Hilbert C-modules - Εθνικόν και Καποδιστριακόν...

Hilbert C∗-modules

operators on Hilbert modulesUnitization

Morita equivalence


M. Anoussis, University of the Aegean

Athens, 2017

M. Anoussis, University of the Aegean Hilbert C∗-modules



Morita equivalence

1 Hilbert C∗-modules

2 operators on Hilbert modules

3 Unitization

4 Morita equivalence




Morita equivalence

C∗-algebras

Definition

A C*-algebra A is a Banach algebra A equipped with an involution

(that is, a map A→ A denoted a 7→ a∗) such that

(a + λb)∗ = a∗ + λb∗

(ab)∗ = b∗a∗,

a∗∗ = a

‖a∗a‖ = ‖a‖2

for all a, b ∈ A and λ ∈ C.




Morita equivalence

C∗-algebras

Examples

Cz∗ = z, ‖z‖ = |z|.X compact, Hausdorff space

C(X) the space of continuous functions on X

g(x) = g(x)‖g‖ = supx∈X |g(x)|.X locally compact, Hausdorff space

C0(X)f ∈ C0(X)⇔ ∀ε > 0, ∃K ⊂ X , K compact : |f(x)| < ε,∀x /∈ K

g(x) = g(x)‖g‖ = supx∈X |g(x)|.




Morita equivalence

C∗-algebras

Examples

B(H) , for H Hilbert space

‖T‖ = supx∈H,‖x‖≤1 ‖Tx‖〈Tx, y〉 = 〈x, T∗y〉.

Theorem (Gelfand-Naimark)

Let A be a C∗-algebra. Then A is isometrically isomorphic to a closed

subalgebra of B(H) for some Hilbert space H.




Morita equivalence

modules

Definition

Let A be a C∗-algebra and E a vector space. E is a left A-module if

there is a map A× E → E denoted by (a, x) 7→ ax s.t. for all a, b ∈ A,

x, y ∈ E , λ ∈ Ca(x + y) = ax + ay

(a + b)x = ax + bx

(ab)x = a(bx)

a(λx) = (λa)x = λ(ax)

1x = x , if A has a unit.

We also say that A acts on E .

A vector space over C is a C-module.




Morita equivalence

modules

Definition

Let A be a C∗-algebra and E a vector space. E is a right A-module if

there is a map E × A→ E denoted by (x, a) 7→ xa s.t. for all a, b ∈ A,

x, y ∈ E , λ ∈ C

(x + y)a = xa + ya

x(a + b) = xa + xb

x(ab) = (xa)b

λ(xa) = (λx)a = x(λa)

x1 = x if A has a unit.




Morita equivalence

bundles

X compact Hausdorff and H a fixed Hilbert space.

For each x ∈ X , consider a subspace Hx of H. Let

E = {ξ : X → H, ξ continuous, ξ(x) ∈ Hx ,∀x ∈ X}

and define

〈ξ, η〉 (x) = (ξ(x), η(x)).

Then

x 7→ (ξ(x), η(x))

is in C(X)and 〈ξ, η〉 is an C(X)-valued ‘‘inner product’’.




Morita equivalence

bundles

Also if f ∈ C(X), define

fξ ∈ E

by

fξ(x) = f(x)ξ(x).

Then E is a C(X)-module and moreover

f 〈ξ, η〉 = 〈fξ, η〉 .

This is the prototypical example of a Hilbert C(X)-module.




Morita equivalence


Kaplansky, 1953

Paschke, 1973

Rieffel, 1974




Morita equivalence


Definition

Let A be a C*-algebra . An inner product A-module is a complex vector

space E such that

(a) E is a right A-module

(b)There is a map

E × E → A : (x, y)→ 〈x, y〉

satisfying

1 〈x, λy + z〉 = λ 〈x, y〉+ 〈x, y〉2 〈x, y · a〉 = 〈x, y〉 a3 〈x, y〉∗ = 〈y, x〉4 〈x, x〉 ≥ 0

5 〈x, x〉 = 0⇒ x = 0 (x, y, z ∈ E, a ∈ A, λ ∈ C).M. Anoussis, University of the Aegean Hilbert C

∗-modules



Morita equivalence


Proposition

In an inner product A-module E, for all x, y ∈ E,

〈y, x〉〈x, y〉 ≤ ‖〈x, x〉‖A〈y, y〉

and

‖〈x, y〉‖2

A≤ ‖〈x, x〉‖

A‖〈y, y〉‖

A.




