CORRESPONDENCES BETWEEN von NEUMANN ALGEBRAS AND

DISCRETE AUTOMORPHISM GROUPS

Hisashi Choda

Department of Mathematics

Osaka Kyoiku University

Osaka, Japan.

1. Introduction. F. Murray and J. von Neumann constructed

examples of factors by means of a measure space construction in [32].

In 195B, T. Turumaru [47] defined an abstract discrete crossed product

of von Neumann algebras and pointed out that the measure space const­

ruction of Murray-von Neumann is the crossed product of an abelian von

Neumann algebra by an ergodic group of freely acting automorphisms.

The discrete crossed products of von Neumann algebras have been used

to construct factors and studied for their own interest, from an alge­

braic point of view by M. Nakamura and Z. Takeda [35] and N. Suzuki

[42] (for some additional references, see [20] and [21]).

Recently, the crossed product of a von Neumann algebra by a

locally compact group was defined by M. Takesaki [44]. Many interest­

ing results in the theory of continuous crossed products of von Neumann

algebras have been obtained by several authors (e.g., [16]1 [30], [31],

[33) and [44], for other references, see [40]).

In this talk, we shall restrict our attention to the discrete

crossed product of von Neumann algebras. We consider the relations

between von Neumann algebras and automorphism groups. We shall treat

only separable Hilbert spaces and discrete countable groups.

2. Discrete crossed products. We begin by stating the defi­

nition of a crossed product (following M. Takesaki [441). Let A be

a von Neumann algebra acting on a Hilbert space Jc/-, G a group of auto­

morphisms of A and G®N the Hilbert space of ~-va1ued square summable

functions on G. Then the space G{g)pJ is identified with /Q/® 12 (G)

or ~gsGN~sg' where {Sg}gSG is an orthonormal basis for the space

12(G) of square summable functions on G defined by

by

42

(g=h) (g;ih) (g,heG) .

Define a faithful normal representation n of A on G~~

-1 (n (a) 1;;) (h) = h (a) E; (h) (aEA, heG, I;;EGriJl'i)

and a unitary representation A of G on G©~ by

(A (g) 1;;) (h) = I;; (g -lh) (g,hsG, I;;sG<OFi).

Then the pair {rr,A} of representations rr of A and A of G is a

covariant representation of {A,G};that is,

A(g)rr(a)A(g)* = rr(g(a» (aEA, gEG).

The von Neumann algebra on GIlF/ generated by rr(A) and A (G) is

called the crossed product of A by G and denoted by G~A. For each

subgroup K of G, the von Neumann algebra on GIi:l}~ generated by 1T(A)

and A (K) is denoted by N (K).

Ostensibly, the crossed product G®A depends also the under­

lying Hilbert space. However, the next proposition assures us that

the algebraic structure of the crossed product is independent of the

Hilbert space. The proposition is used to prove the uniqueness of the

crossed product (within an isomorphism) in several papers (e.g., [38]

for Ill-factors, [21] and [44] for general cases).

Proposition 1. Let Al and AZ be von Neumann algebras and

a an isomorphism of Al onto A2 . Let Gl and G2 be groups of

automorphisms of Al and A2 respectively and ~ an isomorphism of

Gl onto G2 such that

cr(g(a» = ~(g) (a(a»

Then there is an isomorphism a of Gl®Al onto Gj9A2 such that

1T2 (cr(a»

where {rri,Ai } is the covariant representation of {Ai,Gi } cited above.

The discrete crossed products of von Neumann algebras are rela­

tively easy to handle, because we have explicit descriptions of elements

in the crossed product as Fourier expansions or matrices. We shall

describe a matrix representation for each element in the crossed pro­

duct.

