Generalized InversesAcademic Press New York NY USA 1982

[8] J Cui ldquoAdditive Drazin inverse preserversrdquo Linear Algebra andIts Applications vol 426 no 2-3 pp 448ndash453 2007

[9] M Burgos A C Marquez-Garcıa and A Morales-CampoyldquoStrongly preserver problems in Banach algebras and Clowast-algebrasrdquo Linear Algebra and Its Applications vol 437 no 5 pp1183ndash1193 2012

[10] R Harte and M Mbekhta ldquoOn generalized inverses in Clowast-algebrasrdquo Studia Mathematica vol 103 no 1 pp 71ndash77 1992

[11] RHarte andMMbekhta ldquoGeneralized inverses inClowast-algebrasIIrdquo Studia Mathematica vol 106 no 2 pp 129ndash138 1993

[12] M Burgos A C Marquez-Garcıa and A Morales-CampoyldquoLinear maps strongly preserving Moore-Penrose invertibilityrdquoOperators and Matrices vol 6 no 4 pp 819ndash831 2012

[13] M Burgos E A Kaidi A M Campoy A M Peralta and MRamırez ldquoVon Neumann regularity and quadratic conorms inJBlowast-triples and Clowast-algebrasrdquo Acta Mathematica Sinica vol 24no 2 pp 185ndash200 2008

[14] A Fernandez Lopez E Garcıa Rus E Sanchez Campos andM Siles Molina ldquoStrong regularity and generalized inverses inJordan systemsrdquo Communications in Algebra vol 20 no 7 pp1917ndash1936 1992

[15] W Kaup ldquoOn spectral and singular values in JBlowast-triplesrdquoProceedings of the Royal Irish Academy Section A Mathematicaland Physical Sciences vol 96 no 1 pp 95ndash103 1996

[16] W Kaup ldquoOn Grassmannians associated with JBlowast-triplesrdquoMathematische Zeitschrift vol 236 no 3 pp 567ndash584 2001

[17] O Loos Jordan Pairs vol 460 of Lecture Notes in MathematicsSpringer Berlin Germany 1975

[18] A R Sourour ldquoInvertibility preserving linear maps on 119871(119883)rdquoTransactions of the AmericanMathematical Society vol 348 no1 pp 13ndash30 1996

[19] I Kaplansky Algebraic and Analytic Aspects of Operator Alge-bras American Mathematical Society Providence RI USA1970

[20] B Aupetit ldquoSpectrum-preserving linear mappings betweenBanach algebras or Jordan-Banach algebrasrdquo Journal of theLondon Mathematical Society vol 62 no 3 pp 917ndash924 2000

[21] M Bresar A Fosner and P Semrl ldquoA note on invertibilitypreservers on Banach algebrasrdquo Proceedings of the AmericanMathematical Society vol 131 no 12 pp 3833ndash3837 2003

[22] A A Jafarian and A R Sourour ldquoSpectrum-preserving linearmapsrdquo Journal of Functional Analysis vol 66 no 2 pp 255ndash2611986

[23] J-P Kahane and W Zelazko ldquoA characterization of maximalideals in commutative Banach algebrasrdquo Studia Mathematicavol 29 pp 339ndash343 1968

[24] I Kovacs ldquoInvertibility-preserving maps of Clowast-algebras withreal rank zerordquo Abstract and Applied Analysis no 6 pp 685ndash689 2005

[25] B E Johnson ldquoApproximately multiplicative functionalsrdquo Jour-nal of the London Mathematical Society vol 34 no 3 pp 489ndash510 1986

[26] S H Kulkarni and D Sukumar ldquoAlmost multiplicative func-tions on commutative Banach algebrasrdquo Studia Mathematicavol 152 pp 187ndash199 2002

[27] J Alaminos J Extremera and A R Villena ldquoApproximatelyspectrum-preserving mapsrdquo Journal of Functional Analysis vol261 no 1 pp 233ndash266 2011

[28] J Alaminos J Extremera andARVillena ldquoSpectral preserversand approximate spectral preservers on operator algebrasrdquo Inpress

[29] B E Johnson ldquoApproximately multiplicative maps betweenBanach algebrasrdquo Journal of the London Mathematical Societyvol 37 no 2 pp 294ndash316 1988

[30] A Bourhim M Burgos and V S Shulman ldquoLinear mapspreserving the minimum and reduced minimum modulirdquoJournal of Functional Analysis vol 258 no 1 pp 50ndash66 2010


[31] S Heinrich ldquoUltraproducts in Banach space theoryrdquo Journalfur die Reine und Angewandte Mathematik vol 313 pp 72ndash1041980

[32] M Burgos A C Marquez-Garcıa and A Morales-CampoyldquoDetermining Jordan (triple) homomorphisms by invertibilitypreserving conditionsrdquo In press

[33] N Jacobson Structure and Representations of Jordan AlgebrasAmerican Mathematical Society Colloquium Providence RIUSA 1968

[34] M Burgos F J Fernandez-Polo J J Garces JMartınezMorenoand A M Peralta ldquoOrthogonality preservers in Clowast-algebrasJBlowast-algebras and JBlowast-triplesrdquo Journal of Mathematical Analysisand Applications vol 348 no 1 pp 220ndash233 2008

[35] R V Kadison ldquoA generalized Schwarz inequality and algebraicinvariants for operator algebrasrdquoAnnals ofMathematics vol 56pp 494ndash503 1952

[36] V Muller Spectral Theory of Linear Operators and Spectral Sys-tems in Banach Algebras vol 139 of Operator Theory Advancesand Applications Birkhauser Basel Switzerland 2003

[37] M-D Choi D Hadwin E Nordgren H Radjavi and PRosenthal ldquoOn positive linear maps preserving invertibilityrdquoJournal of Functional Analysis vol 59 no 3 pp 462ndash469 1984

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


and 119879(1) commutes with the image of 119879 [4 Theorem42] In [9] the authors showed that the same holds withoutany hypothesis on 119879(1) and even when 119861 is not necessarilyunital

Recall that an element 119886 isin 119860 is called regular if thereexists 119887 isin 119860 (not necessarily unique) such that 119886 = 119886119887119886

and 119887 = 119887119886119887 Notice that the first equality 119886 = 119886119887119886 isa necessary and sufficient condition for 119886 to be regular andthat 119901 = 119886119887 and 119902 = 119887119886 are idempotents in 119860 fulfilling119886119860 = 119901119860 and 119860119886 = 119860119902 For an element 119886 isin 119860 let usconsider the left and right multiplication operators 119871

119886 119909 997891rarr

119886119909 and 119877119886 119909 997891rarr 119909119886 respectively If 119886 is regular then

so are 119871119886

and 119877119886 and thus its ranges 119886119860 = 119871

119886(119860) and

119860119886 = 119877119886(119860) are both closed

Regular elements in a unital 119862lowast-algebra 119860 were studiedby Harte and Mbekhta in [10 11]

An element 119886 isin 119860 has Moore-Penrose inverse 119887 when119886 = 119886119887119886 119887 = 119887119886119887 and the associated idempotents 119886119887 and119887119886 are self-adjoint In the aforementioned papers by Harteand Mbekhta it is shown that every regular element in a 119862lowast-algebra has a Moore-Penrose inverse and that it is uniqueFor a regular element 119886 in a 119862lowast-algebra 119860 119886dagger will denoteits Moore-Penrose inverse The set of all Moore-Penroseinvertible elements in a 119862lowast-algebra 119860 will be denoted by119860daggerLet 119860 and 119861 be unital 119862lowast-algebras A linear map

119879 119860 rarr 119861 strongly preserves Moore-Penrose invert-ibility if 119879(119886

dagger

) = 119879(119886)dagger for every 119886 isin 119860

daggerEvery Jordan lowast-homomorphism strongly preserves Moore-Penrose invertibility and the question is whether or notthe converse holds Some partial positive answers are givenby Mbekhta in [3] and more recently by the authors ofthe present paper in [12] when 119860 has a rich structure ofprojections The problem for general 119862

lowast-algebras remainsopen However we can consider an alternative approach

Recall that the class of119862lowast-algebras is contained in a widerclass of Banach spaces the so called 119869119861lowast-triples in which theconcept of regularity extends the one given for 119862lowast-algebrasA 119869119861lowast-triple is a complex Banach space 119864 together with a

continuous triple product sdot sdot sdot 119864 times 119864 times 119864 rarr 119864 whichis conjugate linear in the middle variable and symmetric andbilinear in the outer variables satisfying that

(a) 119871(119886 119887)119871(119909 119910) = 119871(119909 119910)119871(119886 119887) + 119871(119871(119886 119887)119909 119910) minus

119871(119909 119871(119887 119886)119910) where 119871(119886 119887) is the operatoron 119864 given by 119871(119886 119887)119909 = 119886 119887 119909

(b) 119871(119886 119886) is an hermitian operator with nonnegativespectrum

(c) 119871(119886 119886) = 1198862

For each 119909 in a 119869119861lowast-triple 119864 119876(119909) will stand for the

conjugate linear operator on 119864 defined by 119876(119909)(119910) =

119909 119910 119909An element 119886 in a 119869119861lowast-triple 119864 is called von Neumann

regular if there exists (a unique) 119887 isin 119864 such that 119876(119886)(119887) =

119886 119876(119887)(119886) = 119887 and 119876(119886)119876(119887) = 119876(119887)119876(119886) Theelement 119887 is called the generalized inverse of 119886 and it willbe denoted by 119886

and We will also note by 119864and the set of all

elements in 119864 with generalized inverse We refer to [13ndash17]for basics results on von Neumann regularity in 119869119861lowast-triples

Every 119862lowast-algebra is a 119869119861lowast-triple via the triple product

given by

119909 119910 119911 =

1

2

(119909119910lowast

119911 + 119911119910lowast

119909) (3)

For a 119862lowast-algebra 119860 it is well known that 119860dagger = 119860and and for

every regular element 119886 in 119860 we have 119886and = (119886dagger

)lowast

A linear map 119879 119864 rarr 119865 between 119869119861lowast-triples is a triplehomomorphism if

119879 (119909 119910 119911) = 119879 (119909) 119879 (119910) 119879 (119911) (4)

for every 119909 119910 119911 isin 119864 Every triple homomorphism 119879

119864 rarr 119865 between 119869119861lowast-triples strongly preserves generalizedinvertibility that is 119879(119909

and

) = 119879(119909)and for every 119909 isin 119864

andIn [9] the authors characterized the triple homomorphismbetween 119862

lowast-algebras as the linear maps strongly preservinggeneralized invertibility As a consequence it is proved that aself-adjoint linearmap from a unital119862lowast-algebra 119860 into a119862lowast-algebra 119861 is a triple homomorphism if and only if it stronglypreserves Moore-Penrose invertibility [9 Theorem 35]

The Hua-type descriptions belong to the framework oflinear preserver problemsThis has become an active researcharea in many topics of matrix theory operator theory andBanach algebras theory Some of the most popular preserverproblems are those dealing with determining the linear mapspreserving properties related to invertibility Every Jordanisomorphism 119879 119860 rarr 119861 between unital Banach algebrasis unital and preserves invertibility in both directions [18Proposition 13] or equivalently preserves the spectrum thatis 120590(119879(119886)) = 120590(119886) for every 119886 isin 119860 The celebratedKaplanskyrsquos conjecture [19] reformulated by Aupetit in [20]states that every unital surjective linear map 119879 betweenunital semisimple Banach algebras preserving invertibilityin both directions is a Jordan isomorphism Many partialpositive results are known so far [18 20ndash24] but the generalproblem is still open even in the class of 119862lowast-algebras In thecommutative setting the classical Gleason-Kahane-Zelazkotheorem (see [23]) states that a linear functional 120593 on aunital complex Banach algebra 119860 is multiplicative if andonly if 120593(119886) isin 120590(119886) for all 119886 isin 119860 On the other handJafarian and Sourour proved in [22] that every spectrumpreserving surjective linear map 119879 B(119883) rarr B(119884)

is either an isomorphism or an anti-isomorphism whereB(119883) denotes the Banach algebra of all bounded linearoperators on a Banach space 119883

The authors of [25 26] consider the problem of charac-terizing the approximately multiplicative linear functionalsamong all linear functionals on a commutative Banachalgebra in terms of spectra More recently in [27] (see also[28]) Alaminos Extremera and Villena investigate approxi-mate versions of Kaplanskyrsquos problem by providing approx-imate formulations of [18 22] They considered linear mapsthat approximately preserve spectrum or spectral radius onoperator algebras and established the relationship betweenapproximately preserving spectrum (resp spectral radius)




mult (119879)

= sup 119879 (119886119887) minus 119879 (119886) 119879 (119887) 119886 119887 isin 119860 119886 = 119887 = 1

amult (119879)

= sup 119879 (119886119887) minus 119879 (119887) 119879 (119886) 119886 119887 isin 119860 119886 = 119887 = 1

jmult (119879) = sup 10038171003817100381710038171003817119879 (1198862

) minus 119879(119886)210038171003817100381710038171003817 119886 isin 119860 119886 = 1

(5)




tmult (119879)

= sup 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119886 = 119887 = 119888 = 1

sa (119879) = sup 10038171003817100381710038171003817119879(119886lowast

)

lowast

minus 119879 (119886)

10038171003817100381710038171003817 119886 = 1

(6)



minus1

) = 119879(119886)minus1 into

sup119886=1119886isin119860

minus1

10038171003817100381710038171003817119879 (119886minus1

) minus 119879(119886)minus110038171003817100381710038171003817lt 120576 (7)


) = 119879(119886)119866 into

sup119886=1119886isin119860

119866

10038171003817100381710038171003817119879 (119886119866

) minus 119879(119886)11986610038171003817100381710038171003817lt 120576 (8)



dagger



sup119886=1119886isin119860

dagger

10038171003817100381710038171003817119879 (119886dagger

) minus 119879(119886)dagger10038171003817100381710038171003817lt 120576 (9)






120574 (119879) = inf 119879119909 dist (119909 ker (119879)) ge 1 (10)


119888 (119886) =

10038171003817100381710038171003817119886dagger10038171003817100381710038171003817

minus1 (11)


sup119886=1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120576 (12)





defined as follows

(119883119899)

U=

ℓinfin

(N 119883119899)

NU

(13)

where ℓinfin



119899isin 119883119899


infinnorm and

NU = 119909119899119899isinN

isin ℓinfin

(N 119883119899) lim

U

1003817100381710038171003817119909119899

1003817100381710038171003817= 0 (14)


U=

ℓinfin


119899] the



x = limU

1003817100381710038171003817119909119899

1003817100381710038171003817 (15)






U 119884

U) given by

T (x) = [119879119899(119909119899)] (16)

for every T = [119879119899] isin B(119883 119884)

U and x = [119909119899] isin



U 119884



U)





equivalent


119899] such that 119906

119899isin 119860minus1





119899]




limU

1003817100381710038171003817119886119899119887119899minus 1100381710038171003817

1003817= 0

limU

1003817100381710038171003817119887119899119886119899minus 1100381710038171003817

1003817= 0

(17)


119877 = 119899 isin N

1003817100381710038171003817119886119899119887119899minus 1100381710038171003817

1003817lt 120575 isin U

119871 = 119899 isin N 1003817100381710038171003817119887119899119886119899minus 1100381710038171003817

1003817lt 120575 isin U

(18)


119899



10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817119887119899(119886119899119887119899)

minus110038171003817100381710038171003817le1003817100381710038171003817119887119899

