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Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
Hilbert C∗-modules
M. Anoussis, University of the Aegean
Athens, 2017
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
1 Hilbert C∗-modules
2 operators on Hilbert modules
3 Unitization
4 Morita equivalence
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
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)|.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
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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’’.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
Hilbert C∗-modules
Kaplansky, 1953
Paschke, 1973
Rieffel, 1974
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
Hilbert C∗-modules
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
Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
Hilbert C∗-modules
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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
Hilbert C∗-modules
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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
Hilbert C∗-modules
Corollary
If E is an inner product A-module, then
‖x · a‖E≤ ‖x‖
E‖a‖
A.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
Hilbert C∗-modules
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).
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
Hilbert C∗-modules
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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
Hilbert C∗-modules
Examples(A x
y λ
)A n× n
x n× 1
y 1× n
and
λ 1× 1.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
Hilbert C∗-modules
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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
Hilbert C∗-modules
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 ′
⟩.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
Hilbert C∗-modules
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).
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
Hilbert C∗-modules
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}.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
Hilbert C∗-modules
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).
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
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Hilbert C∗-modules
If F is a submodule of E , then we may have
F ⊕ F⊥ 6= E.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
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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 .
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
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operators
Proposition
Let T be an adjointable map. Then
1 T is a linear module map.
2 T is bounded.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
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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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
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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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
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∗.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
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
Hilbert C∗-modules
operators on Hilbert modulesUnitization
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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
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〉 .
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
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〉 .
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
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 .
M. Anoussis, University of the Aegean Hilbert C∗-modules
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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).
M. Anoussis, University of the Aegean Hilbert C∗-modules
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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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
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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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
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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 ).
M. Anoussis, University of the Aegean Hilbert C∗-modules
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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}.
M. Anoussis, University of the Aegean Hilbert C∗-modules
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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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
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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 .
M. Anoussis, University of the Aegean Hilbert C∗-modules
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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.
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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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
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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).
M. Anoussis, University of the Aegean Hilbert C∗-modules
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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 ).
M. Anoussis, University of the Aegean Hilbert C∗-modules
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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.
M. Anoussis, University of the Aegean Hilbert C∗-modules
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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).
M. Anoussis, University of the Aegean Hilbert C∗-modules
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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 .
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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
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
Morita equivalence
Morita equivalence
Definition
The C∗-algebras A and B are Morita equivalent if there is an A− B
imprimitivity bimodule E .
M. Anoussis, University of the Aegean Hilbert C∗-modules
Hilbert C∗-modules
operators on Hilbert modulesUnitization
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).
M. Anoussis, University of the Aegean Hilbert C∗-modules