Morita equivalence


Proposition

If E is a inner product A-module, we write

‖x‖E

= ‖〈x, x〉‖1/2

A (x ∈ E).

This is a norm on E.

Definition

A Hilbert C*-module over A is an inner product A-module such that

(E, ‖·‖E) is complete.




Morita equivalence


Corollary

If E is an inner product A-module, then

‖x · a‖E≤ ‖x‖

E‖a‖

A.




Morita equivalence


Examples

A Hilbert space H is a left Hilbert C∗-module over C with inner

product

C 〈x, y〉 = (x, y)

(where (, ) is the inner product on H which is antilinear in the

second variable).

A Hilbert space H is a right Hilbert C∗-module over C with inner

product

〈x, y〉C = (y, x)

(where (, ) is the inner product on H which is antilinear in the

second variable).




Morita equivalence


Examples

Any C*-algebra A is a Hilbert C*-module over A with 〈a, b〉 = a∗band a · b = ab.

Any closed ideal J of A is an A-submodule, hence a Hilbert

C∗-module over A.




Morita equivalence


Examples(A x

y λ

)A n× n

x n× 1

y 1× n

and

λ 1× 1.




Morita equivalence


Examples

E =

{(0 x

0 0

): x n× 1

}⟨(

0 x

0 0

),

(0 x ′

0 0

)⟩C

=

(0 x

0 0

)∗(0 x ′

0 0

)=

(0 0

x∗ 0

)(0 x ′

0 0

)=

(0 0

0 x∗x ′

).

E is a right Hilbert C∗-module over C.




Morita equivalence


Examples

E =

{(0 x

0 0

): x n× 1

}⟨(

0 x

0 0

),

(0 x ′

0 0

)⟩=

(0 x

0 0

)(0 x ′

0 0

)∗=

(0 x

0 0

)(0 0

(x ′)∗ 0

)=

(x(x ′)∗ 0

0 0

).

E is a left Hilbert C∗-module over M(n,C).⟨Ax, x ′

⟩= Ax(x

′)∗ = A(x(x′)∗) = A

⟨x, x ′

⟩.




Morita equivalence


Examples

The direct sum⊕

n

k=1Ek of finitely many Hilbert C*-modules over the

same C*-algebra A is the vector space direct sum equipped with

coordinate-wise inner product and module action:

〈(xk), (yk)〉E

=n∑

k=1

〈xk , xk〉Ekand (xk) · a = (xk · a).




Morita equivalence


Examples

The direct sum⊕

Ek of a sequence of Hilbert C*-modules over a fixed

C*-algebra A is defined to be

E =⊕

Ek =

{x = (xk) ∈∏

k

Ek :∑

k

〈xk , xk〉Ekconverges in the norm of A}.




Morita equivalence


Examples

The standard C*-module over a C*-algebra A, sometimes denotedHA,

is the direct sum⊕

EK , where each Ek equals the Hilbert C*-module A.

Thus

{x = (xk) : xk ∈ A :

∑k

x∗k xk converges in the norm of A}.

Thus, in case A = C, the standard module is just `2(N).




Morita equivalence


If F is a submodule of E , then we may have

F ⊕ F⊥ 6= E.




Morita equivalence

operators

Definition

Let A be a C∗-algebra and E a Hilbert C∗-module over A. A map

T : E → E is called adjointable if there exists a map T∗ : E → E such

that 〈Tx, y〉 = 〈x, T∗y〉 for all x, y in E .

remark

If follows from the definition that if T is adjointable, T∗ is adjointable and

〈T∗x, y〉 = 〈x, Ty〉. That is (T∗)∗ = T .