1003817100381710038171003817

10038171003817100381710038171003817(119886119899119887119899)

minus110038171003817100381710038171003817

le

1003817100381710038171003817119887119899

1003817100381710038171003817

1 minus

10038171003817100381710038171 minus 119886119899119887119899

1003817100381710038171003817

le

1003817100381710038171003817119887119899

1003817100381710038171003817

1 minus 120575

(19)






we choose

1198861015840

119899=

119886119899 if 119899 isin 119868

1 if 119899 notin 119868(20)

Clearly [119886119899] = [119886

1015840

119899])



119899] such that 119887

119899 =


119887119899=

119886119899

1003817100381710038171003817119886119899

1003817100381710038171003817

if 119886119899

= 0

1 if 119886119899= 0

(21)








ucomm (119879) = sup119886=1

119879 (119886) 119879 (1) minus 119879 (1) 119879 (119886) (22)







119899] 119860

Urarr 119861

U Then

limU

ucomm (119879119899) = ucomm (T) (23)


[119886119899] with 119886



= limU

1003817100381710038171003817119879119899(119886119899) 119879119899(1) minus 119879

119899(1) 119879119899(119886119899)

1003817100381710038171003817le lim

Uucomm (119879

119899)

(24)

and hence

ucomm (T) le limU

ucomm (119879119899) (25)


119886119899 = 1 such that

ucomm (119879119899) minus

1

119899

lt

1003817100381710038171003817119879119899(119886119899) 119879119899(1) minus 119879

119899(1) 119879119899(119886119899)

1003817100381710038171003817 (26)



limU

ucomm (119879119899)

le limU

1003817100381710038171003817119879119899(119886119899) 119879119899(1) minus 119879

119899(1) 119879119899(119886119899)

1003817100381710038171003817


(27)



sup119886=1119886isin119860

minus1

10038171003817100381710038171003817119879 (119886minus1

) minus 119879(119886)minus110038171003817100381710038171003817lt 120575 (28)

implies that

jmult (119879 (1) 119879) lt 120576 ucomm (119879) lt 120576 (29)



119899119899isinN of


(i) 119879119899 lt 119870

0

(ii) sup119886=1

119879119899(119886minus1

) minus 119879119899(119886)minus1

lt 1119899(iii) jmult(119879

119899(1)119879119899) ge 1205760or ucomm(119879

119899) ge 1205760

Consider that T = [119879119899] 119860

Urarr 119861




119886minus1


As10038171003817100381710038171003817119879119899(119886minus1

119899)

10038171003817100381710038171003817le

1003817100381710038171003817119879119899

1003817100381710038171003817

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817lt 1198700120572

10038171003817100381710038171003817119879119899(119886minus1

119899) minus 119879119899(119886119899)

minus110038171003817100381710038171003817lt 1

(30)

we get

10038171003817100381710038171003817119879119899(119886119899)

minus110038171003817100381710038171003817lt 1198700120572 + 1 (31)




1003817

= limU

10038171003817100381710038171003817119879119899(119886minus1

119899) minus 119879119899(119886119899)

minus110038171003817100381710038171003817le lim

U

1

119899

= 0

(32)






jmult (119879119899(1) 119879119899)


ucomm (119879119899)

(33)

Consequently



(34)





(119886minus1


)

minus1

)

minus1

= 119886 minus 119886119887119886 (35)





119906minus1




(119906minus1


)

minus1

)

minus1

= 119906 minus 119906 (120582minus1

119906) 119906 = 119906 minus 120582minus1

1199063

(36)




and 0 lt |120582| lt 119879(119906)119866

minus2 gives

119879 (119906) minus 120582minus1

119879(119906)3

= (119879(119906)119866

minus (119879 (119906) minus 120582119879(119906)119866

)

119866

)

119866

(37)




we get

119879 (119906) minus 120582minus1

119879(119906)3

= (119879 (119906minus1


)

119866

)

119866

= 119879(119906minus1


)

minus1

)

119866

= 119879((119906minus1


)

minus1

)

minus1

)

= 119879 (119906 minus 120582minus1

1199063

) = 119879 (119906) minus 120582minus1

119879 (1199063

)

(38)

Hence 119879(1199063



3119879 (119909) = 119879(1)2119879 (119909) + 119879 (119909) 119879(1)2 + 119879 (1) 119879 (119909) 119879 (1)

3119879 (1199092

) = 119879(119909)2

119879 (1) + 119879 (1) 119879(119909)2 + 119879 (119909) 119879 (1) 119879 (119909) (39)





sup119886=1119886isin119860

119866

10038171003817100381710038171003817119879 (119886119866

) minus 119879(119886)11986610038171003817100381710038171003817lt 120575 (40)

implies that

jmult (119879 (1) 119879) lt 120576 ucomm (119879) lt 120576 (41)


119899] where 119887


N and 119887119866


and b119866 = [119887119866



Urarr






119879 (119886) 119879 (119886minus1

) = 119879(1)2




119899 119860 rarr 119861 satisfying

that 119879119899 119879119899(1)minus1


10038171003817100381710038171003817119879119899(119886) 119879119899(119886minus1

) minus 119879119899(1)210038171003817100381710038171003817lt

1

119899








minus1



10038171003817100381710038171003817119879 (119886) 119879 (119886

minus1

) minus 119879(1)210038171003817100381710038171003817lt 120575 (44)

implies that






119909 119910 119911 = sum

1205722

=1

sum

1205734

=1

120572120573(119909 + 120572119910 + 120573119911)

[3]

for119909 119910 119911 isin 119864

(46)



119864 = 1198642(119890) oplus 119864

1(119890) oplus 119864

0(119890) (47)



119909 119890 119910 and involution 119909♯






2(119890) If 119886 is



+(119887minus1

minus119886)minus1


(119886minus1

+ (119887minus1

minus 119886)

minus1

)

minus1

= 119886 minus 119880119886(119887) (48)


2

∘ 119909 (see [33] (11))




119860)2(119890) =

119890119890lowast

119860119890lowast


|120582| lt 119906and


lowast

119860119890lowast


and and 119906and

minus (119906 minus

120582119906and

)and in 119890119890

lowast

119860119890lowast



119906[3]

= (119906and

minus (119906 minus 120582119906and

)

and

)

and

(49)





119879 (119906) minus 120582minus1

119879(119906)[3]

= (119879(119906)and

minus (119879 (119906) minus 120582119879(119906)and

)

and

)

and

= 119879 (119906) minus 120582minus1

119879 (1199063

)

(50)




3


we obtain

119879 (1198863

) + 3120572119879 (1198862

) + 31205722

119879 (119886) + 1205723

119879 (1)

= 119879(119886)[3]

+ 1205723

119879 (1) + 2120572 119879 (119886) 119879 (119886) 119879 (1)

+ 120572 119879 (119886) 119879 (1) 119879 (119886) + 21205722

119879 (1) 119879 (1) 119879 (119886)

+ 1205722

119879 (1) 119879 (119886) 119879 (1)

(51)






tmult (119879) = sup 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119886 = 119887 = 119888 = 1

sa (119879) = sup119886=1

10038171003817100381710038171003817119879(119886lowast

)

lowast

minus 119879 (119886)

10038171003817100381710038171003817

(52)



119879119869(119886 119887) = 119879 (119886 ∘ 119887) minus 119879 (119886) ∘ 119879 (119887)

119879119879(119886 119887 119888) = 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119879lowast(119886) = 119879(119886

lowast

)

lowast

minus 119879 (119886)

(53)


119879 (119886 119887 119888)

= 119879 ((119886 ∘ 119887lowast

) ∘ 119888) + 119879 (119886 ∘ (119887lowast

∘ 119888)) minus 119879 ((119886 ∘ 119888) ∘ 119887lowast

)

= 119879 (119886 ∘ 119887lowast

) ∘ 119879 (119888) + 119879119869(119886 ∘ 119887lowast

119888)

+ 119879 (119886) ∘ 119879 (119887lowast

∘ 119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879 (119886 ∘ 119888) ∘ 119879 (119887lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

= (119879 (119886) ∘ 119879 (119887lowast

)) ∘ 119879 (119888) + 119879119869(119886 119887lowast

) ∘ 119879 (119888)

+ 119879119869(119886 ∘ 119887lowast

119888) + 119879 (119886) ∘ (119879 (119887lowast

) ∘ 119879 (119888))

+ 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus (119879 (119886) ∘ 119879 (119888)) ∘ 119879 (119887lowast

) minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

)

minus 119879119869(119886 ∘ 119888 119887

lowast

)

= 119879 (119886) 119879(119887lowast

)

lowast

119879 (119888) + 119879119869(119886 119887lowast

) ∘ 119879 (119888)

+ 119879119869(119886 ∘ 119887lowast

119888) + 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

(54)

Therefore

119879119879(119886 119887 119888)

= 119879 (119886) 119879lowast(119887) 119879 (119888) + 119879

119869(119886 119887lowast

) ∘ 119879 (119888) + 119879119869(119886 ∘ 119887lowast

119888)

+ 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

(55)

This implies that

tmult (119879) le 1198792sa (119879) + 3 (119879 + 1) jmult (119879) (56)



limU

tmult (119879119899) = tmult (T)

limU

sa (119879119899) = sa (T)

(57)



119899] 119860

Urarr 119861




sup119886=1119886=119886

lowast

10038171003817100381710038171003817119879 (119886and

) minus 119879(119886)and10038171003817100381710038171003817lt 120575 (58)

implies that

tmult (119879) lt 120576 (59)


lowast


and sa(119879(1)lowast119879) lt 120576


sup119886=1119886isin119860

dagger

10038171003817100381710038171003817119879 (119886dagger


imply that

tmult (119879) lt 120576 (61)





1003817100381710038171003817119879119899

1003817100381710038171003817lt 1198700 sup

119886=1

10038171003817100381710038171003817119879119899(119886dagger

) minus 119879119899(119886)dagger10038171003817100381710038171003817lt

1

119899

sa (119879119899) lt

1

119899

tmult (119879119899) ge 1205760

(62)

Then T = [119879119899] 119860

Urarr 119861







119898(119886) = inf 119886119909 119909 isin 119860 119909 = 1

119902 (119886) = inf 119909119886 119909 isin 119860 119909 = 1

(63)


119898(119886) = 119902 (119886) =

10038171003817100381710038171003817119886minus110038171003817100381710038171003817

minus1

(64)

Moreover

119898(119886)119898 (119887) le 119898 (119886119887) le 119886119898 (119887)

|119898 (119886) minus 119898 (119887)| le 119886 minus 119887

(65)




119888 (119886) =

inf 119886119909 dist (119909 ker (119871119886)) ge 1 if 119886 = 0

infin if 119886 = 0

(66)


119888 (119886) =

10038171003817100381710038171003817119886dagger10038171003817100381710038171003817

minus1

(67)







1 = 119888 (119879 (1)) = 119888 (119879 (1 + 1205721198860)) = 119888 (1 + 120572119886

0) (68)


1 le 119888 (1 + 1198941199051198860) 1 le 119888 (1 minus 119905119886

0) (69)


0are




1 + 119900 (119905) le 119888 (1 + 119894119905119886) (as 119905 rarr 0) (70)


1 + 119900 (119905) le 119888 (1 + 119894119905119886) = 119888 (1 + 119894119905119879 (119886)) (as 119905 rarr 0) (71)



120582cap 120590(119909) = 0 If 120582 notin 120590(119909)

then 120582 notin 120590119870(119909) where 120590



120590119870(119886) = 120582 isin C lim

120583rarr120582

119888 (119886 minus 120583) = 0 (72)



lim120583rarr120582

119888 (119879 (119909) minus 120583) gt 0 (73)




120597120590 (119879 (119909)) sube 120590119870(119879 (119909)) sube 120590 (119909) (74)




120597120590 (119879 (119906)) ⫋ T 119879 (119906) le 1

10038171003817100381710038171003817119879(119906)minus110038171003817100381710038171003817= 119888(119879 (119906))

minus1

= 119888(119906)minus1

= 1

(75)


1 = 119879 (119906) 119879(119906)lowast

= 119879 (119906) 119879 (119906lowast

)

= (1 + 119894119905119879 (119886) minus 1

2

1199052

119879 (1198862



2

1199052

119879 (1198862


(76)






lowast


119888(1 + 120572119909)2

= 119888(119887 + 120572119910)

2

= 119888 (119887119887lowast

+ |120572|2

119910119910lowast

) (77)

Hence

lim|120572|rarr0

119888 (119887119887lowast

+ |120572|2

119910119910lowast

) = lim|120572|rarr0

119888(1 + 120572119909)2

= 1 = 119888 (119887119887lowast

)

(78)


119888(1 + 120572119909)2

= 119888 (119887119887lowast

+ |120572| 119910119910lowast

) ge 1 minus |120572|210038171003817100381710038171199101003817100381710038171003817

2

(79)

and therefore

119888 (1 + 119894119905119909) ge (1 minus 119905210038171003817100381710038171199101003817100381710038171003817

2

)

12

119888 (1 minus 119905119909) ge (1 minus 11990521003817100381710038171003817119910

1003817100381710038171003817

2

)

12

(80)




minus1


119898(119878 (119909)) = 119898 (119887minus1

119879 (119909)) le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119898 (119879 (119909))


(81)


minus1


119898(1 + 119894119905119910) le 119888 (1 + 119894119905119878minus1

(119910)) (82)


119860dagger and 119906 isin 119860


(119906119909) (119909dagger

119906minus1

) (119906119909) = 119906119909

(119909dagger

119906minus1

) (119906119909) (119909dagger

119906minus1

) = 119909dagger

119906minus1

(83)


1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le 119888 (119906119909)

le

10038171003817100381710038171003817119909dagger

119906minus1

119906119909

10038171003817100381710038171003817

10038171003817100381710038171003817119906119909119909dagger

119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le

119906

10038171003817100381710038171003817119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

(84)


119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909)) ge

1

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

ge

1

119887

10038171003817100381710038171003817119879(119909)dagger10038171003817100381710038171003817

= 119887minus1

119888 (119879 (119909))

119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909))

le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119879 (1)

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

le 119887 |

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2

119888 (119879 (119909))

(85)



119887minus1

119888 (119909) le 119888 (119878 (119909)) le 119887

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2


(86)


120597120590 (119878 (119909)) sub 120590119870(119878 (119909)) sube 120590 (119909) (87)


120597120590 (119878 (119909) + 120583119878 (119910)) sub 120590119870(119878 (119909) + 120583119878 (119910))

sub 120590 (119909 + 120583119910) sub R cup 120583R(88)



120590 (119878 (119909) + 120583119878 (119910)) sub R cup 120583R (89)


minus1

)119887minus1


10038171003817100381710038171003817119909minus110038171003817100381710038171003817 (90)


)

10038171003817100381710038171003817=

10038171003817100381710038171003817119909minus110038171003817100381710038171003817=

10038171003817100381710038171003817119878 (119909minus1

) 119887minus110038171003817100381710038171003817 (91)

or equivalently1003817100381710038171003817119910

1003817100381710038171003817=

10038171003817100381710038171003817119910119887minus110038171003817100381710038171003817 (92)



119888 (a) = 10038171003817100381710038171003817aminus11003817100381710038171003817

1003817

minus1

= limU

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817

minus1

= limU

119888 (119886119899) (93)





sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 119879 (1) minus 1 lt 120575 (94)

imply that

jmult (119879) lt 120576 sa (119879) lt 120576 (95)