Then is an isometric

43

linear mapping of )a! onto the subspace I::I~ Sg of G~lq. For each x

in the algebra L(GiI>!4) of all operators on G®;:l, we represent x

by a matrix (Xg,h)g,hsG ' where

x h = J *xJh (g,hsG). g, 9

For each asA, we have

ll(a)g,h

and, for each kEG,

A(k}g,h = 6~h

(g,hsG) ;

(g,heG) •

Modifying Zeller-Meier [49, Proposition 8.4], we have the following

proposition.

Proposition 2. For each subgroup K of G, the von Neumann

algebra N(K) is the set of all elements in L(Gq;I}1) with the matrix

form

xg,h = g-1(x(9h- l »

for some A-valued function x(g) on G such that XK(g)x(g) x(g) ,

where is the characteristic function of K.

An expectation on a von Neumann algebra is a useful tool for

investigating the relation between the von Neumann algebra and its sub­

algebras (cf. [34], [46] and [48]). Let C be a von Neumann algebra

and D a von Neumann subalgebra of C. Then a positive linear mapping

e of C onto D is called an expectation of C if e satisfies

that

e (1) 1

and

e(cd) = e(c)d (csC, dsD).

By using Arveson's expectation [3, 6.1.3), we have the follow­

ing proposition (cf. [9, Proposition 2).

Proposition 3. For each subgroup K of G, there exists a

faithful normal expectation e of G®A onto N(K) such that

e(A(g» = 0 (giK) •

In particular, there exists a faithful normal expectation e

of G®A onto TI(A) such that e(A(g» = 0 for all g t 1.

Conversely, this property characterizes a crossed product [11,

44

Theorem 4] (also, cf. [15, Proposition 4.1.21):

Theorem 4. Let M be a von Neumann algebra, A a von Neumann

subalgebra of M and G a group of automorphisms of A. Assume that

satisfies the following three conditions; (M, A, G)

(1) there exists a unitary representation ug of G

such that g; Ad u (gEG), where Ad u (a) ; u au * (aEA), g g g g

(2) there exists a faithful normal expectation e of

A such that e(u) ; 0 for all g ~ I, g (3) M is generated by A and u G.

in M

M onto

Then M is isomorphic to the crossed product G@A by a mapping ~

such that

~(a)

and

If a group G is freely acting on A (cf. §3), then, for each

gEG, the dependent element e(ug ) of g is O. Therefore, we may

weaken assumption (2) for a freely acting automorphism group of A in

Theorem 4 as follows ([12, Corollary 5]):

(2') there exists a faithful normal expectation of t1 onto

A.

Theorem 4 is fundamental for the sequel. Some Ill-factor versions of

this theorem can be found in [20] and [42]. Some generalizations of

this theorem for discrete crossed products with factor sets are given

in [131 and [41]. A continuous crossed product version is Landstad's

characterization of crossed products by locally compact groups [31,

Theorem 1].

3. Freely acting automorphisms. The free action of automor­

phisms of von Neumann algebras played an important role from the be­

ginning in the theory of von Neumann algebras. The notion of free

action is used, especially, in connection with crossed products. In

this section, we shall consider the free action of automorphisms of

von Neumann algebras.

Let A be a von Neumann algebra and ~ an automorphism of

A. Then an element a of A is called a dependent element of ~

if

ab Ci(b)a (bEA)

45

[14]. Dependent elements have the following properties (cf. [4, §2],

[8, Theorem 1) and [14)):

Proposition 5. Let Ci be an automorphism of a von Neumann

algebra A. Then a dependent element a of Ci has the following

properties;

and

(1) a*a '" aa*

(2) a*a is a central element of A

(3) ala) = a

(4) a* is a dependent element. of -1 a

An automorphism a of a von Nemnann algebra A is said to be

freely acting on A if there are no nonzero dependent elements of

a. In other words, an element aEA having the property ab ~ a{b)a

for all bEA is 0 [27]. Suppose that an automorphism Ci of A is

spatial, that is, Ci = Ad u for some unitafY operator u. Then the

automorphism a is freely acting on A if and only if uA'o A to} [8, Proposition IJ. Hence, the automorphism a is freely acting on