Morita equivalence

operators

Proposition

Let T be an adjointable map. Then

1 T is a linear module map.

2 T is bounded.




Morita equivalence

operators

proof

Linearity:

If x, y, z ∈ E and λ, µ ∈ C we have:

〈T(λx + µy), z〉 = 〈λx + µy, T∗z〉 = λ 〈x, T∗z〉+ µ 〈y, T∗z〉 =

λ 〈Tx, z〉+ µ 〈Ty, z〉 = 〈λT(x) + µT(y), z〉and so T(λx + µy) = λT(x) + µT(y).

T is a module map:

If x, y ∈ E and a ∈ A we have:

〈T(xa), y〉 = 〈xa, T∗y〉 = a∗ 〈x, T∗y〉 = a

∗ 〈T(x), y〉 =

〈T(x)a, y〉and so T(xa) = T(x)a.




Morita equivalence

operators

T is bounded: Let {xn}n∈N be a sequence in E . Assume there exist

x, z ∈ E such that xn → x and Txn → z. Let y ∈ E . We have:

〈T(xn), y〉 → 〈z, y〉

and also

〈Txn, y〉 = 〈xn, T∗y〉 →

〈x, T∗y〉 = 〈Tx, y〉

and so T(x) = z. Hence T is bounded by the Closed Graph

Theorem.




Morita equivalence

operators

Proposition

Let T and S be adjointable operators and λ ∈ C. Then

1 (T + S)∗ = T∗ + S∗.

2 (λT)∗ = λT .

3 TS is adjointable and (TS)∗ = S∗T∗.




Morita equivalence

operators

Proposition

The algebra L(E) of adjointable operators is a C∗-algebra.

proof

‖T∗T‖ ≤ ‖T∗‖‖T‖

and

‖T∗T‖ ≥ supx∈E,‖x‖≤1

{〈T∗Tx, x〉} = supx∈E,‖x‖≤1

{〈Tx, Tx〉} = ‖T‖2.

It follows that

‖T‖ ≤ ‖T∗‖

and since T∗∗ = T we obtain ‖T∗‖ = ‖T‖.M. Anoussis, University of the Aegean Hilbert C

∗-modules



Morita equivalence

By the inequality above we then have:

‖T‖2 ≤ ‖T∗T‖ ≤ ‖T∗‖‖T‖ = ‖T‖2 ⇒ ‖T‖2 = ‖T∗T‖.

We show that L(E) is complete. Let {Tn}n∈N be a Cauchy sequence in

L(E). Since the space of bounded linear operators on E is a Banach

space, {Tn}n∈N converges to a linear operator T and {T∗n }n∈Nconverges to a linear operator T . We show that T is adjointable and

T∗ = T . We have for y ∈ E :

〈Tx, y〉 = lim 〈Tnx, y〉 = lim 〈x, T∗n y〉 =⟨x, Ty

⟩.

So, T∗ = T and L(E) is complete.




Morita equivalence

compact operators

Definition

Let A be a C∗-algebra and E a Hilbert C∗-module over A. Let x, y in E .

Define the map Θx,y : E → E by:

Θx,y(z) = x 〈y, z〉 .




Morita equivalence

compact operators

Proposition

Let A be a C∗-algebra and E a Hilbert C∗-module over A. Then for

every x, y in E the map Θx,y : E → E is adjointable and

Θ∗x,y = Θy,x .

proof For z,w ∈ E we have:

〈Θx,y z,w〉 = 〈x 〈y, z〉 ,w〉 = 〈y, z〉∗ 〈x,w〉 =

〈z, y〉〈x,w〉 = 〈z, y 〈x,w〉〉 = 〈z,Θy,xw〉 .