[119887119899] where 119887


N and 119887dagger



119899]





(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119879

119899(1) minus 1 lt

1119899(iii) jmult(119879

119899) ge 1205760or sa(119879

119899) ge 1205760


Urarr 119861



1003817100381710038171003817119879119899(1) minus 1

1003817100381710038171003817le lim

U

1

119899

= 0 (96)




minus1


We know that

119888 (119886119899) minus

1

119899


Hence 119879119899(119886119899) isin 119860




|119888 (T (a)) minus 119888 (a)| = limU

1003816100381610038161003816119888 (119879119899(119886119899)) minus 119888 (119886

119899)

1003816100381610038161003816le lim

U

1

119899

= 0

(98)



119884sube 119879(119861

119883) whereas






condition

sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 (99)

implies that




0 1205760



(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119902(119879

119899) ge 119870

minus1

0


119879119899) ge 1205760or sa(119879

119899(1)lowast

119879119899) ge 1205760


119899] preserves


119902 (T) = limU119902 (119879119899) ge 119870

minus1

0gt 0 (101)




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












mult (119879)

= sup 119879 (119886119887) minus 119879 (119886) 119879 (119887) 119886 119887 isin 119860 119886 = 119887 = 1

amult (119879)

= sup 119879 (119886119887) minus 119879 (119887) 119879 (119886) 119886 119887 isin 119860 119886 = 119887 = 1

jmult (119879) = sup 10038171003817100381710038171003817119879 (1198862

) minus 119879(119886)210038171003817100381710038171003817 119886 isin 119860 119886 = 1

(5)




tmult (119879)

= sup 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119886 = 119887 = 119888 = 1

sa (119879) = sup 10038171003817100381710038171003817119879(119886lowast

)

lowast

minus 119879 (119886)

10038171003817100381710038171003817 119886 = 1

(6)



minus1

) = 119879(119886)minus1 into

sup119886=1119886isin119860

minus1

10038171003817100381710038171003817119879 (119886minus1

) minus 119879(119886)minus110038171003817100381710038171003817lt 120576 (7)


) = 119879(119886)119866 into

sup119886=1119886isin119860

119866

10038171003817100381710038171003817119879 (119886119866

) minus 119879(119886)11986610038171003817100381710038171003817lt 120576 (8)



dagger



sup119886=1119886isin119860

dagger

10038171003817100381710038171003817119879 (119886dagger

) minus 119879(119886)dagger10038171003817100381710038171003817lt 120576 (9)






120574 (119879) = inf 119879119909 dist (119909 ker (119879)) ge 1 (10)


119888 (119886) =

10038171003817100381710038171003817119886dagger10038171003817100381710038171003817

minus1 (11)


sup119886=1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120576 (12)





defined as follows

(119883119899)

U=

ℓinfin

(N 119883119899)

NU

(13)

where ℓinfin



119899isin 119883119899


infinnorm and

NU = 119909119899119899isinN

isin ℓinfin

(N 119883119899) lim

U

1003817100381710038171003817119909119899

1003817100381710038171003817= 0 (14)


U=

ℓinfin


119899] the



x = limU

1003817100381710038171003817119909119899

1003817100381710038171003817 (15)






U 119884

U) given by

T (x) = [119879119899(119909119899)] (16)

for every T = [119879119899] isin B(119883 119884)

U and x = [119909119899] isin



U 119884



U)





equivalent


119899] such that 119906

119899isin 119860minus1





119899]




limU

1003817100381710038171003817119886119899119887119899minus 1100381710038171003817

1003817= 0

limU

1003817100381710038171003817119887119899119886119899minus 1100381710038171003817

1003817= 0

(17)


119877 = 119899 isin N

1003817100381710038171003817119886119899119887119899minus 1100381710038171003817

1003817lt 120575 isin U

119871 = 119899 isin N 1003817100381710038171003817119887119899119886119899minus 1100381710038171003817

1003817lt 120575 isin U

(18)


119899



10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817119887119899(119886119899119887119899)

minus110038171003817100381710038171003817le1003817100381710038171003817119887119899

1003817100381710038171003817

10038171003817100381710038171003817(119886119899119887119899)

minus110038171003817100381710038171003817

le

1003817100381710038171003817119887119899

1003817100381710038171003817

1 minus

10038171003817100381710038171 minus 119886119899119887119899

1003817100381710038171003817

le

1003817100381710038171003817119887119899

1003817100381710038171003817

1 minus 120575

(19)






we choose

1198861015840

119899=

119886119899 if 119899 isin 119868

1 if 119899 notin 119868(20)

Clearly [119886119899] = [119886

1015840

119899])



119899] such that 119887

119899 =


119887119899=

119886119899

1003817100381710038171003817119886119899

1003817100381710038171003817

if 119886119899

= 0

1 if 119886119899= 0

(21)








ucomm (119879) = sup119886=1

119879 (119886) 119879 (1) minus 119879 (1) 119879 (119886) (22)







119899] 119860

Urarr 119861

U Then

limU

ucomm (119879119899) = ucomm (T) (23)


[119886119899] with 119886



= limU

1003817100381710038171003817119879119899(119886119899) 119879119899(1) minus 119879

119899(1) 119879119899(119886119899)

1003817100381710038171003817le lim

Uucomm (119879

119899)

(24)

and hence

ucomm (T) le limU

ucomm (119879119899) (25)


119886119899 = 1 such that

ucomm (119879119899) minus

1

119899

lt

1003817100381710038171003817119879119899(119886119899) 119879119899(1) minus 119879

119899(1) 119879119899(119886119899)

1003817100381710038171003817 (26)



limU

ucomm (119879119899)

le limU

1003817100381710038171003817119879119899(119886119899) 119879119899(1) minus 119879

119899(1) 119879119899(119886119899)

1003817100381710038171003817


(27)



sup119886=1119886isin119860

minus1

10038171003817100381710038171003817119879 (119886minus1

) minus 119879(119886)minus110038171003817100381710038171003817lt 120575 (28)

implies that

jmult (119879 (1) 119879) lt 120576 ucomm (119879) lt 120576 (29)



119899119899isinN of


(i) 119879119899 lt 119870

0

(ii) sup119886=1

119879119899(119886minus1

) minus 119879119899(119886)minus1

lt 1119899(iii) jmult(119879

119899(1)119879119899) ge 1205760or ucomm(119879

119899) ge 1205760

Consider that T = [119879119899] 119860

Urarr 119861




119886minus1


As10038171003817100381710038171003817119879119899(119886minus1

119899)

10038171003817100381710038171003817le

1003817100381710038171003817119879119899

1003817100381710038171003817

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817lt 1198700120572

10038171003817100381710038171003817119879119899(119886minus1

119899) minus 119879119899(119886119899)

minus110038171003817100381710038171003817lt 1

(30)

we get

10038171003817100381710038171003817119879119899(119886119899)

minus110038171003817100381710038171003817lt 1198700120572 + 1 (31)




1003817

= limU

10038171003817100381710038171003817119879119899(119886minus1

119899) minus 119879119899(119886119899)

minus110038171003817100381710038171003817le lim

U

1

119899

= 0

(32)






jmult (119879119899(1) 119879119899)


ucomm (119879119899)

(33)

Consequently



(34)





(119886minus1


)

minus1

)

minus1

= 119886 minus 119886119887119886 (35)





119906minus1




(119906minus1


)

minus1

)

minus1

= 119906 minus 119906 (120582minus1

119906) 119906 = 119906 minus 120582minus1

1199063

(36)




and 0 lt |120582| lt 119879(119906)119866

minus2 gives

119879 (119906) minus 120582minus1

119879(119906)3

= (119879(119906)119866

minus (119879 (119906) minus 120582119879(119906)119866

)

119866

)

119866

(37)




we get

119879 (119906) minus 120582minus1

119879(119906)3

= (119879 (119906minus1


)

119866

)

119866

= 119879(119906minus1


)

minus1

)

119866

= 119879((119906minus1


)

minus1

)

minus1

)

= 119879 (119906 minus 120582minus1

1199063

) = 119879 (119906) minus 120582minus1

119879 (1199063

)

(38)

Hence 119879(1199063



3119879 (119909) = 119879(1)2119879 (119909) + 119879 (119909) 119879(1)2 + 119879 (1) 119879 (119909) 119879 (1)

3119879 (1199092

) = 119879(119909)2

119879 (1) + 119879 (1) 119879(119909)2 + 119879 (119909) 119879 (1) 119879 (119909) (39)





sup119886=1119886isin119860

119866

10038171003817100381710038171003817119879 (119886119866

) minus 119879(119886)11986610038171003817100381710038171003817lt 120575 (40)

implies that

jmult (119879 (1) 119879) lt 120576 ucomm (119879) lt 120576 (41)


119899] where 119887


N and 119887119866


and b119866 = [119887119866



Urarr






119879 (119886) 119879 (119886minus1

) = 119879(1)2




119899 119860 rarr 119861 satisfying

that 119879119899 119879119899(1)minus1


10038171003817100381710038171003817119879119899(119886) 119879119899(119886minus1

) minus 119879119899(1)210038171003817100381710038171003817lt

1

119899








minus1



10038171003817100381710038171003817119879 (119886) 119879 (119886

minus1

) minus 119879(1)210038171003817100381710038171003817lt 120575 (44)

implies that






119909 119910 119911 = sum

1205722

=1

sum

1205734

=1

120572120573(119909 + 120572119910 + 120573119911)

[3]

for119909 119910 119911 isin 119864

(46)



119864 = 1198642(119890) oplus 119864

1(119890) oplus 119864

0(119890) (47)



119909 119890 119910 and involution 119909♯






2(119890) If 119886 is



+(119887minus1

minus119886)minus1


(119886minus1

+ (119887minus1

minus 119886)

minus1

)

minus1

= 119886 minus 119880119886(119887) (48)


2

∘ 119909 (see [33] (11))




119860)2(119890) =

119890119890lowast

119860119890lowast


|120582| lt 119906and


lowast

119860119890lowast


and and 119906and

minus (119906 minus

120582119906and

)and in 119890119890

lowast

119860119890lowast



119906[3]

= (119906and

minus (119906 minus 120582119906and

)

and

)

and

(49)





119879 (119906) minus 120582minus1

119879(119906)[3]

= (119879(119906)and

minus (119879 (119906) minus 120582119879(119906)and

)

and

)

and

= 119879 (119906) minus 120582minus1

119879 (1199063

)

(50)




3


we obtain

119879 (1198863

) + 3120572119879 (1198862

) + 31205722

119879 (119886) + 1205723

119879 (1)

= 119879(119886)[3]

+ 1205723

119879 (1) + 2120572 119879 (119886) 119879 (119886) 119879 (1)

+ 120572 119879 (119886) 119879 (1) 119879 (119886) + 21205722

119879 (1) 119879 (1) 119879 (119886)

+ 1205722

119879 (1) 119879 (119886) 119879 (1)

(51)






tmult (119879) = sup 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119886 = 119887 = 119888 = 1

sa (119879) = sup119886=1

10038171003817100381710038171003817119879(119886lowast

)

lowast

minus 119879 (119886)

10038171003817100381710038171003817

(52)



119879119869(119886 119887) = 119879 (119886 ∘ 119887) minus 119879 (119886) ∘ 119879 (119887)

119879119879(119886 119887 119888) = 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119879lowast(119886) = 119879(119886

lowast

)

lowast

minus 119879 (119886)

(53)


119879 (119886 119887 119888)

= 119879 ((119886 ∘ 119887lowast

) ∘ 119888) + 119879 (119886 ∘ (119887lowast

∘ 119888)) minus 119879 ((119886 ∘ 119888) ∘ 119887lowast

)

= 119879 (119886 ∘ 119887lowast

) ∘ 119879 (119888) + 119879119869(119886 ∘ 119887lowast

119888)

+ 119879 (119886) ∘ 119879 (119887lowast

∘ 119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879 (119886 ∘ 119888) ∘ 119879 (119887lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

= (119879 (119886) ∘ 119879 (119887lowast

)) ∘ 119879 (119888) + 119879119869(119886 119887lowast

) ∘ 119879 (119888)

+ 119879119869(119886 ∘ 119887lowast

119888) + 119879 (119886) ∘ (119879 (119887lowast

) ∘ 119879 (119888))

+ 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus (119879 (119886) ∘ 119879 (119888)) ∘ 119879 (119887lowast

) minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

)

minus 119879119869(119886 ∘ 119888 119887

lowast

)

= 119879 (119886) 119879(119887lowast

)

lowast

119879 (119888) + 119879119869(119886 119887lowast

) ∘ 119879 (119888)

+ 119879119869(119886 ∘ 119887lowast

119888) + 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

(54)

Therefore

119879119879(119886 119887 119888)

= 119879 (119886) 119879lowast(119887) 119879 (119888) + 119879

119869(119886 119887lowast

) ∘ 119879 (119888) + 119879119869(119886 ∘ 119887lowast

119888)

+ 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

(55)

This implies that

tmult (119879) le 1198792sa (119879) + 3 (119879 + 1) jmult (119879) (56)



limU

tmult (119879119899) = tmult (T)

limU

sa (119879119899) = sa (T)

(57)



119899] 119860

Urarr 119861




sup119886=1119886=119886

lowast

10038171003817100381710038171003817119879 (119886and

) minus 119879(119886)and10038171003817100381710038171003817lt 120575 (58)

implies that

tmult (119879) lt 120576 (59)


lowast


and sa(119879(1)lowast119879) lt 120576


sup119886=1119886isin119860

dagger

10038171003817100381710038171003817119879 (119886dagger


imply that

tmult (119879) lt 120576 (61)





1003817100381710038171003817119879119899

1003817100381710038171003817lt 1198700 sup

119886=1

10038171003817100381710038171003817119879119899(119886dagger

) minus 119879119899(119886)dagger10038171003817100381710038171003817lt

1

119899

sa (119879119899) lt

1

119899

tmult (119879119899) ge 1205760

(62)

Then T = [119879119899] 119860

Urarr 119861







119898(119886) = inf 119886119909 119909 isin 119860 119909 = 1

119902 (119886) = inf 119909119886 119909 isin 119860 119909 = 1

(63)


119898(119886) = 119902 (119886) =

10038171003817100381710038171003817119886minus110038171003817100381710038171003817

minus1

(64)

Moreover

119898(119886)119898 (119887) le 119898 (119886119887) le 119886119898 (119887)

|119898 (119886) minus 119898 (119887)| le 119886 minus 119887

(65)




119888 (119886) =

inf 119886119909 dist (119909 ker (119871119886)) ge 1 if 119886 = 0

infin if 119886 = 0

(66)


119888 (119886) =

10038171003817100381710038171003817119886dagger10038171003817100381710038171003817

minus1

(67)







1 = 119888 (119879 (1)) = 119888 (119879 (1 + 1205721198860)) = 119888 (1 + 120572119886

0) (68)


1 le 119888 (1 + 1198941199051198860) 1 le 119888 (1 minus 119905119886

0) (69)


0are




1 + 119900 (119905) le 119888 (1 + 119894119905119886) (as 119905 rarr 0) (70)


1 + 119900 (119905) le 119888 (1 + 119894119905119886) = 119888 (1 + 119894119905119879 (119886)) (as 119905 rarr 0) (71)



120582cap 120590(119909) = 0 If 120582 notin 120590(119909)

then 120582 notin 120590119870(119909) where 120590



120590119870(119886) = 120582 isin C lim

120583rarr120582

119888 (119886 minus 120583) = 0 (72)



lim120583rarr120582

119888 (119879 (119909) minus 120583) gt 0 (73)