A if and only if the automorphism Ad u of A' is freely acting on

A' ([25, Lemma 91, [26, Lemma l.l]). The following decomposition theo­

rem of automorphisms is well known:

Kallman's Theorem ([27, Theorem 1.11J). Let a be an auto­

morphism of a von Neumann algebra A. Then there exists a unique cen­

tral projection p in A such that

a(p) = p,

a is inner on

and

Ci is freely acting on A(l_p)'

We shall call the projection p in the Kallman's theorem a

central projection inducing the inner part of a and denote it by

p(a). Then, by Proposition 5, we have the following proposition (cf.

[8) :

Proposition 6. Let A be a von Neumann algebra and G a group

of automorphisms of A. If an automorphism a of A satisfies ag~ga

for all g£G, then g(p(a») = p(a) for all g£G.

46

Corollary ([7J, [45]). Let G be an outer automorphism group

of a von Neumann algebra A. If there exists an automorphism group H

of A with the following properties;

(I) gh hg (gsG, hcR)

and

(2) H is ergodic on A~A',

then G is freely acting on A (that is, every g (f 1) in G is

freely acting on A).

In particular, an ergodic and abelian automorphism group of an

abelian von Neumann algebra is freely acting.

The notion of free action of an automorphism group is charac­

terized in terms of crossed products by Y. Raga and Z. Takeda [25] and

Y. Haga [24]:

Proposition 7. Let G be a group of automorphisms of a von

Neumann algebra A. Then the group G is freely acting if and only

if

Gi&lA 1\ 'IT (Al I c 'IT (A) .

4. Full groups. R. A. Dye introduced the notion of full groups

for automorphism groups on abelian von Neumann algebras in [18]. And

Y. Raga and Z. Takeda generalized the notion for automorphism groups

of general von Neumann algebras in [25J.

Let A be a von Neumann algebra

phisms of A. Then we shall denote by -1 phisms a of A having sUPgcG pIa g)

group determined by G. The full group

(cf. [18] and [25J):

Proposition 8.

and G a group of automor­

[G] the set of all automor­

= 1. We call [G] the full

has the following properties

(1) [G] is again a group of automorphisms of A.

(2) [[G]] = [GJ.

(3) The elements a of [GJ are precisely those automorph­

isms of A having a representation

(asAl,

-1 where gnCG, v is a unitary operator in A and {Pn} (resp. {gn (Pn)})

is a family of mutually orthogonal central projections having sum 1.

Suppose that the group G is freely acting on A. For each

aE [GI, let

Pg

Then we have that

-1 g(p(g a))

47

(gEG) •

(aEA)

for some unitary operator v in A. We may assume that each as[G]

is an inner automorphism of G~A induced by a unitary operator

-1 LgEG A(g)TI(p(g a)v)

for some unitary operator v in A. Conversely an automorphism of

this type is in [Gj. Therefore, we have the following theorem [25,

Theorem 11 :

Theorem 9. An automorphism a of A belongs to [GI if and

only if a can be extended to an inner automorphism of G@A, identify­

ing A with TI(A).

We shall denote by [Gl z the set of all automorphisms ex of

A having the following form

a (a) = Ln Pngn (a) (aEAl,

where gnEG, and {Pn} (resp. {g~l(Pn)}) is a family of mutually or­

thogonal central projections having sum 1 (cf. [251). We shall call

[Gl z the Z-full group determined by G.

H. A. Dye introduced equivalent relations among automorphism

groups in [181. Let Al and A2 be von Neumann algebras, and Gl and G2 groups of automorphisms of Al and A2 respectively. Then

two groups Gl and G2 are called equivalent if there exists -1

an isomorphism a of Al onto A2 such that [G1I = [0 G2ol. In

the case where Al = A2 , two groups Gl and G2 are called equiva­

lent if [G1I = [G2]. In terms of crossed products, we shall give a

characterization of the equivalence among groups of freely acting auto­

morphisms of an abelian von Neumann algebra; and we shall give a chara­

cterization of weak eqtiFalence (cf. [51 and [6]).