Morita equivalence

compact operators

Proposition

Let A be a C∗-algebra and E a Hilbert C∗-module over A. The closed

linear span of the set {Θx,y : x ∈ E, y ∈ E} is a closed ideal in L(E). We

call it the algebra of compact operators on E and denote it by K(E).

proof Let T ∈ L(E) and x, y ∈ E . We have:

TΘx,y = ΘTx,y

and

Θx,yT = Θx,T∗y .




Morita equivalence

Example

Let H be a Hilbert space, A = C and consider the Hilbert space H as a

Hilbert C∗-module over A. Then the algebra of adjointable operators on

the Hilbert C∗-module H over A is the algebra of bounded linear

operators on H, and the algebra of compact operators on the Hilbert

C∗-module H over A is the algebra of compact operators on the Hilbert

space H.

Θx,y z = x 〈y, z〉 = x(z, y) = (x ⊗ y)(z).




Morita equivalence

Example

Let A be a C∗-algebra and consider the Hilbert C∗-module A over A.

Consider the map La : A→ A defined by La(x) = ax . Then La is

adjointable with adjoint La∗ and ||La|| = 1. Thus the map a → La is an

isometric homomorphism from A onto a closed C∗-subalgebra ImL of

L(E). Since Θa,b = Lab∗ , ImL contains K(A). On the other hand, if

a ∈ A and {ui}i∈I is a contractive approximate identity for A, we have

Luia → La and since Luia is in K(A) we see that La is in K(A). Thus ImL is

contained in K(A). We conclude that K(A) = ImL and so K(A) is

isomorphic to A.




Morita equivalence

Example

Let A be a unital C∗-algebra and consider the Hilbert C∗-module A

over A. Let T be an adjointable operator on A. Then

T(a) = T(1a) = T(1)a and T = LT(1). The map a → La is an

isomorphism from A onto L(E). Hence we have L(E) = K(E) ' A.




Morita equivalence

Unitization

Definition

Let X be a locally compact Hausdorff space. A compactification of X is

a compact Hausdorff space Y and an injective map i : X → Y such

that i is a homeomorphism onto a dense, open subset of Y .

Example

If X is a locally compact Hausdorff space the one point

compactification of X is a compactification. The Stone-Cech

compactification βX of X is also a compactification.




Morita equivalence

Unitization

Let X be a locally compact Hausdorff space and Y a compactification

of X . If i : X → Y is the embedding of X into Y , define:

i∗ : C0(X)→ C(Y ) by

i∗f(y) =

0 if y /∈ i(X)

f(x) if y = i(x) ∈ i(X)

Then

i∗C0(X) = {f ∈ C(Y ) : f(x) = 0, x /∈ i(X)}

and is an ideal of C(Y ).




Morita equivalence

Unitization

Let J be an ideal in C(Y ). There exists an open set U ⊆ Y such that

J = {f ∈ C(Y ) : f(x) = 0, x /∈ U}.

Then i(X) dense in Y implies that i(X) ∩ U 6= ∅ and

i∗C0(X) ∩ J 6= {0}.




Morita equivalence

Unitization

Definition

Let A be a C∗-algebra and I an ideal of A. The ideal I is essential if

I ∩ J 6= {0} for every ideal J of A, J 6= {0}.

Proposition

The following are equivalent for an ideal I of A.

1 I is essential.

2 If a ∈ A and aI = {0} then a = 0.




Morita equivalence

Unitization

Example

Let X be a compact Hausdorff space. Consider the C∗-algebra C(X).

If I is an ideal of C(X) there exists an open set U such that

I = {f ∈ C(X) : f(x) = 0, x /∈ U}. The ideal I is essential if and only if U

is dense in X .




Morita equivalence

Unitization

Definition

A unitization of a C∗-algebra A is a unital C∗-algebra B and an injective

homomorphism i : A→ B such that i(A) is an essential ideal in B.

remark

If A is unital and B is a unitization of A, then A = B.

proof Let 1 be the unit of A and b the unit of B. If a ∈ A we have

(b − 1)a = ba − 1a = 0 and hence (b − 1)A = {0}. By Proposition

b = 1 and so A = B.