120597120590 (119879 (119909)) sube 120590119870(119879 (119909)) sube 120590 (119909) (74)




120597120590 (119879 (119906)) ⫋ T 119879 (119906) le 1

10038171003817100381710038171003817119879(119906)minus110038171003817100381710038171003817= 119888(119879 (119906))

minus1

= 119888(119906)minus1

= 1

(75)


1 = 119879 (119906) 119879(119906)lowast

= 119879 (119906) 119879 (119906lowast

)

= (1 + 119894119905119879 (119886) minus 1

2

1199052

119879 (1198862



2

1199052

119879 (1198862


(76)






lowast


119888(1 + 120572119909)2

= 119888(119887 + 120572119910)

2

= 119888 (119887119887lowast

+ |120572|2

119910119910lowast

) (77)

Hence

lim|120572|rarr0

119888 (119887119887lowast

+ |120572|2

119910119910lowast

) = lim|120572|rarr0

119888(1 + 120572119909)2

= 1 = 119888 (119887119887lowast

)

(78)


119888(1 + 120572119909)2

= 119888 (119887119887lowast

+ |120572| 119910119910lowast

) ge 1 minus |120572|210038171003817100381710038171199101003817100381710038171003817

2

(79)

and therefore

119888 (1 + 119894119905119909) ge (1 minus 119905210038171003817100381710038171199101003817100381710038171003817

2

)

12

119888 (1 minus 119905119909) ge (1 minus 11990521003817100381710038171003817119910

1003817100381710038171003817

2

)

12

(80)




minus1


119898(119878 (119909)) = 119898 (119887minus1

119879 (119909)) le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119898 (119879 (119909))


(81)


minus1


119898(1 + 119894119905119910) le 119888 (1 + 119894119905119878minus1

(119910)) (82)


119860dagger and 119906 isin 119860


(119906119909) (119909dagger

119906minus1

) (119906119909) = 119906119909

(119909dagger

119906minus1

) (119906119909) (119909dagger

119906minus1

) = 119909dagger

119906minus1

(83)


1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le 119888 (119906119909)

le

10038171003817100381710038171003817119909dagger

119906minus1

119906119909

10038171003817100381710038171003817

10038171003817100381710038171003817119906119909119909dagger

119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le

119906

10038171003817100381710038171003817119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

(84)


119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909)) ge

1

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

ge

1

119887

10038171003817100381710038171003817119879(119909)dagger10038171003817100381710038171003817

= 119887minus1

119888 (119879 (119909))

119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909))

le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119879 (1)

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

le 119887 |

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2

119888 (119879 (119909))

(85)



119887minus1

119888 (119909) le 119888 (119878 (119909)) le 119887

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2


(86)


120597120590 (119878 (119909)) sub 120590119870(119878 (119909)) sube 120590 (119909) (87)


120597120590 (119878 (119909) + 120583119878 (119910)) sub 120590119870(119878 (119909) + 120583119878 (119910))

sub 120590 (119909 + 120583119910) sub R cup 120583R(88)



120590 (119878 (119909) + 120583119878 (119910)) sub R cup 120583R (89)


minus1

)119887minus1


10038171003817100381710038171003817119909minus110038171003817100381710038171003817 (90)


)

10038171003817100381710038171003817=

10038171003817100381710038171003817119909minus110038171003817100381710038171003817=

10038171003817100381710038171003817119878 (119909minus1

) 119887minus110038171003817100381710038171003817 (91)

or equivalently1003817100381710038171003817119910

1003817100381710038171003817=

10038171003817100381710038171003817119910119887minus110038171003817100381710038171003817 (92)



119888 (a) = 10038171003817100381710038171003817aminus11003817100381710038171003817

1003817

minus1

= limU

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817

minus1

= limU

119888 (119886119899) (93)





sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 119879 (1) minus 1 lt 120575 (94)

imply that

jmult (119879) lt 120576 sa (119879) lt 120576 (95)


[119887119899] where 119887


N and 119887dagger



119899]





(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119879

119899(1) minus 1 lt

1119899(iii) jmult(119879

119899) ge 1205760or sa(119879

119899) ge 1205760


Urarr 119861



1003817100381710038171003817119879119899(1) minus 1

1003817100381710038171003817le lim

U

1

119899

= 0 (96)




minus1


We know that

119888 (119886119899) minus

1

119899


Hence 119879119899(119886119899) isin 119860




|119888 (T (a)) minus 119888 (a)| = limU

1003816100381610038161003816119888 (119879119899(119886119899)) minus 119888 (119886

119899)

1003816100381610038161003816le lim

U

1

119899

= 0

(98)



119884sube 119879(119861

119883) whereas






condition

sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 (99)

implies that




0 1205760



(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119902(119879

119899) ge 119870

minus1

0


119879119899) ge 1205760or sa(119879

119899(1)lowast

119879119899) ge 1205760


119899] preserves


119902 (T) = limU119902 (119879119899) ge 119870

minus1

0gt 0 (101)




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














U 119884

U) given by

T (x) = [119879119899(119909119899)] (16)

for every T = [119879119899] isin B(119883 119884)

U and x = [119909119899] isin



U 119884



U)





equivalent


119899] such that 119906

119899isin 119860minus1





119899]




limU

1003817100381710038171003817119886119899119887119899minus 1100381710038171003817

1003817= 0

limU

1003817100381710038171003817119887119899119886119899minus 1100381710038171003817

1003817= 0

(17)


119877 = 119899 isin N

1003817100381710038171003817119886119899119887119899minus 1100381710038171003817

1003817lt 120575 isin U

119871 = 119899 isin N 1003817100381710038171003817119887119899119886119899minus 1100381710038171003817

1003817lt 120575 isin U

(18)


119899



10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817=

10038171003817100381710038171003817119887119899(119886119899119887119899)

minus110038171003817100381710038171003817le1003817100381710038171003817119887119899

1003817100381710038171003817

10038171003817100381710038171003817(119886119899119887119899)

minus110038171003817100381710038171003817

le

1003817100381710038171003817119887119899

1003817100381710038171003817

1 minus

10038171003817100381710038171 minus 119886119899119887119899

1003817100381710038171003817

le

1003817100381710038171003817119887119899

1003817100381710038171003817

1 minus 120575

(19)






we choose

1198861015840

119899=

119886119899 if 119899 isin 119868

1 if 119899 notin 119868(20)

Clearly [119886119899] = [119886

1015840

119899])



119899] such that 119887

119899 =


119887119899=

119886119899

1003817100381710038171003817119886119899

1003817100381710038171003817

if 119886119899

= 0

1 if 119886119899= 0

(21)








ucomm (119879) = sup119886=1

119879 (119886) 119879 (1) minus 119879 (1) 119879 (119886) (22)







119899] 119860

Urarr 119861

U Then

limU

ucomm (119879119899) = ucomm (T) (23)


[119886119899] with 119886



= limU

1003817100381710038171003817119879119899(119886119899) 119879119899(1) minus 119879

119899(1) 119879119899(119886119899)

1003817100381710038171003817le lim

Uucomm (119879

119899)

(24)

and hence

ucomm (T) le limU

ucomm (119879119899) (25)


119886119899 = 1 such that

ucomm (119879119899) minus

1

119899

lt

1003817100381710038171003817119879119899(119886119899) 119879119899(1) minus 119879

119899(1) 119879119899(119886119899)

1003817100381710038171003817 (26)



limU

ucomm (119879119899)

le limU

1003817100381710038171003817119879119899(119886119899) 119879119899(1) minus 119879

119899(1) 119879119899(119886119899)

1003817100381710038171003817


(27)



sup119886=1119886isin119860

minus1

10038171003817100381710038171003817119879 (119886minus1

) minus 119879(119886)minus110038171003817100381710038171003817lt 120575 (28)

implies that

jmult (119879 (1) 119879) lt 120576 ucomm (119879) lt 120576 (29)



119899119899isinN of


(i) 119879119899 lt 119870

0

(ii) sup119886=1

119879119899(119886minus1

) minus 119879119899(119886)minus1

lt 1119899(iii) jmult(119879

119899(1)119879119899) ge 1205760or ucomm(119879

119899) ge 1205760

Consider that T = [119879119899] 119860

Urarr 119861




119886minus1


As10038171003817100381710038171003817119879119899(119886minus1

119899)

10038171003817100381710038171003817le

1003817100381710038171003817119879119899

1003817100381710038171003817

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817lt 1198700120572

10038171003817100381710038171003817119879119899(119886minus1

119899) minus 119879119899(119886119899)

minus110038171003817100381710038171003817lt 1

(30)

we get

10038171003817100381710038171003817119879119899(119886119899)

minus110038171003817100381710038171003817lt 1198700120572 + 1 (31)




1003817

= limU

10038171003817100381710038171003817119879119899(119886minus1

119899) minus 119879119899(119886119899)

minus110038171003817100381710038171003817le lim

U

1

119899

= 0

(32)






jmult (119879119899(1) 119879119899)


ucomm (119879119899)

(33)

Consequently



(34)





(119886minus1


)

minus1

)

minus1

= 119886 minus 119886119887119886 (35)





119906minus1




(119906minus1


)

minus1

)

minus1

= 119906 minus 119906 (120582minus1

119906) 119906 = 119906 minus 120582minus1

1199063

(36)




and 0 lt |120582| lt 119879(119906)119866

minus2 gives

119879 (119906) minus 120582minus1

119879(119906)3

= (119879(119906)119866

minus (119879 (119906) minus 120582119879(119906)119866

)

119866

)

119866

(37)




we get

119879 (119906) minus 120582minus1

119879(119906)3

= (119879 (119906minus1


)

119866

)

119866

= 119879(119906minus1


)

minus1

)

119866

= 119879((119906minus1


)

minus1

)

minus1

)

= 119879 (119906 minus 120582minus1

1199063

) = 119879 (119906) minus 120582minus1

119879 (1199063

)

(38)

Hence 119879(1199063



3119879 (119909) = 119879(1)2119879 (119909) + 119879 (119909) 119879(1)2 + 119879 (1) 119879 (119909) 119879 (1)

3119879 (1199092

) = 119879(119909)2

119879 (1) + 119879 (1) 119879(119909)2 + 119879 (119909) 119879 (1) 119879 (119909) (39)





sup119886=1119886isin119860

119866

10038171003817100381710038171003817119879 (119886119866

) minus 119879(119886)11986610038171003817100381710038171003817lt 120575 (40)

implies that

jmult (119879 (1) 119879) lt 120576 ucomm (119879) lt 120576 (41)


119899] where 119887


N and 119887119866


and b119866 = [119887119866



Urarr






119879 (119886) 119879 (119886minus1

) = 119879(1)2




119899 119860 rarr 119861 satisfying

that 119879119899 119879119899(1)minus1


10038171003817100381710038171003817119879119899(119886) 119879119899(119886minus1

) minus 119879119899(1)210038171003817100381710038171003817lt

1

119899








minus1



10038171003817100381710038171003817119879 (119886) 119879 (119886

minus1

) minus 119879(1)210038171003817100381710038171003817lt 120575 (44)

implies that






119909 119910 119911 = sum

1205722

=1

sum

1205734

=1

120572120573(119909 + 120572119910 + 120573119911)

[3]

for119909 119910 119911 isin 119864

(46)



119864 = 1198642(119890) oplus 119864

1(119890) oplus 119864

0(119890) (47)



119909 119890 119910 and involution 119909♯






2(119890) If 119886 is



+(119887minus1

minus119886)minus1


(119886minus1

+ (119887minus1

minus 119886)

minus1

)

minus1

= 119886 minus 119880119886(119887) (48)


2

∘ 119909 (see [33] (11))




119860)2(119890) =

119890119890lowast

119860119890lowast


|120582| lt 119906and


lowast

119860119890lowast


and and 119906and

minus (119906 minus

120582119906and

)and in 119890119890

lowast

119860119890lowast



119906[3]

= (119906and

minus (119906 minus 120582119906and

)

and

)

and

(49)





119879 (119906) minus 120582minus1

119879(119906)[3]

= (119879(119906)and

minus (119879 (119906) minus 120582119879(119906)and

)

and

)

and

= 119879 (119906) minus 120582minus1

119879 (1199063

)

(50)




3


we obtain

119879 (1198863

) + 3120572119879 (1198862

) + 31205722

119879 (119886) + 1205723

119879 (1)

= 119879(119886)[3]

+ 1205723

119879 (1) + 2120572 119879 (119886) 119879 (119886) 119879 (1)

+ 120572 119879 (119886) 119879 (1) 119879 (119886) + 21205722

119879 (1) 119879 (1) 119879 (119886)

+ 1205722

119879 (1) 119879 (119886) 119879 (1)

(51)






tmult (119879) = sup 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119886 = 119887 = 119888 = 1

sa (119879) = sup119886=1

10038171003817100381710038171003817119879(119886lowast

)

lowast

minus 119879 (119886)

10038171003817100381710038171003817

(52)



119879119869(119886 119887) = 119879 (119886 ∘ 119887) minus 119879 (119886) ∘ 119879 (119887)

119879119879(119886 119887 119888) = 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119879lowast(119886) = 119879(119886

lowast

)

lowast

minus 119879 (119886)

(53)


119879 (119886 119887 119888)

= 119879 ((119886 ∘ 119887lowast

) ∘ 119888) + 119879 (119886 ∘ (119887lowast

∘ 119888)) minus 119879 ((119886 ∘ 119888) ∘ 119887lowast

)

= 119879 (119886 ∘ 119887lowast

) ∘ 119879 (119888) + 119879119869(119886 ∘ 119887lowast

119888)

+ 119879 (119886) ∘ 119879 (119887lowast

∘ 119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879 (119886 ∘ 119888) ∘ 119879 (119887lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

= (119879 (119886) ∘ 119879 (119887lowast

)) ∘ 119879 (119888) + 119879119869(119886 119887lowast

) ∘ 119879 (119888)

+ 119879119869(119886 ∘ 119887lowast

119888) + 119879 (119886) ∘ (119879 (119887lowast

) ∘ 119879 (119888))

+ 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus (119879 (119886) ∘ 119879 (119888)) ∘ 119879 (119887lowast

) minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

)

minus 119879119869(119886 ∘ 119888 119887

lowast

)

= 119879 (119886) 119879(119887lowast

)

lowast

119879 (119888) + 119879119869(119886 119887lowast

) ∘ 119879 (119888)

+ 119879119869(119886 ∘ 119887lowast

119888) + 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

(54)

Therefore

119879119879(119886 119887 119888)

= 119879 (119886) 119879lowast(119887) 119879 (119888) + 119879

119869(119886 119887lowast

) ∘ 119879 (119888) + 119879119869(119886 ∘ 119887lowast

119888)

+ 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

(55)

This implies that

tmult (119879) le 1198792sa (119879) + 3 (119879 + 1) jmult (119879) (56)



limU

tmult (119879119899) = tmult (T)

limU

sa (119879119899) = sa (T)

(57)



119899] 119860

Urarr 119861




sup119886=1119886=119886

lowast

10038171003817100381710038171003817119879 (119886and

) minus 119879(119886)and10038171003817100381710038171003817lt 120575 (58)

implies that

tmult (119879) lt 120576 (59)


lowast


and sa(119879(1)lowast119879) lt 120576


sup119886=1119886isin119860

dagger

10038171003817100381710038171003817119879 (119886dagger


imply that

tmult (119879) lt 120576 (61)