Theorem 10. Let A be an abelian von Neumann algebra, and

Gl and G2 be groups of freely acting automorphisms of A. Then a

necessary and sufficient condition that Gl and G2 are equivalent

is that there exists an isomorphism a of Gl~A onto G2®A such

that

TI2 (a) (aEA) .

48

In fact, suppose that Then each

belongs to [GIl. For each g2EG2' let

vg2

= EgSGI

A~g)1fl{P(g~lg».

Then g ..... vg is a unitary representation of G2 in Gl@A and GfA

is generated by 1f l (A) and vG . Therefore, by Theorem 4, we have

the desired isomorphism of Gl® ft. onto Gl::JA. The converse is an im­mediate consequence of Theorem 9.

Theorem 11. Let Al and A2 be abelian von Neumann algebras,

and Gl and G2 be groups of freely acting automorphisms of Al and

A2 respectively. Then a necessary and sufficient condition that Gl and G2 are weakly equivalent is that there exists an isomorphism a

of Gli9Al onto G£lJA2 such that O(1f l (Al )) 1f2 (A2 ).

W. Krieger, in [28], gives, in terms of Krieger factors, a simi­

lar characterization of weak equivalence. In [29], he also gets a

strong version of Theorem 11 for singly generated groups.

5. Correspondences between subgroups and subalgebras in a cro­

ssed product. In [35], M. Nakamura and Z. Takeda showed that there is

a one-to-one correspondence between the class of subgroups of a group

and the class of intermediate subalgebras of the crossed product of a

III-factor by the group. And Y. Haga and Z. Takeda showed in [25] for

finite von Neumann algebras that there is a one-to-one correspondence

between the class of full subgroups and the class of intermediate sub­

algebras of a crossed product, as a generalization of the Dye corre­

spondence [19, Proposition 6.11. The conservation of types in the

Dye-Haga-Takeda correspondence was obtained in [19, Proposition 6.11,

[10, 'l'heorem 241 and [23, Theorem 4.101.

In this section, we shall consider a correspondence between

subgroups and intermediate subalgebras of a crossed product for a gener­

al von Neumann algebra. Detailed accounts of the material we shall

discuss in this section and the next section can be found in [9].

We shall need the following lemma, which is a variant of (15,

Lemma 1.5.6] and [25, Lemma 5].

Lemma. Let A be a von Neumann algebra, B a von Neumann sub­

algebra of A with B'nA (B, C a von Neumann subalgebra of A contain­

ing Band e an expectation of A onto C. If a unitary operator

u in A satisfies the condition uBu* = B, then e(u) has the

49

following properties;

(1) e(u) is a partial isometry,

(2) the initial projection p and the final projection q of e(u) are contained in the center of B, and

(3) e (u) = up = quo

Let

isms of A.

ing 1T (A)

A be a von Neumann algebra and G a group of automorph­

We shall call a von Neumann subalgebra of G®A contain­

an intermediate subalgebra of G®A.

Theorem 12. Let G be a group of freely acting automorphisms

of a von Neumann algebra A. If there exists a group H of automor­

phisms of G®A having the following properties;

(1) h(1T(A» = 1T(A) (hEH),

(2) H is ergodic on 1T(A riA' )

and

(3) h(A(g» = A(g) (gEG, hEH),

then there exists a ono-to~ne correspondence between the class of sub­

groups of G and the class of H-invariant intermediate subalgebras

B having a faithful normal expectation of G®A onto B, obtained by

associating with each subgroup K the intermediate subalgebra N(K)

and, with the intermediate subalgebra B, the subgroup {gEG; A{g)SB}

(denoted by K(B».

In fact, by proposition 3, for each subgroup K of G, N(K)

is an H-invariant intermediate subalgebra of G®A having a faithful

normal expectation of G0A onto N(K); and we have K(N(K» = K.