Morita equivalence

Unitization

Example

Let A be a C∗-algebra without unit. Set A1 = A⊕ C. Define

(a, λ)(b, µ) = (ab + µa + λb, λµ) and (a, λ)∗ = (a∗, λ). Consider

the embedding L : A→ K(A). (La is the operator defined by

La(x) = ax for x in A). Define L : A1 → L(A) by L((a, λ)) = La + λI.

Then, the image of A1 by L is closed in L(A) and so it is a C∗-algebra.

Define the norm on A1 by ‖(a, λ)‖ = ‖La + λI‖. Then, A1 with this

norm is a C∗-algebra and is a unitization of A.




Morita equivalence

Unitization

Example

Let H be a Hilbert space and K (H) the algebra of compact operators

on H. Then the subalgebra K (H) + CI of B(H) is closed in B(H) and is a

unitization of K (H).

Example

Let A be a C∗-algebra and consider the Hilbert C∗-module A over A.

Consider the map L : A→ K(A). Then L(A) is a unitization of A. One

has to show that K(A) is an essential ideal of L(A). Let T ∈ L(A) and

assume that TΘx,y = 0 for every x, y ∈ A. Then ΘTx,y = 0 for every

x, y ∈ A which implies that Tx = 0 for every x ∈ A and so T = 0. It

follows that K(A) is an essential ideal of L(A).




Morita equivalence

Unitization

Example

Let X be a non compact locally compact Hausdorff space and Y a

compactification of X . If i : X → Y is the embedding of X into Y ,

define: i∗ : C0(X)→ C(Y ) by

i∗f(y) =

0 if y /∈ i(X)

f(x) if y = i(x) ∈ i(X)

Then, C(Y ) and i∗ is a unitization of C0(X). If Y is the one-point

compactification of X , then C0(X)1 = C(Y ).




Morita equivalence

Unitization

Definition

A unitization (B, i) of a C∗-algebra A is maximal if whenever C is a

C∗-algebra, j : A→ C a homomorphism such that j(A) is an essential

ideal of C, then there exists an homomorphism φ : C → B such that

φj = i.

It is not obvious from the definition that a maximal unitization of a

C∗-algebra exists.




Morita equivalence

Unitization

Theorem

Let A be a C∗-algebra. The C∗-algebra (L(A), i) (where i(a) = La) is a

maximal unitization of A. Moreover it (B, j) is another maximal

unitization, there exists an isomorphism φ : B → L(A) such that φj = i .

Definition

We will refer to L(A) as the multiplier algebra of A and denote it by

M(A).




Morita equivalence

Unitization

Corollary

Let E be a Hilbert C∗-module. Then M(K(E)) = L(E).

Proposition

1 Let H be a Hilbert space. Then M(K (H)) = B(H).

2 Let T be a locally compact Hausdorff space. Then

M(C0(T)) = Cb(T) = C(βT) where Cb(T) is the space of

bounded continuous functions on T and βT is the Stone Cech

compactification of T .




Morita equivalence

Morita equivalence

Definition

Let A, B be C∗-algebras. An A− B imprimitivity bimodule E is an A− B

bimodule s.t.

E is a full left Hilbert C∗-module over A and a full right Hilbert

C∗-module over B.

For x, y ∈ E , a ∈ A and b ∈ B we have:

A 〈xb, y〉 =A 〈x, yb∗〉

〈ax, y〉B

= 〈x, a∗y〉B

For x, y, z ∈ E

A 〈x, y〉 z = x 〈y, z〉B




Morita equivalence

Morita equivalence

Definition

The C∗-algebras A and B are Morita equivalent if there is an A− B

imprimitivity bimodule E .




Morita equivalence

Morita equivalence

Example

A Hilbert space H is a K (H)− C imprimitivity bimodule with

K(H) 〈h, k〉 = h⊗ k

where h⊗ k(l) = h(l, k).




Morita equivalence

Morita equivalence

Proposition

A full Hilbert C∗-module E over B is a K(E)− B imprimitivity bimodule

with

K(E) 〈x, y〉 = Θx,y .


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