1003817100381710038171003817119879119899

1003817100381710038171003817lt 1198700 sup

119886=1

10038171003817100381710038171003817119879119899(119886dagger

) minus 119879119899(119886)dagger10038171003817100381710038171003817lt

1

119899

sa (119879119899) lt

1

119899

tmult (119879119899) ge 1205760

(62)

Then T = [119879119899] 119860

Urarr 119861







119898(119886) = inf 119886119909 119909 isin 119860 119909 = 1

119902 (119886) = inf 119909119886 119909 isin 119860 119909 = 1

(63)


119898(119886) = 119902 (119886) =

10038171003817100381710038171003817119886minus110038171003817100381710038171003817

minus1

(64)

Moreover

119898(119886)119898 (119887) le 119898 (119886119887) le 119886119898 (119887)

|119898 (119886) minus 119898 (119887)| le 119886 minus 119887

(65)




119888 (119886) =

inf 119886119909 dist (119909 ker (119871119886)) ge 1 if 119886 = 0

infin if 119886 = 0

(66)


119888 (119886) =

10038171003817100381710038171003817119886dagger10038171003817100381710038171003817

minus1

(67)







1 = 119888 (119879 (1)) = 119888 (119879 (1 + 1205721198860)) = 119888 (1 + 120572119886

0) (68)


1 le 119888 (1 + 1198941199051198860) 1 le 119888 (1 minus 119905119886

0) (69)


0are




1 + 119900 (119905) le 119888 (1 + 119894119905119886) (as 119905 rarr 0) (70)


1 + 119900 (119905) le 119888 (1 + 119894119905119886) = 119888 (1 + 119894119905119879 (119886)) (as 119905 rarr 0) (71)



120582cap 120590(119909) = 0 If 120582 notin 120590(119909)

then 120582 notin 120590119870(119909) where 120590



120590119870(119886) = 120582 isin C lim

120583rarr120582

119888 (119886 minus 120583) = 0 (72)



lim120583rarr120582

119888 (119879 (119909) minus 120583) gt 0 (73)




120597120590 (119879 (119909)) sube 120590119870(119879 (119909)) sube 120590 (119909) (74)




120597120590 (119879 (119906)) ⫋ T 119879 (119906) le 1

10038171003817100381710038171003817119879(119906)minus110038171003817100381710038171003817= 119888(119879 (119906))

minus1

= 119888(119906)minus1

= 1

(75)


1 = 119879 (119906) 119879(119906)lowast

= 119879 (119906) 119879 (119906lowast

)

= (1 + 119894119905119879 (119886) minus 1

2

1199052

119879 (1198862



2

1199052

119879 (1198862


(76)






lowast


119888(1 + 120572119909)2

= 119888(119887 + 120572119910)

2

= 119888 (119887119887lowast

+ |120572|2

119910119910lowast

) (77)

Hence

lim|120572|rarr0

119888 (119887119887lowast

+ |120572|2

119910119910lowast

) = lim|120572|rarr0

119888(1 + 120572119909)2

= 1 = 119888 (119887119887lowast

)

(78)


119888(1 + 120572119909)2

= 119888 (119887119887lowast

+ |120572| 119910119910lowast

) ge 1 minus |120572|210038171003817100381710038171199101003817100381710038171003817

2

(79)

and therefore

119888 (1 + 119894119905119909) ge (1 minus 119905210038171003817100381710038171199101003817100381710038171003817

2

)

12

119888 (1 minus 119905119909) ge (1 minus 11990521003817100381710038171003817119910

1003817100381710038171003817

2

)

12

(80)




minus1


119898(119878 (119909)) = 119898 (119887minus1

119879 (119909)) le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119898 (119879 (119909))


(81)


minus1


119898(1 + 119894119905119910) le 119888 (1 + 119894119905119878minus1

(119910)) (82)


119860dagger and 119906 isin 119860


(119906119909) (119909dagger

119906minus1

) (119906119909) = 119906119909

(119909dagger

119906minus1

) (119906119909) (119909dagger

119906minus1

) = 119909dagger

119906minus1

(83)


1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le 119888 (119906119909)

le

10038171003817100381710038171003817119909dagger

119906minus1

119906119909

10038171003817100381710038171003817

10038171003817100381710038171003817119906119909119909dagger

119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le

119906

10038171003817100381710038171003817119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

(84)


119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909)) ge

1

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

ge

1

119887

10038171003817100381710038171003817119879(119909)dagger10038171003817100381710038171003817

= 119887minus1

119888 (119879 (119909))

119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909))

le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119879 (1)

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

le 119887 |

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2

119888 (119879 (119909))

(85)



119887minus1

119888 (119909) le 119888 (119878 (119909)) le 119887

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2


(86)


120597120590 (119878 (119909)) sub 120590119870(119878 (119909)) sube 120590 (119909) (87)


120597120590 (119878 (119909) + 120583119878 (119910)) sub 120590119870(119878 (119909) + 120583119878 (119910))

sub 120590 (119909 + 120583119910) sub R cup 120583R(88)



120590 (119878 (119909) + 120583119878 (119910)) sub R cup 120583R (89)


minus1

)119887minus1


10038171003817100381710038171003817119909minus110038171003817100381710038171003817 (90)


)

10038171003817100381710038171003817=

10038171003817100381710038171003817119909minus110038171003817100381710038171003817=

10038171003817100381710038171003817119878 (119909minus1

) 119887minus110038171003817100381710038171003817 (91)

or equivalently1003817100381710038171003817119910

1003817100381710038171003817=

10038171003817100381710038171003817119910119887minus110038171003817100381710038171003817 (92)



119888 (a) = 10038171003817100381710038171003817aminus11003817100381710038171003817

1003817

minus1

= limU

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817

minus1

= limU

119888 (119886119899) (93)





sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 119879 (1) minus 1 lt 120575 (94)

imply that

jmult (119879) lt 120576 sa (119879) lt 120576 (95)


[119887119899] where 119887


N and 119887dagger



119899]





(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119879

119899(1) minus 1 lt

1119899(iii) jmult(119879

119899) ge 1205760or sa(119879

119899) ge 1205760


Urarr 119861



1003817100381710038171003817119879119899(1) minus 1

1003817100381710038171003817le lim

U

1

119899

= 0 (96)




minus1


We know that

119888 (119886119899) minus

1

119899


Hence 119879119899(119886119899) isin 119860




|119888 (T (a)) minus 119888 (a)| = limU

1003816100381610038161003816119888 (119879119899(119886119899)) minus 119888 (119886

119899)

1003816100381610038161003816le lim

U

1

119899

= 0

(98)



119884sube 119879(119861

119883) whereas






condition

sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 (99)

implies that




0 1205760



(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119902(119879

119899) ge 119870

minus1

0


119879119899) ge 1205760or sa(119879

119899(1)lowast

119879119899) ge 1205760


119899] preserves


119902 (T) = limU119902 (119879119899) ge 119870

minus1

0gt 0 (101)




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











limU

ucomm (119879119899)

le limU

1003817100381710038171003817119879119899(119886119899) 119879119899(1) minus 119879

119899(1) 119879119899(119886119899)

1003817100381710038171003817


(27)



sup119886=1119886isin119860

minus1

10038171003817100381710038171003817119879 (119886minus1

) minus 119879(119886)minus110038171003817100381710038171003817lt 120575 (28)

implies that

jmult (119879 (1) 119879) lt 120576 ucomm (119879) lt 120576 (29)



119899119899isinN of


(i) 119879119899 lt 119870

0

(ii) sup119886=1

119879119899(119886minus1

) minus 119879119899(119886)minus1

lt 1119899(iii) jmult(119879

119899(1)119879119899) ge 1205760or ucomm(119879

119899) ge 1205760

Consider that T = [119879119899] 119860

Urarr 119861




119886minus1


As10038171003817100381710038171003817119879119899(119886minus1

119899)

10038171003817100381710038171003817le

1003817100381710038171003817119879119899

1003817100381710038171003817

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817lt 1198700120572

10038171003817100381710038171003817119879119899(119886minus1

119899) minus 119879119899(119886119899)

minus110038171003817100381710038171003817lt 1

(30)

we get

10038171003817100381710038171003817119879119899(119886119899)

minus110038171003817100381710038171003817lt 1198700120572 + 1 (31)




1003817

= limU

10038171003817100381710038171003817119879119899(119886minus1

119899) minus 119879119899(119886119899)

minus110038171003817100381710038171003817le lim

U

1

119899

= 0

(32)






jmult (119879119899(1) 119879119899)


ucomm (119879119899)

(33)

Consequently



(34)





(119886minus1


)

minus1

)

minus1

= 119886 minus 119886119887119886 (35)





119906minus1




(119906minus1


)

minus1

)

minus1

= 119906 minus 119906 (120582minus1

119906) 119906 = 119906 minus 120582minus1

1199063

(36)




and 0 lt |120582| lt 119879(119906)119866

minus2 gives

119879 (119906) minus 120582minus1

119879(119906)3

= (119879(119906)119866

minus (119879 (119906) minus 120582119879(119906)119866

)

119866

)

119866

(37)




we get

119879 (119906) minus 120582minus1

119879(119906)3

= (119879 (119906minus1


)

119866

)

119866

= 119879(119906minus1


)

minus1

)

119866

= 119879((119906minus1


)

minus1

)

minus1

)

= 119879 (119906 minus 120582minus1

1199063

) = 119879 (119906) minus 120582minus1

119879 (1199063

)

(38)

Hence 119879(1199063



3119879 (119909) = 119879(1)2119879 (119909) + 119879 (119909) 119879(1)2 + 119879 (1) 119879 (119909) 119879 (1)

3119879 (1199092

) = 119879(119909)2

119879 (1) + 119879 (1) 119879(119909)2 + 119879 (119909) 119879 (1) 119879 (119909) (39)





sup119886=1119886isin119860

119866

10038171003817100381710038171003817119879 (119886119866

) minus 119879(119886)11986610038171003817100381710038171003817lt 120575 (40)

implies that

jmult (119879 (1) 119879) lt 120576 ucomm (119879) lt 120576 (41)


119899] where 119887


N and 119887119866


and b119866 = [119887119866



Urarr






119879 (119886) 119879 (119886minus1

) = 119879(1)2




119899 119860 rarr 119861 satisfying

that 119879119899 119879119899(1)minus1


10038171003817100381710038171003817119879119899(119886) 119879119899(119886minus1

) minus 119879119899(1)210038171003817100381710038171003817lt

1

119899








minus1



10038171003817100381710038171003817119879 (119886) 119879 (119886

minus1

) minus 119879(1)210038171003817100381710038171003817lt 120575 (44)

implies that






119909 119910 119911 = sum

1205722

=1

sum

1205734

=1

120572120573(119909 + 120572119910 + 120573119911)

[3]

for119909 119910 119911 isin 119864

(46)



119864 = 1198642(119890) oplus 119864

1(119890) oplus 119864

0(119890) (47)



119909 119890 119910 and involution 119909♯






2(119890) If 119886 is



+(119887minus1

minus119886)minus1


(119886minus1

+ (119887minus1

minus 119886)

minus1

)

minus1

= 119886 minus 119880119886(119887) (48)


2

∘ 119909 (see [33] (11))




119860)2(119890) =

119890119890lowast

119860119890lowast


|120582| lt 119906and


lowast

119860119890lowast


and and 119906and

minus (119906 minus

120582119906and

)and in 119890119890

lowast

119860119890lowast



119906[3]

= (119906and

minus (119906 minus 120582119906and

)

and

)

and

(49)





119879 (119906) minus 120582minus1

119879(119906)[3]

= (119879(119906)and

minus (119879 (119906) minus 120582119879(119906)and

)

and

)

and

= 119879 (119906) minus 120582minus1

119879 (1199063

)

(50)




3


we obtain

119879 (1198863

) + 3120572119879 (1198862

) + 31205722

119879 (119886) + 1205723

119879 (1)

= 119879(119886)[3]

+ 1205723

119879 (1) + 2120572 119879 (119886) 119879 (119886) 119879 (1)

+ 120572 119879 (119886) 119879 (1) 119879 (119886) + 21205722

119879 (1) 119879 (1) 119879 (119886)

+ 1205722

119879 (1) 119879 (119886) 119879 (1)

(51)






tmult (119879) = sup 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119886 = 119887 = 119888 = 1

sa (119879) = sup119886=1

10038171003817100381710038171003817119879(119886lowast

)

lowast

minus 119879 (119886)

10038171003817100381710038171003817

(52)



119879119869(119886 119887) = 119879 (119886 ∘ 119887) minus 119879 (119886) ∘ 119879 (119887)

119879119879(119886 119887 119888) = 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119879lowast(119886) = 119879(119886

lowast

)

lowast

minus 119879 (119886)

(53)


119879 (119886 119887 119888)

= 119879 ((119886 ∘ 119887lowast

) ∘ 119888) + 119879 (119886 ∘ (119887lowast

∘ 119888)) minus 119879 ((119886 ∘ 119888) ∘ 119887lowast

)

= 119879 (119886 ∘ 119887lowast

) ∘ 119879 (119888) + 119879119869(119886 ∘ 119887lowast

119888)

+ 119879 (119886) ∘ 119879 (119887lowast

∘ 119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879 (119886 ∘ 119888) ∘ 119879 (119887lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

= (119879 (119886) ∘ 119879 (119887lowast

)) ∘ 119879 (119888) + 119879119869(119886 119887lowast

) ∘ 119879 (119888)

+ 119879119869(119886 ∘ 119887lowast

119888) + 119879 (119886) ∘ (119879 (119887lowast

) ∘ 119879 (119888))

+ 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus (119879 (119886) ∘ 119879 (119888)) ∘ 119879 (119887lowast

) minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

)

minus 119879119869(119886 ∘ 119888 119887

lowast

)

= 119879 (119886) 119879(119887lowast

)

lowast

119879 (119888) + 119879119869(119886 119887lowast

) ∘ 119879 (119888)

+ 119879119869(119886 ∘ 119887lowast

119888) + 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

(54)

Therefore

119879119879(119886 119887 119888)

= 119879 (119886) 119879lowast(119887) 119879 (119888) + 119879

119869(119886 119887lowast

) ∘ 119879 (119888) + 119879119869(119886 ∘ 119887lowast

119888)

+ 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

(55)

This implies that

tmult (119879) le 1198792sa (119879) + 3 (119879 + 1) jmult (119879) (56)



limU

tmult (119879119899) = tmult (T)

limU

sa (119879119899) = sa (T)

(57)



119899] 119860

Urarr 119861




sup119886=1119886=119886

lowast

10038171003817100381710038171003817119879 (119886and

) minus 119879(119886)and10038171003817100381710038171003817lt 120575 (58)

implies that

tmult (119879) lt 120576 (59)


lowast


and sa(119879(1)lowast119879) lt 120576


sup119886=1119886isin119860

dagger

10038171003817100381710038171003817119879 (119886dagger


imply that

tmult (119879) lt 120576 (61)





1003817100381710038171003817119879119899

1003817100381710038171003817lt 1198700 sup

119886=1

10038171003817100381710038171003817119879119899(119886dagger

) minus 119879119899(119886)dagger10038171003817100381710038171003817lt

1

119899

sa (119879119899) lt

1

119899

tmult (119879119899) ge 1205760

(62)

Then T = [119879119899] 119860

Urarr 119861







119898(119886) = inf 119886119909 119909 isin 119860 119909 = 1

119902 (119886) = inf 119909119886 119909 isin 119860 119909 = 1

(63)