Conversely, let B be an H-invariant intermediate subalgebra of G®A

having a faithful normal expectation e of G®A onto B. For each

gsG, by the preceding lemma and freeness of action of G, there exists

a central projection Pg in A such that e(A(g» n(pg)A(g). On

the other hand, by the property that B 'n GII;)A C B, we have that

h(e(x» = e(h(x» (xEG®A, hEH)

(cf. [IS, Theorem loS.S]). lIenee, Pg= 0 or 1 for each gsG. Then

e(A(g» = 0 for all A(g)iB. Therefore, we have e(G0A) C N(K(B».

Corollary. Let A be a factor and G a group of outer auto-

rnorphisms of

the class of

B of G®A

A. Then there exists a one-to-one correspondence between

subgroups of G and the class of intermediate subalgebras

having a faithful normal expectation of G®A onto B, by

50

the same association as in Theorem 12.

The Corollary is a generalization of [12, Theorem 5] and [35,

Theorem 2] and also has another generalization, which is proved, in a

manner similar to that of the proof of Theorem 12.

Theorem 13. Let A be a von Neumann algebra and G a group

of ( not necesSarily freely acting) automorphisms of A. Then there

exists a one-to-one correspondence between the class of subgroups of

G and the class of intermediate subalgebras B of G@A having a

faithful normal expectation e of G®A onto B such that e(A(g))=O

for all A(g)iB, by the same association as in Theorem 12.

6. Galois correspondences. In this section, we shall consider

a Galois correspondence. Throughout this section, we shall use a von

Neumann algebra A, a group G of freely acting automorphisms of A

and the fixed point algebra B of A under G. We shall call a von

Neumann subalgebra of A containing B an intermediate subalgebra

of A.

M. Nakamura and Z. Takeda introduced a Galois theory for 11 1 -

factors and finite groups of outer automorphisms by observing the close

analogy between the theories of classical simple rings and of 11 1 -

factors ([36], [37] and others). M. Henle [26] established, among

other results, a Galois correspondence between the class of subgroups

of G and a class of intermediate subalgebras of A, under the condi­

tion that there exists a family of mutually orthogonal projections

Pg (gtG) in A such that LgSG Pg = 1 and that g(Ph) = Phg- I (g,hsG).

As a consequence of a generalization of the Dye correspondence, Y. Haga

and Z. Takeda established a Galois correspondence between the class of

Z-full subgroups of Z-full group determined by G and the class of

intermediate subalgebras of A, under certain conditions [25].

A' ,

We shall consider the following conditions;

(A) there exists a faithful normal expectation of B' onto

(B) there exists a unitary representation u of g G such

that 9 = Ad ug for all gsG,

of B'

for all

gh hg

(e) under the condition (B), there exists an isomorphism

onto G«lA' such that e(A') = Tf(A')

gsG, and

and that e (u ) = A (g) g

e

(D) there exists a group II of automorphisms of A such that

for all gsG and hsH and that H is ergodic on the center

51

of A.

The condition (A) is an algebraic property. Under the condition

(B), the condition (A) is equivalent to the condition (Cl (as follows

from Proposition 3 and Theorem 4).

Under the conditions (B) and (C), an automorphism of A leav­

ing B pointwise fixed is in [Gl z [26, Propos~tion 2.11. Then, by

Proposition 6, we have the following;

Proposition 14. Under the conditions (B), (C) and (D), if an

automorphism et of A satisfies the conditions,

het = cth (hsH)

and

a(xl x (x£B) I

then a is in G.

This proposition is a discrete version of [2, Theorem 111.3.3

(i) I .

Theorem 15. Suppose the conditions (Al and (D) hold. Then

there exists a one-to-one Galois correspondence between the class of

subgroups of G and the class of H-invariant intermediate subalgebras

C of A having a faithful normal expectation of B' onto C'.

In fact, we may assume that there exists a cyclic and separat­

ing vector ~ for A, since the conditions (A) and (D) are algebraic.