119898(119886) = 119902 (119886) =

10038171003817100381710038171003817119886minus110038171003817100381710038171003817

minus1

(64)

Moreover

119898(119886)119898 (119887) le 119898 (119886119887) le 119886119898 (119887)

|119898 (119886) minus 119898 (119887)| le 119886 minus 119887

(65)




119888 (119886) =

inf 119886119909 dist (119909 ker (119871119886)) ge 1 if 119886 = 0

infin if 119886 = 0

(66)


119888 (119886) =

10038171003817100381710038171003817119886dagger10038171003817100381710038171003817

minus1

(67)







1 = 119888 (119879 (1)) = 119888 (119879 (1 + 1205721198860)) = 119888 (1 + 120572119886

0) (68)


1 le 119888 (1 + 1198941199051198860) 1 le 119888 (1 minus 119905119886

0) (69)


0are




1 + 119900 (119905) le 119888 (1 + 119894119905119886) (as 119905 rarr 0) (70)


1 + 119900 (119905) le 119888 (1 + 119894119905119886) = 119888 (1 + 119894119905119879 (119886)) (as 119905 rarr 0) (71)



120582cap 120590(119909) = 0 If 120582 notin 120590(119909)

then 120582 notin 120590119870(119909) where 120590



120590119870(119886) = 120582 isin C lim

120583rarr120582

119888 (119886 minus 120583) = 0 (72)



lim120583rarr120582

119888 (119879 (119909) minus 120583) gt 0 (73)




120597120590 (119879 (119909)) sube 120590119870(119879 (119909)) sube 120590 (119909) (74)




120597120590 (119879 (119906)) ⫋ T 119879 (119906) le 1

10038171003817100381710038171003817119879(119906)minus110038171003817100381710038171003817= 119888(119879 (119906))

minus1

= 119888(119906)minus1

= 1

(75)


1 = 119879 (119906) 119879(119906)lowast

= 119879 (119906) 119879 (119906lowast

)

= (1 + 119894119905119879 (119886) minus 1

2

1199052

119879 (1198862



2

1199052

119879 (1198862


(76)






lowast


119888(1 + 120572119909)2

= 119888(119887 + 120572119910)

2

= 119888 (119887119887lowast

+ |120572|2

119910119910lowast

) (77)

Hence

lim|120572|rarr0

119888 (119887119887lowast

+ |120572|2

119910119910lowast

) = lim|120572|rarr0

119888(1 + 120572119909)2

= 1 = 119888 (119887119887lowast

)

(78)


119888(1 + 120572119909)2

= 119888 (119887119887lowast

+ |120572| 119910119910lowast

) ge 1 minus |120572|210038171003817100381710038171199101003817100381710038171003817

2

(79)

and therefore

119888 (1 + 119894119905119909) ge (1 minus 119905210038171003817100381710038171199101003817100381710038171003817

2

)

12

119888 (1 minus 119905119909) ge (1 minus 11990521003817100381710038171003817119910

1003817100381710038171003817

2

)

12

(80)




minus1


119898(119878 (119909)) = 119898 (119887minus1

119879 (119909)) le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119898 (119879 (119909))


(81)


minus1


119898(1 + 119894119905119910) le 119888 (1 + 119894119905119878minus1

(119910)) (82)


119860dagger and 119906 isin 119860


(119906119909) (119909dagger

119906minus1

) (119906119909) = 119906119909

(119909dagger

119906minus1

) (119906119909) (119909dagger

119906minus1

) = 119909dagger

119906minus1

(83)


1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le 119888 (119906119909)

le

10038171003817100381710038171003817119909dagger

119906minus1

119906119909

10038171003817100381710038171003817

10038171003817100381710038171003817119906119909119909dagger

119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le

119906

10038171003817100381710038171003817119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

(84)


119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909)) ge

1

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

ge

1

119887

10038171003817100381710038171003817119879(119909)dagger10038171003817100381710038171003817

= 119887minus1

119888 (119879 (119909))

119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909))

le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119879 (1)

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

le 119887 |

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2

119888 (119879 (119909))

(85)



119887minus1

119888 (119909) le 119888 (119878 (119909)) le 119887

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2


(86)


120597120590 (119878 (119909)) sub 120590119870(119878 (119909)) sube 120590 (119909) (87)


120597120590 (119878 (119909) + 120583119878 (119910)) sub 120590119870(119878 (119909) + 120583119878 (119910))

sub 120590 (119909 + 120583119910) sub R cup 120583R(88)



120590 (119878 (119909) + 120583119878 (119910)) sub R cup 120583R (89)


minus1

)119887minus1


10038171003817100381710038171003817119909minus110038171003817100381710038171003817 (90)


)

10038171003817100381710038171003817=

10038171003817100381710038171003817119909minus110038171003817100381710038171003817=

10038171003817100381710038171003817119878 (119909minus1

) 119887minus110038171003817100381710038171003817 (91)

or equivalently1003817100381710038171003817119910

1003817100381710038171003817=

10038171003817100381710038171003817119910119887minus110038171003817100381710038171003817 (92)



119888 (a) = 10038171003817100381710038171003817aminus11003817100381710038171003817

1003817

minus1

= limU

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817

minus1

= limU

119888 (119886119899) (93)





sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 119879 (1) minus 1 lt 120575 (94)

imply that

jmult (119879) lt 120576 sa (119879) lt 120576 (95)


[119887119899] where 119887


N and 119887dagger



119899]





(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119879

119899(1) minus 1 lt

1119899(iii) jmult(119879

119899) ge 1205760or sa(119879

119899) ge 1205760


Urarr 119861



1003817100381710038171003817119879119899(1) minus 1

1003817100381710038171003817le lim

U

1

119899

= 0 (96)




minus1


We know that

119888 (119886119899) minus

1

119899


Hence 119879119899(119886119899) isin 119860




|119888 (T (a)) minus 119888 (a)| = limU

1003816100381610038161003816119888 (119879119899(119886119899)) minus 119888 (119886

119899)

1003816100381610038161003816le lim

U

1

119899

= 0

(98)



119884sube 119879(119861

119883) whereas






condition

sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 (99)

implies that




0 1205760



(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119902(119879

119899) ge 119870

minus1

0


119879119899) ge 1205760or sa(119879

119899(1)lowast

119879119899) ge 1205760


119899] preserves


119902 (T) = limU119902 (119879119899) ge 119870

minus1

0gt 0 (101)




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












we get

119879 (119906) minus 120582minus1

119879(119906)3

= (119879 (119906minus1


)

119866

)

119866

= 119879(119906minus1


)

minus1

)

119866

= 119879((119906minus1


)

minus1

)

minus1

)

= 119879 (119906 minus 120582minus1

1199063

) = 119879 (119906) minus 120582minus1

119879 (1199063

)

(38)

Hence 119879(1199063



3119879 (119909) = 119879(1)2119879 (119909) + 119879 (119909) 119879(1)2 + 119879 (1) 119879 (119909) 119879 (1)

3119879 (1199092

) = 119879(119909)2

119879 (1) + 119879 (1) 119879(119909)2 + 119879 (119909) 119879 (1) 119879 (119909) (39)





sup119886=1119886isin119860

119866

10038171003817100381710038171003817119879 (119886119866

) minus 119879(119886)11986610038171003817100381710038171003817lt 120575 (40)

implies that

jmult (119879 (1) 119879) lt 120576 ucomm (119879) lt 120576 (41)


119899] where 119887


N and 119887119866


and b119866 = [119887119866



Urarr






119879 (119886) 119879 (119886minus1

) = 119879(1)2




119899 119860 rarr 119861 satisfying

that 119879119899 119879119899(1)minus1


10038171003817100381710038171003817119879119899(119886) 119879119899(119886minus1

) minus 119879119899(1)210038171003817100381710038171003817lt

1

119899








minus1



10038171003817100381710038171003817119879 (119886) 119879 (119886

minus1

) minus 119879(1)210038171003817100381710038171003817lt 120575 (44)

implies that






119909 119910 119911 = sum

1205722

=1

sum

1205734

=1

120572120573(119909 + 120572119910 + 120573119911)

[3]

for119909 119910 119911 isin 119864

(46)



119864 = 1198642(119890) oplus 119864

1(119890) oplus 119864

0(119890) (47)



119909 119890 119910 and involution 119909♯






2(119890) If 119886 is



+(119887minus1

minus119886)minus1


(119886minus1

+ (119887minus1

minus 119886)

minus1

)

minus1

= 119886 minus 119880119886(119887) (48)


2

∘ 119909 (see [33] (11))




119860)2(119890) =

119890119890lowast

119860119890lowast


|120582| lt 119906and


lowast

119860119890lowast


and and 119906and

minus (119906 minus

120582119906and

)and in 119890119890

lowast

119860119890lowast



119906[3]

= (119906and

minus (119906 minus 120582119906and

)

and

)

and

(49)





119879 (119906) minus 120582minus1

119879(119906)[3]

= (119879(119906)and

minus (119879 (119906) minus 120582119879(119906)and

)

and

)

and

= 119879 (119906) minus 120582minus1

119879 (1199063

)

(50)




3


we obtain

119879 (1198863

) + 3120572119879 (1198862

) + 31205722

119879 (119886) + 1205723

119879 (1)

= 119879(119886)[3]

+ 1205723

119879 (1) + 2120572 119879 (119886) 119879 (119886) 119879 (1)

+ 120572 119879 (119886) 119879 (1) 119879 (119886) + 21205722

119879 (1) 119879 (1) 119879 (119886)

+ 1205722

119879 (1) 119879 (119886) 119879 (1)

(51)






tmult (119879) = sup 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119886 = 119887 = 119888 = 1

sa (119879) = sup119886=1

10038171003817100381710038171003817119879(119886lowast

)

lowast

minus 119879 (119886)

10038171003817100381710038171003817

(52)



119879119869(119886 119887) = 119879 (119886 ∘ 119887) minus 119879 (119886) ∘ 119879 (119887)

119879119879(119886 119887 119888) = 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119879lowast(119886) = 119879(119886

lowast

)

lowast

minus 119879 (119886)

(53)


119879 (119886 119887 119888)

= 119879 ((119886 ∘ 119887lowast

) ∘ 119888) + 119879 (119886 ∘ (119887lowast

∘ 119888)) minus 119879 ((119886 ∘ 119888) ∘ 119887lowast

)

= 119879 (119886 ∘ 119887lowast

) ∘ 119879 (119888) + 119879119869(119886 ∘ 119887lowast

119888)

+ 119879 (119886) ∘ 119879 (119887lowast

∘ 119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879 (119886 ∘ 119888) ∘ 119879 (119887lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

= (119879 (119886) ∘ 119879 (119887lowast

)) ∘ 119879 (119888) + 119879119869(119886 119887lowast

) ∘ 119879 (119888)

+ 119879119869(119886 ∘ 119887lowast

119888) + 119879 (119886) ∘ (119879 (119887lowast

) ∘ 119879 (119888))

+ 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus (119879 (119886) ∘ 119879 (119888)) ∘ 119879 (119887lowast

) minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

)

minus 119879119869(119886 ∘ 119888 119887

lowast

)

= 119879 (119886) 119879(119887lowast

)

lowast

119879 (119888) + 119879119869(119886 119887lowast

) ∘ 119879 (119888)

+ 119879119869(119886 ∘ 119887lowast

119888) + 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

(54)

Therefore

119879119879(119886 119887 119888)

= 119879 (119886) 119879lowast(119887) 119879 (119888) + 119879

119869(119886 119887lowast

) ∘ 119879 (119888) + 119879119869(119886 ∘ 119887lowast

119888)

+ 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

(55)

This implies that

tmult (119879) le 1198792sa (119879) + 3 (119879 + 1) jmult (119879) (56)



limU

tmult (119879119899) = tmult (T)

limU

sa (119879119899) = sa (T)

(57)



119899] 119860

Urarr 119861




sup119886=1119886=119886

lowast

10038171003817100381710038171003817119879 (119886and

) minus 119879(119886)and10038171003817100381710038171003817lt 120575 (58)

implies that

tmult (119879) lt 120576 (59)


lowast


and sa(119879(1)lowast119879) lt 120576


sup119886=1119886isin119860

dagger

10038171003817100381710038171003817119879 (119886dagger


imply that

tmult (119879) lt 120576 (61)





1003817100381710038171003817119879119899

1003817100381710038171003817lt 1198700 sup

119886=1

10038171003817100381710038171003817119879119899(119886dagger

) minus 119879119899(119886)dagger10038171003817100381710038171003817lt

1

119899

sa (119879119899) lt

1

119899

tmult (119879119899) ge 1205760

(62)

Then T = [119879119899] 119860

Urarr 119861







119898(119886) = inf 119886119909 119909 isin 119860 119909 = 1

119902 (119886) = inf 119909119886 119909 isin 119860 119909 = 1

(63)


119898(119886) = 119902 (119886) =

10038171003817100381710038171003817119886minus110038171003817100381710038171003817

minus1

(64)

Moreover

119898(119886)119898 (119887) le 119898 (119886119887) le 119886119898 (119887)

|119898 (119886) minus 119898 (119887)| le 119886 minus 119887

(65)




119888 (119886) =

inf 119886119909 dist (119909 ker (119871119886)) ge 1 if 119886 = 0

infin if 119886 = 0

(66)


119888 (119886) =

10038171003817100381710038171003817119886dagger10038171003817100381710038171003817

minus1

(67)







1 = 119888 (119879 (1)) = 119888 (119879 (1 + 1205721198860)) = 119888 (1 + 120572119886

0) (68)


1 le 119888 (1 + 1198941199051198860) 1 le 119888 (1 minus 119905119886

0) (69)


0are




1 + 119900 (119905) le 119888 (1 + 119894119905119886) (as 119905 rarr 0) (70)


1 + 119900 (119905) le 119888 (1 + 119894119905119886) = 119888 (1 + 119894119905119879 (119886)) (as 119905 rarr 0) (71)



120582cap 120590(119909) = 0 If 120582 notin 120590(119909)

then 120582 notin 120590119870(119909) where 120590



120590119870(119886) = 120582 isin C lim

120583rarr120582

119888 (119886 minus 120583) = 0 (72)



lim120583rarr120582

119888 (119879 (119909) minus 120583) gt 0 (73)




120597120590 (119879 (119909)) sube 120590119870(119879 (119909)) sube 120590 (119909) (74)




120597120590 (119879 (119906)) ⫋ T 119879 (119906) le 1

10038171003817100381710038171003817119879(119906)minus110038171003817100381710038171003817= 119888(119879 (119906))

minus1

= 119888(119906)minus1

= 1

(75)


1 = 119879 (119906) 119879(119906)lowast

= 119879 (119906) 119879 (119906lowast

)

= (1 + 119894119905119879 (119886) minus 1

2

1199052

119879 (1198862



2

1199052

119879 (1198862


(76)






lowast


119888(1 + 120572119909)2

= 119888(119887 + 120572119910)

2

= 119888 (119887119887lowast

+ |120572|2

119910119910lowast

) (77)

Hence

lim|120572|rarr0

119888 (119887119887lowast

+ |120572|2

119910119910lowast

) = lim|120572|rarr0

119888(1 + 120572119909)2

= 1 = 119888 (119887119887lowast

)

(78)


119888(1 + 120572119909)2

= 119888 (119887119887lowast

+ |120572| 119910119910lowast

) ge 1 minus |120572|210038171003817100381710038171199101003817100381710038171003817

2

(79)

and therefore

119888 (1 + 119894119905119909) ge (1 minus 119905210038171003817100381710038171199101003817100381710038171003817

2

)

12

119888 (1 minus 119905119909) ge (1 minus 11990521003817100381710038171003817119910

1003817100381710038171003817

2

)

12

(80)




minus1


119898(119878 (119909)) = 119898 (119887minus1

119879 (119909)) le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119898 (119879 (119909))


(81)


minus1


119898(1 + 119894119905119910) le 119888 (1 + 119894119905119878minus1

(119910)) (82)


119860dagger and 119906 isin 119860


(119906119909) (119909dagger

119906minus1

) (119906119909) = 119906119909

(119909dagger

119906minus1

) (119906119909) (119909dagger

119906minus1

) = 119909dagger

119906minus1

(83)


1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le 119888 (119906119909)

le

10038171003817100381710038171003817119909dagger

119906minus1

119906119909

10038171003817100381710038171003817

10038171003817100381710038171003817119906119909119909dagger

119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le

119906

10038171003817100381710038171003817119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

(84)


119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909)) ge

1

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

ge

1

119887

10038171003817100381710038171003817119879(119909)dagger10038171003817100381710038171003817

= 119887minus1

119888 (119879 (119909))

119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909))

le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119879 (1)

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

le 119887 |

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2

119888 (119879 (119909))

(85)



119887minus1

119888 (119909) le 119888 (119878 (119909)) le 119887

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2


(86)


120597120590 (119878 (119909)) sub 120590119870(119878 (119909)) sube 120590 (119909) (87)


120597120590 (119878 (119909) + 120583119878 (119910)) sub 120590119870(119878 (119909) + 120583119878 (119910))

sub 120590 (119909 + 120583119910) sub R cup 120583R(88)



120590 (119878 (119909) + 120583119878 (119910)) sub R cup 120583R (89)


minus1

)119887minus1


10038171003817100381710038171003817119909minus110038171003817100381710038171003817 (90)


)

10038171003817100381710038171003817=

10038171003817100381710038171003817119909minus110038171003817100381710038171003817=

10038171003817100381710038171003817119878 (119909minus1

) 119887minus110038171003817100381710038171003817 (91)

or equivalently1003817100381710038171003817119910

1003817100381710038171003817=

10038171003817100381710038171003817119910119887minus110038171003817100381710038171003817 (92)



119888 (a) = 10038171003817100381710038171003817aminus11003817100381710038171003817

1003817

minus1

= limU

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817

minus1

= limU

119888 (119886119899) (93)





sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 119879 (1) minus 1 lt 120575 (94)

imply that

jmult (119879) lt 120576 sa (119879) lt 120576 (95)


[119887119899] where 119887


N and 119887dagger



119899]





(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119879

119899(1) minus 1 lt

1119899(iii) jmult(119879

119899) ge 1205760or sa(119879

119899) ge 1205760


Urarr 119861



1003817100381710038171003817119879119899(1) minus 1

1003817100381710038171003817le lim

U

1

119899

= 0 (96)




minus1


We know that

119888 (119886119899) minus

1

119899


Hence 119879119899(119886119899) isin 119860




|119888 (T (a)) minus 119888 (a)| = limU

1003816100381610038161003816119888 (119879119899(119886119899)) minus 119888 (119886

119899)

1003816100381610038161003816le lim

U

1

119899

= 0

(98)



119884sube 119879(119861

119883) whereas






condition

sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 (99)

implies that




0 1205760



(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119902(119879

119899) ge 119870

minus1

0


119879119899) ge 1205760or sa(119879

119899(1)lowast

119879119899) ge 1205760


119899] preserves


119902 (T) = limU119902 (119879119899) ge 119870

minus1

0gt 0 (101)




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119860)2(119890) =

119890119890lowast

119860119890lowast


|120582| lt 119906and


lowast

119860119890lowast


and and 119906and

minus (119906 minus

120582119906and

)and in 119890119890

lowast

119860119890lowast



119906[3]

= (119906and

minus (119906 minus 120582119906and

)

and

)

and

(49)





119879 (119906) minus 120582minus1

119879(119906)[3]

= (119879(119906)and

minus (119879 (119906) minus 120582119879(119906)and

)

and

)

and

= 119879 (119906) minus 120582minus1

119879 (1199063

)

(50)




3


we obtain

119879 (1198863

) + 3120572119879 (1198862

) + 31205722

119879 (119886) + 1205723

119879 (1)

= 119879(119886)[3]

+ 1205723

119879 (1) + 2120572 119879 (119886) 119879 (119886) 119879 (1)

+ 120572 119879 (119886) 119879 (1) 119879 (119886) + 21205722

119879 (1) 119879 (1) 119879 (119886)

+ 1205722

119879 (1) 119879 (119886) 119879 (1)

(51)






tmult (119879) = sup 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119886 = 119887 = 119888 = 1

sa (119879) = sup119886=1

10038171003817100381710038171003817119879(119886lowast

)

lowast

minus 119879 (119886)

10038171003817100381710038171003817

(52)



119879119869(119886 119887) = 119879 (119886 ∘ 119887) minus 119879 (119886) ∘ 119879 (119887)

119879119879(119886 119887 119888) = 119879 (119886 119887 119888) minus 119879 (119886) 119879 (119887) 119879 (119888)

119879lowast(119886) = 119879(119886

lowast

)

lowast

minus 119879 (119886)

(53)


119879 (119886 119887 119888)

= 119879 ((119886 ∘ 119887lowast

) ∘ 119888) + 119879 (119886 ∘ (119887lowast

∘ 119888)) minus 119879 ((119886 ∘ 119888) ∘ 119887lowast

)

= 119879 (119886 ∘ 119887lowast

) ∘ 119879 (119888) + 119879119869(119886 ∘ 119887lowast

119888)

+ 119879 (119886) ∘ 119879 (119887lowast

∘ 119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879 (119886 ∘ 119888) ∘ 119879 (119887lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

= (119879 (119886) ∘ 119879 (119887lowast

)) ∘ 119879 (119888) + 119879119869(119886 119887lowast

) ∘ 119879 (119888)

+ 119879119869(119886 ∘ 119887lowast

119888) + 119879 (119886) ∘ (119879 (119887lowast

) ∘ 119879 (119888))

+ 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus (119879 (119886) ∘ 119879 (119888)) ∘ 119879 (119887lowast

) minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

)

minus 119879119869(119886 ∘ 119888 119887

lowast

)

= 119879 (119886) 119879(119887lowast

)

lowast

119879 (119888) + 119879119869(119886 119887lowast

) ∘ 119879 (119888)

+ 119879119869(119886 ∘ 119887lowast

119888) + 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

(54)

Therefore

119879119879(119886 119887 119888)

= 119879 (119886) 119879lowast(119887) 119879 (119888) + 119879

119869(119886 119887lowast

) ∘ 119879 (119888) + 119879119869(119886 ∘ 119887lowast

119888)

+ 119879 (119886) ∘ 119879119869(119887lowast

119888) + 119879119869(119886 119887lowast

∘ 119888)

minus 119879119869(119886 119888) ∘ 119879 (119887

lowast

) minus 119879119869(119886 ∘ 119888 119887

lowast

)

(55)

This implies that

tmult (119879) le 1198792sa (119879) + 3 (119879 + 1) jmult (119879) (56)



limU

tmult (119879119899) = tmult (T)

limU

sa (119879119899) = sa (T)

(57)



119899] 119860

Urarr 119861




sup119886=1119886=119886

lowast

10038171003817100381710038171003817119879 (119886and

) minus 119879(119886)and10038171003817100381710038171003817lt 120575 (58)

implies that

tmult (119879) lt 120576 (59)


lowast


and sa(119879(1)lowast119879) lt 120576


sup119886=1119886isin119860

dagger

10038171003817100381710038171003817119879 (119886dagger


imply that

tmult (119879) lt 120576 (61)





1003817100381710038171003817119879119899

1003817100381710038171003817lt 1198700 sup

119886=1

10038171003817100381710038171003817119879119899(119886dagger

) minus 119879119899(119886)dagger10038171003817100381710038171003817lt

1

119899

sa (119879119899) lt

1

119899

tmult (119879119899) ge 1205760

(62)

Then T = [119879119899] 119860

Urarr 119861







119898(119886) = inf 119886119909 119909 isin 119860 119909 = 1

119902 (119886) = inf 119909119886 119909 isin 119860 119909 = 1

(63)


119898(119886) = 119902 (119886) =

10038171003817100381710038171003817119886minus110038171003817100381710038171003817

minus1

(64)

Moreover

119898(119886)119898 (119887) le 119898 (119886119887) le 119886119898 (119887)

|119898 (119886) minus 119898 (119887)| le 119886 minus 119887

(65)




119888 (119886) =

inf 119886119909 dist (119909 ker (119871119886)) ge 1 if 119886 = 0

infin if 119886 = 0

(66)


119888 (119886) =

10038171003817100381710038171003817119886dagger10038171003817100381710038171003817

minus1

(67)







1 = 119888 (119879 (1)) = 119888 (119879 (1 + 1205721198860)) = 119888 (1 + 120572119886

0) (68)


1 le 119888 (1 + 1198941199051198860) 1 le 119888 (1 minus 119905119886

0) (69)


0are




1 + 119900 (119905) le 119888 (1 + 119894119905119886) (as 119905 rarr 0) (70)


1 + 119900 (119905) le 119888 (1 + 119894119905119886) = 119888 (1 + 119894119905119879 (119886)) (as 119905 rarr 0) (71)



120582cap 120590(119909) = 0 If 120582 notin 120590(119909)

then 120582 notin 120590119870(119909) where 120590



120590119870(119886) = 120582 isin C lim

120583rarr120582

119888 (119886 minus 120583) = 0 (72)



lim120583rarr120582

119888 (119879 (119909) minus 120583) gt 0 (73)




120597120590 (119879 (119909)) sube 120590119870(119879 (119909)) sube 120590 (119909) (74)




120597120590 (119879 (119906)) ⫋ T 119879 (119906) le 1

10038171003817100381710038171003817119879(119906)minus110038171003817100381710038171003817= 119888(119879 (119906))

minus1

= 119888(119906)minus1

= 1

(75)


1 = 119879 (119906) 119879(119906)lowast

= 119879 (119906) 119879 (119906lowast

)

= (1 + 119894119905119879 (119886) minus 1

2

1199052

119879 (1198862



2

1199052

119879 (1198862


(76)






lowast


119888(1 + 120572119909)2

= 119888(119887 + 120572119910)

2

= 119888 (119887119887lowast

+ |120572|2

119910119910lowast

) (77)

Hence

lim|120572|rarr0

119888 (119887119887lowast

+ |120572|2

119910119910lowast

) = lim|120572|rarr0

119888(1 + 120572119909)2

= 1 = 119888 (119887119887lowast

)

(78)


119888(1 + 120572119909)2

= 119888 (119887119887lowast

+ |120572| 119910119910lowast

) ge 1 minus |120572|210038171003817100381710038171199101003817100381710038171003817

2

(79)

and therefore

119888 (1 + 119894119905119909) ge (1 minus 119905210038171003817100381710038171199101003817100381710038171003817

2

)

12

119888 (1 minus 119905119909) ge (1 minus 11990521003817100381710038171003817119910

1003817100381710038171003817

2

)

12

(80)




minus1


119898(119878 (119909)) = 119898 (119887minus1

119879 (119909)) le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119898 (119879 (119909))


(81)


minus1


119898(1 + 119894119905119910) le 119888 (1 + 119894119905119878minus1

(119910)) (82)


119860dagger and 119906 isin 119860


(119906119909) (119909dagger

119906minus1

) (119906119909) = 119906119909

(119909dagger

119906minus1

) (119906119909) (119909dagger

119906minus1

) = 119909dagger

119906minus1

(83)


1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le 119888 (119906119909)

le

10038171003817100381710038171003817119909dagger

119906minus1

119906119909

10038171003817100381710038171003817

10038171003817100381710038171003817119906119909119909dagger

119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le

119906

10038171003817100381710038171003817119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

(84)


119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909)) ge

1

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

ge

1

119887

10038171003817100381710038171003817119879(119909)dagger10038171003817100381710038171003817

= 119887minus1

119888 (119879 (119909))

119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909))

le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119879 (1)

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

le 119887 |

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2

119888 (119879 (119909))

(85)



119887minus1

119888 (119909) le 119888 (119878 (119909)) le 119887

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2


(86)


120597120590 (119878 (119909)) sub 120590119870(119878 (119909)) sube 120590 (119909) (87)


120597120590 (119878 (119909) + 120583119878 (119910)) sub 120590119870(119878 (119909) + 120583119878 (119910))

sub 120590 (119909 + 120583119910) sub R cup 120583R(88)



120590 (119878 (119909) + 120583119878 (119910)) sub R cup 120583R (89)


minus1

)119887minus1


10038171003817100381710038171003817119909minus110038171003817100381710038171003817 (90)


)

10038171003817100381710038171003817=

10038171003817100381710038171003817119909minus110038171003817100381710038171003817=

10038171003817100381710038171003817119878 (119909minus1

) 119887minus110038171003817100381710038171003817 (91)

or equivalently1003817100381710038171003817119910

1003817100381710038171003817=

10038171003817100381710038171003817119910119887minus110038171003817100381710038171003817 (92)



119888 (a) = 10038171003817100381710038171003817aminus11003817100381710038171003817

1003817

minus1

= limU

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817

minus1

= limU

119888 (119886119899) (93)





sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 119879 (1) minus 1 lt 120575 (94)

imply that

jmult (119879) lt 120576 sa (119879) lt 120576 (95)


[119887119899] where 119887


N and 119887dagger



119899]





(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119879

119899(1) minus 1 lt

1119899(iii) jmult(119879

119899) ge 1205760or sa(119879

119899) ge 1205760


Urarr 119861



1003817100381710038171003817119879119899(1) minus 1

1003817100381710038171003817le lim

U

1

119899

= 0 (96)




minus1


We know that

119888 (119886119899) minus

1

119899


Hence 119879119899(119886119899) isin 119860




|119888 (T (a)) minus 119888 (a)| = limU

1003816100381610038161003816119888 (119879119899(119886119899)) minus 119888 (119886

119899)

1003816100381610038161003816le lim

U

1

119899

= 0

(98)



119884sube 119879(119861

119883) whereas






condition

sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 (99)

implies that




0 1205760



(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119902(119879

119899) ge 119870

minus1

0


119879119899) ge 1205760or sa(119879

119899(1)lowast

119879119899) ge 1205760


119899] preserves


119902 (T) = limU119902 (119879119899) ge 119870

minus1

0gt 0 (101)




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899] 119860

Urarr 119861




sup119886=1119886=119886

lowast

10038171003817100381710038171003817119879 (119886and

) minus 119879(119886)and10038171003817100381710038171003817lt 120575 (58)

implies that

tmult (119879) lt 120576 (59)


lowast


and sa(119879(1)lowast119879) lt 120576


sup119886=1119886isin119860

dagger

10038171003817100381710038171003817119879 (119886dagger


imply that

tmult (119879) lt 120576 (61)





1003817100381710038171003817119879119899

1003817100381710038171003817lt 1198700 sup

119886=1

10038171003817100381710038171003817119879119899(119886dagger

) minus 119879119899(119886)dagger10038171003817100381710038171003817lt

1

119899

sa (119879119899) lt

1

119899

tmult (119879119899) ge 1205760

(62)