Considering the canonical unitary implementation u of the automor­

phism group of A with respect to ~ in [1] (also, cf. [22]), the

condition (B) is satisfied by G. Put HO = {8 o Ad uh e- l ; h£H}, where

e is the mapping in the condition (Cl. Then HO is a group of auto­

morphisms of G@A' having the properties (1), (2) and (3) in Theorem

12. The theorem follows now from Theorem 12.

Corollary 1. Suppose the condition (D) holds. If the group

G is finite, then there exists the same Galois correspondence as in

Theorem 15.

Corollary 2. Suppose the condition (A) holds. If the algebra

A is a factor, then there exists a one-to-one Galois correspondence

between the class of subgroups of G and the class of intermediate

subalgebras C of A having a faithful normal expectation of B'

52

onto C'.

Corollary 2 leads us to the following;

Proposition 16. Suppose the conditions (Al and (B) hold. Then

there exists a one-to-one Galois correspondence between the class of

subgroups of G and the class of intermediate subalgebras C of A

having a faithful normal expectation e of B' onto C' such that

e{ug ) = 0 for all u g t C'.

This proposition is a slight generalization of a theorem of

Henle [26, Theorem 3.11. It is valid also for a group G which is

not necessarily freely acting, if the following condition (A') is as­

sumed in place of the condition (A):

(A') Under the condition (B), there exists a faithful normal

expectation e of B' onto A' such that e(ug

) = 0 for all 9 t 1.

1. H.

2. H.

3. W.

4. H.

5. H.

6.

7.

References

Araki, Some properties of modular conjugation operator of von Neumann algebras and a non commutative Radon-Nikodym theorem with a chain rule, Pacific J. Math., 50(1974), 309-354.

Araki, D. Kastler, M. Takesaki and R. Haag, Extention of KMS states and chemical potential, Commun. Math. Phys., 53(1977), 97-134.

Arveson, Analyticity in operator algebras, Amer. J. Math., 89 (1967), 578-642.

Behncke, Automorphisms of crossed products, Tohoku Hath. J., 21(1969), 580-600.

Choda, On the crossed product of abelian von Neumann algebras, I., Proc. Japan Acad., 43(1967), 111-116.

On the crossed product of abelian von Neumann algebras, II., Proc. Japan Acad., 43(1967), 198-201.

On ergodic and abelian automorphism groups of von Neumann algebras, proc. Japan Acad., 47(1971), 982-985.

8.

9.

____ -,~.' On freely acting automorphisms of operator algebras, KOdai Math. Sem. Rep., 26(1974), 1-21.

______ ,' A Galois correspondence in a von Nernuann algebra, Preprint (1976) •

10. M. Choda, On types over von Neumann subalgebras and the Dye corre­spondence, Publ. RIMS, Kyoto Univ., 9(1973), 45-60.

11. _____ , Normal expectations and crossed products of von Neumann algebras, Proc. Japan Acad., 50(1974), 738-742.

12. Correspondence between subgroups and subalgebras in the -----compact crossed product of a von Neumann algebra, Math. Japon.,

21(1976), 51-59. 13. _____ , Some relations of Ill-factors on free groups, Preprint (1

977) • 14. M. Choda, I. Kasahara and R. Nakamoto, Dependent elements of an

automorphism of a C*-algebra, Proc. Japan Acad., 48(1972}, 561-565.

15. A. Connes, Une classification des facteurs de type III, Ann. Sci.

53

Ecole Norm. Sup., 6(1973), 133-252. 16. A. Connes and M. Takesaki, The flow of weights on factors of type

III, Preprint, UCLA, (1975). 17. J. Dixmicr, Les A1gebres d'Operateurs dans l'Espace Hi1bertien,

Gauthier-Villars, Paris, 1957. 18. H. A. Dye, On groups of measure preserving transformations, I.,

Amer. J. Math., 81(1959), 119-159. 19. , On groups of measure preserving transformations, II., Amer.