Then T = [119879119899] 119860

Urarr 119861







119898(119886) = inf 119886119909 119909 isin 119860 119909 = 1

119902 (119886) = inf 119909119886 119909 isin 119860 119909 = 1

(63)


119898(119886) = 119902 (119886) =

10038171003817100381710038171003817119886minus110038171003817100381710038171003817

minus1

(64)

Moreover

119898(119886)119898 (119887) le 119898 (119886119887) le 119886119898 (119887)

|119898 (119886) minus 119898 (119887)| le 119886 minus 119887

(65)




119888 (119886) =

inf 119886119909 dist (119909 ker (119871119886)) ge 1 if 119886 = 0

infin if 119886 = 0

(66)


119888 (119886) =

10038171003817100381710038171003817119886dagger10038171003817100381710038171003817

minus1

(67)







1 = 119888 (119879 (1)) = 119888 (119879 (1 + 1205721198860)) = 119888 (1 + 120572119886

0) (68)


1 le 119888 (1 + 1198941199051198860) 1 le 119888 (1 minus 119905119886

0) (69)


0are




1 + 119900 (119905) le 119888 (1 + 119894119905119886) (as 119905 rarr 0) (70)


1 + 119900 (119905) le 119888 (1 + 119894119905119886) = 119888 (1 + 119894119905119879 (119886)) (as 119905 rarr 0) (71)



120582cap 120590(119909) = 0 If 120582 notin 120590(119909)

then 120582 notin 120590119870(119909) where 120590



120590119870(119886) = 120582 isin C lim

120583rarr120582

119888 (119886 minus 120583) = 0 (72)



lim120583rarr120582

119888 (119879 (119909) minus 120583) gt 0 (73)




120597120590 (119879 (119909)) sube 120590119870(119879 (119909)) sube 120590 (119909) (74)




120597120590 (119879 (119906)) ⫋ T 119879 (119906) le 1

10038171003817100381710038171003817119879(119906)minus110038171003817100381710038171003817= 119888(119879 (119906))

minus1

= 119888(119906)minus1

= 1

(75)


1 = 119879 (119906) 119879(119906)lowast

= 119879 (119906) 119879 (119906lowast

)

= (1 + 119894119905119879 (119886) minus 1

2

1199052

119879 (1198862



2

1199052

119879 (1198862


(76)






lowast


119888(1 + 120572119909)2

= 119888(119887 + 120572119910)

2

= 119888 (119887119887lowast

+ |120572|2

119910119910lowast

) (77)

Hence

lim|120572|rarr0

119888 (119887119887lowast

+ |120572|2

119910119910lowast

) = lim|120572|rarr0

119888(1 + 120572119909)2

= 1 = 119888 (119887119887lowast

)

(78)


119888(1 + 120572119909)2

= 119888 (119887119887lowast

+ |120572| 119910119910lowast

) ge 1 minus |120572|210038171003817100381710038171199101003817100381710038171003817

2

(79)

and therefore

119888 (1 + 119894119905119909) ge (1 minus 119905210038171003817100381710038171199101003817100381710038171003817

2

)

12

119888 (1 minus 119905119909) ge (1 minus 11990521003817100381710038171003817119910

1003817100381710038171003817

2

)

12

(80)




minus1


119898(119878 (119909)) = 119898 (119887minus1

119879 (119909)) le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119898 (119879 (119909))


(81)


minus1


119898(1 + 119894119905119910) le 119888 (1 + 119894119905119878minus1

(119910)) (82)


119860dagger and 119906 isin 119860


(119906119909) (119909dagger

119906minus1

) (119906119909) = 119906119909

(119909dagger

119906minus1

) (119906119909) (119909dagger

119906minus1

) = 119909dagger

119906minus1

(83)


1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le 119888 (119906119909)

le

10038171003817100381710038171003817119909dagger

119906minus1

119906119909

10038171003817100381710038171003817

10038171003817100381710038171003817119906119909119909dagger

119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le

119906

10038171003817100381710038171003817119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

(84)


119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909)) ge

1

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

ge

1

119887

10038171003817100381710038171003817119879(119909)dagger10038171003817100381710038171003817

= 119887minus1

119888 (119879 (119909))

119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909))

le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119879 (1)

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

le 119887 |

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2

119888 (119879 (119909))

(85)



119887minus1

119888 (119909) le 119888 (119878 (119909)) le 119887

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2


(86)


120597120590 (119878 (119909)) sub 120590119870(119878 (119909)) sube 120590 (119909) (87)


120597120590 (119878 (119909) + 120583119878 (119910)) sub 120590119870(119878 (119909) + 120583119878 (119910))

sub 120590 (119909 + 120583119910) sub R cup 120583R(88)



120590 (119878 (119909) + 120583119878 (119910)) sub R cup 120583R (89)


minus1

)119887minus1


10038171003817100381710038171003817119909minus110038171003817100381710038171003817 (90)


)

10038171003817100381710038171003817=

10038171003817100381710038171003817119909minus110038171003817100381710038171003817=

10038171003817100381710038171003817119878 (119909minus1

) 119887minus110038171003817100381710038171003817 (91)

or equivalently1003817100381710038171003817119910

1003817100381710038171003817=

10038171003817100381710038171003817119910119887minus110038171003817100381710038171003817 (92)



119888 (a) = 10038171003817100381710038171003817aminus11003817100381710038171003817

1003817

minus1

= limU

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817

minus1

= limU

119888 (119886119899) (93)





sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 119879 (1) minus 1 lt 120575 (94)

imply that

jmult (119879) lt 120576 sa (119879) lt 120576 (95)


[119887119899] where 119887


N and 119887dagger



119899]





(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119879

119899(1) minus 1 lt

1119899(iii) jmult(119879

119899) ge 1205760or sa(119879

119899) ge 1205760


Urarr 119861



1003817100381710038171003817119879119899(1) minus 1

1003817100381710038171003817le lim

U

1

119899

= 0 (96)




minus1


We know that

119888 (119886119899) minus

1

119899


Hence 119879119899(119886119899) isin 119860




|119888 (T (a)) minus 119888 (a)| = limU

1003816100381610038161003816119888 (119879119899(119886119899)) minus 119888 (119886

119899)

1003816100381610038161003816le lim

U

1

119899

= 0

(98)



119884sube 119879(119861

119883) whereas






condition

sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 (99)

implies that




0 1205760



(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119902(119879

119899) ge 119870

minus1

0


119879119899) ge 1205760or sa(119879

119899(1)lowast

119879119899) ge 1205760


119899] preserves


119902 (T) = limU119902 (119879119899) ge 119870

minus1

0gt 0 (101)




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1 + 119900 (119905) le 119888 (1 + 119894119905119886) (as 119905 rarr 0) (70)


1 + 119900 (119905) le 119888 (1 + 119894119905119886) = 119888 (1 + 119894119905119879 (119886)) (as 119905 rarr 0) (71)



120582cap 120590(119909) = 0 If 120582 notin 120590(119909)

then 120582 notin 120590119870(119909) where 120590



120590119870(119886) = 120582 isin C lim

120583rarr120582

119888 (119886 minus 120583) = 0 (72)



lim120583rarr120582

119888 (119879 (119909) minus 120583) gt 0 (73)




120597120590 (119879 (119909)) sube 120590119870(119879 (119909)) sube 120590 (119909) (74)




120597120590 (119879 (119906)) ⫋ T 119879 (119906) le 1

10038171003817100381710038171003817119879(119906)minus110038171003817100381710038171003817= 119888(119879 (119906))

minus1

= 119888(119906)minus1

= 1

(75)


1 = 119879 (119906) 119879(119906)lowast

= 119879 (119906) 119879 (119906lowast

)

= (1 + 119894119905119879 (119886) minus 1

2

1199052

119879 (1198862



2

1199052

119879 (1198862


(76)






lowast


119888(1 + 120572119909)2

= 119888(119887 + 120572119910)

2

= 119888 (119887119887lowast

+ |120572|2

119910119910lowast

) (77)

Hence

lim|120572|rarr0

119888 (119887119887lowast

+ |120572|2

119910119910lowast

) = lim|120572|rarr0

119888(1 + 120572119909)2

= 1 = 119888 (119887119887lowast

)

(78)


119888(1 + 120572119909)2

= 119888 (119887119887lowast

+ |120572| 119910119910lowast

) ge 1 minus |120572|210038171003817100381710038171199101003817100381710038171003817

2

(79)

and therefore

119888 (1 + 119894119905119909) ge (1 minus 119905210038171003817100381710038171199101003817100381710038171003817

2

)

12

119888 (1 minus 119905119909) ge (1 minus 11990521003817100381710038171003817119910

1003817100381710038171003817

2

)

12

(80)




minus1


119898(119878 (119909)) = 119898 (119887minus1

119879 (119909)) le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119898 (119879 (119909))


(81)


minus1


119898(1 + 119894119905119910) le 119888 (1 + 119894119905119878minus1

(119910)) (82)


119860dagger and 119906 isin 119860


(119906119909) (119909dagger

119906minus1

) (119906119909) = 119906119909

(119909dagger

119906minus1

) (119906119909) (119909dagger

119906minus1

) = 119909dagger

119906minus1

(83)


1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le 119888 (119906119909)

le

10038171003817100381710038171003817119909dagger

119906minus1

119906119909

10038171003817100381710038171003817

10038171003817100381710038171003817119906119909119909dagger

119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

le

119906

10038171003817100381710038171003817119906minus110038171003817100381710038171003817

1003817100381710038171003817119909dagger119906minus11003817100381710038171003817

(84)


119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909)) ge

1

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

ge

1

119887

10038171003817100381710038171003817119879(119909)dagger10038171003817100381710038171003817

= 119887minus1

119888 (119879 (119909))

119888 (119878 (119909)) = 119888 (119887minus1

119879 (119909))

le

10038171003817100381710038171003817119887minus110038171003817100381710038171003817119879 (1)

10038171003817100381710038171003817119879(119909)dagger

119887

10038171003817100381710038171003817

le 119887 |

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2

119888 (119879 (119909))

(85)



119887minus1

119888 (119909) le 119888 (119878 (119909)) le 119887

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2


(86)


120597120590 (119878 (119909)) sub 120590119870(119878 (119909)) sube 120590 (119909) (87)


120597120590 (119878 (119909) + 120583119878 (119910)) sub 120590119870(119878 (119909) + 120583119878 (119910))

sub 120590 (119909 + 120583119910) sub R cup 120583R(88)



120590 (119878 (119909) + 120583119878 (119910)) sub R cup 120583R (89)


minus1

)119887minus1


10038171003817100381710038171003817119909minus110038171003817100381710038171003817 (90)


)

10038171003817100381710038171003817=

10038171003817100381710038171003817119909minus110038171003817100381710038171003817=

10038171003817100381710038171003817119878 (119909minus1

) 119887minus110038171003817100381710038171003817 (91)

or equivalently1003817100381710038171003817119910

1003817100381710038171003817=

10038171003817100381710038171003817119910119887minus110038171003817100381710038171003817 (92)



119888 (a) = 10038171003817100381710038171003817aminus11003817100381710038171003817

1003817

minus1

= limU

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817

minus1

= limU

119888 (119886119899) (93)





sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 119879 (1) minus 1 lt 120575 (94)

imply that

jmult (119879) lt 120576 sa (119879) lt 120576 (95)


[119887119899] where 119887


N and 119887dagger



119899]





(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119879

119899(1) minus 1 lt

1119899(iii) jmult(119879

119899) ge 1205760or sa(119879

119899) ge 1205760


Urarr 119861



1003817100381710038171003817119879119899(1) minus 1

1003817100381710038171003817le lim

U

1

119899

= 0 (96)




minus1


We know that

119888 (119886119899) minus

1

119899


Hence 119879119899(119886119899) isin 119860




|119888 (T (a)) minus 119888 (a)| = limU

1003816100381610038161003816119888 (119879119899(119886119899)) minus 119888 (119886

119899)

1003816100381610038161003816le lim

U

1

119899

= 0

(98)



119884sube 119879(119861

119883) whereas






condition

sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 (99)

implies that




0 1205760



(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119902(119879

119899) ge 119870

minus1

0


119879119899) ge 1205760or sa(119879

119899(1)lowast

119879119899) ge 1205760


119899] preserves


119902 (T) = limU119902 (119879119899) ge 119870

minus1

0gt 0 (101)




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119887minus1

119888 (119909) le 119888 (119878 (119909)) le 119887

10038171003817100381710038171003817119887minus110038171003817100381710038171003817

2


(86)


120597120590 (119878 (119909)) sub 120590119870(119878 (119909)) sube 120590 (119909) (87)


120597120590 (119878 (119909) + 120583119878 (119910)) sub 120590119870(119878 (119909) + 120583119878 (119910))

sub 120590 (119909 + 120583119910) sub R cup 120583R(88)



120590 (119878 (119909) + 120583119878 (119910)) sub R cup 120583R (89)


minus1

)119887minus1


10038171003817100381710038171003817119909minus110038171003817100381710038171003817 (90)


)

10038171003817100381710038171003817=

10038171003817100381710038171003817119909minus110038171003817100381710038171003817=

10038171003817100381710038171003817119878 (119909minus1

) 119887minus110038171003817100381710038171003817 (91)

or equivalently1003817100381710038171003817119910

1003817100381710038171003817=

10038171003817100381710038171003817119910119887minus110038171003817100381710038171003817 (92)



119888 (a) = 10038171003817100381710038171003817aminus11003817100381710038171003817

1003817

minus1

= limU

10038171003817100381710038171003817119886minus1

119899

10038171003817100381710038171003817

minus1

= limU

119888 (119886119899) (93)





sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 119879 (1) minus 1 lt 120575 (94)

imply that

jmult (119879) lt 120576 sa (119879) lt 120576 (95)


[119887119899] where 119887


N and 119887dagger



119899]





(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119879

119899(1) minus 1 lt

1119899(iii) jmult(119879

119899) ge 1205760or sa(119879

119899) ge 1205760


Urarr 119861



1003817100381710038171003817119879119899(1) minus 1

1003817100381710038171003817le lim

U

1

119899

= 0 (96)




minus1


We know that

119888 (119886119899) minus

1

119899


Hence 119879119899(119886119899) isin 119860




|119888 (T (a)) minus 119888 (a)| = limU

1003816100381610038161003816119888 (119879119899(119886119899)) minus 119888 (119886

119899)

1003816100381610038161003816le lim

U

1

119899

= 0

(98)



119884sube 119879(119861

119883) whereas






condition

sup119886=1119886isin119860

minus1

|119888 (119879 (119886)) minus 119888 (119886)| lt 120575 (99)

implies that




0 1205760



(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119902(119879

119899) ge 119870

minus1

0


119879119899) ge 1205760or sa(119879

119899(1)lowast

119879119899) ge 1205760


119899] preserves


119902 (T) = limU119902 (119879119899) ge 119870

minus1

0gt 0 (101)




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 1205760



(i) 119879119899 le 119870

0

(ii) sup119886=1119886isin119860

minus1|119888(119879119899(119886)) minus 119888(119886)| lt 1119899 119902(119879

119899) ge 119870

minus1

0


119879119899) ge 1205760or sa(119879

119899(1)lowast

119879119899) ge 1205760


119899] preserves


119902 (T) = limU119902 (119879119899) ge 119870

minus1

0gt 0 (101)




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article Approximate Preservers on Banach Algebras...

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