--J. Math., 85(1963), 551-576. 20. V. Ya. Go1odets, Crossed products of von Neumann algebras, Russian

Math. Surveys, 26(1971), 1-50. 21. A. Guichardet, Systemes dynamiques non cornrntatifs, Soc. Math.

France, 1974. 22. U. Haagcrup, The standard form of von Neumann algebras, Math.

Scand., 37(1975), 271-283. 23. Y. Haga, On suba1gebras of a cross product von Neumann algebra,

T5hoku Math. J., 25(1973), 291-305. 24. , Crossed products of von Neumann algebras by compact groups,

-----Tohoku Math. J., 28(1976), 511-522. 25. Y. Haga and Z. Takeda, Correspondence between subgroups and sub­

algebras in a cross product von Neumann algebra, TOhoku Math. J., 24(1972), 167-190.

26. H. Henle, Galois theory of W*-a1gebras, Preprint. 27. R. R. Kallman, A generalization of free action, Duke Math. J.,

36(1969}, 781-789. 28. W. Krieger, On constructing non-*isomorphic hyperfinite factors

of type III, J. Functional Analysis, 6(1970), 97-109. 29. , On ergodic flows and the isomorphism of factors, Preprint. 30. A. Kishimoto, Remarks on compact automorphism groups of a certain

von Neumann algebra, Preprint (1976). 31. M. Landstad, Duality theory of covariant systems, Trondheim Univ.,

Preprint (1974). 32. F. J. Hurray and J. von Neumann, On rings of operators, Ann. Hath.,

37 (1936), 116-229. 33. Y. Nakagami, Essential spectrum res) of a dual action on a von

Neumann algebra, Preprint (1976). 34. H. Nakamura and T. Turumaru, Expectations in an operator algebra,

Tohoku Math. J., 6(1954), 182-188. 35. M. Nakamura and Z. Takeda, On some elementary properties of the

crossed products of von Neumann algebras, Proc. Japan Acad., 34(1958}, 489-494.

36. and ,A Galois theory for finite factors, Proc. Japan --Acad. ,36(1960), 258-260.

37. and , On the fundamental theorem of the Galois theory -----for finite factors, Proc. Japan Acad., 36(1960), 313-318.

38. T. Saito, The direct product and the crossed product of rings of operators, Tohoku Math. J., 11(1959), 299-304.

39. I. M. Singer, Automorphisms of finite factors, Amer. J. Math., 77 (1955), 117-133.

40. S. Strati1a, D. Voicu1escu and L. Zsid6, On crossed products, Rev. Roum Math. P~res et Appl., 21(1976), 1411-14449 and 22 (1977), 83-117.

41. C. E. Sutherland, Cohomology and extensions of von Neumann alge­bras, II., Univ. of Oslo, Preprint (1976).

42. N. Suzuki, Crossed products of rings of operators, TOhoku Math. J., 11(1959), 113-124.

43. H. Takai, On a Fourier expansion in continuous crossed products, Publ. RIMS', Kyoto Univ., 11 (1976), 849-880.

44. M. Takesaki, Duality for crossed products and the structure of von Neumann algebras of type III, Acta Math., 131(1973), 249-310.

45. P. K. Tarn, On ergodic abelian M-group, Proc. Japan Acad., 47(1971), 456-457.

46. J. Torniyarna, Tensor products and projections of norm one in von Neumann algebras, Lecture Note at Univ. of Copenhagen, (1970).

47. T. Tururnaru, Crossed product of operator algebra, T5hoku Math. J., 10(1958), 355-365.

48. H. Urnegaki, Conditional expectation in an operator algebra, III., Kodai Math. Sern. Rep., 11(1959), 51-64.

49. G. Zeller-Meier, produ~s croises d'une C*-algebre par un groupe d'autornorphisrns, J. de Math. Pure et Appl., 47(1968), 101-239.