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```Hilbert C ∗ -modules
Saeid Zahmatkesh
Department of Mathematics
Faculty of Science
King Mongkut’s University of Technology Thonburi (KMUTT)
Bangkok 10140, THAILAND
[email protected]
November 30, 2016
Saeid Zahmatkesh
Hilbert C ∗ -modules
Definitions and Examples
The idea behind the introducing Hilbert C ∗ -modules is to generalize the
concept of Hilbert spaces, replacing the field of scalars by a C ∗ -algebra.
• Let X be a (complex) vector space and A a C ∗ -algebra. If there is a
bilinear map (module action)
X×A→X
(x, a) 7→ x · a
such that
λ(x · a) = (λx) · a = x · (λa) (x ∈ X, a ∈ A, λ ∈ C),
then X is called a (right) A-module and denoted by XA .
Example 1. Let Mn (A) be the vector space of all n × n matrices with
entries in a C ∗ -algebra A. Then it is a (right) A-module with the action
(ai,j ) · a := (ai,j a).
Saeid Zahmatkesh
Hilbert C ∗ -modules
Definitions and Examples
Definition
An (right) inner product A-module is an (right) A-module together with
a map h·, ·iA : X × X → A (A-valued inner product) such that
1
hx, λy + µziA = λhx, yiA + µhx, ziA ;
2
hx, y · aiA = hx, yiA a;
3
hx, yi∗A = hy, xiA ;
4
hx, xiA ≥ 0 (a positive element of A);
5
hx, xiA = 0 ⇐⇒ x = 0,
for all x, y, z ∈ X and λ, µ ∈ C.
• It follows from (1), (2), and (3) in above that
I := span{hx, yiA : x, y ∈ X}
is and ideal of A.
Saeid Zahmatkesh
Hilbert C ∗ -modules
Definitions and Examples
There is a norm on an inner product A-module X induced by its A-valued
inner product h·, ·iA :
kxkA := khx, xiA k1/2 ,
such that kx · akA ≤ kxkA kak for all x ∈ X and a ∈ A. Now:
Definition
A Hilbert A-module is an inner product A-module X which is complete in
the norm k · kA . It is called a full Hilbert A-module if the ideal
I := span{hx, yiA : x, y ∈ X}
is dense in A.
• Note that a left A-module, a left A-valued inner product A h·, ·i, and a
left Hilbert A-module can be defined in a similar way.
Saeid Zahmatkesh
Hilbert C ∗ -modules
Definitions and Examples
Example 2. Any Hilbert space H is actually a Hilbert module over C
(Hilbert C-module), with:
h · λ := λh (the usual sacalar multiplication); and
hh, kiC := hk|hi (the usual Hilbert space inner product),
for h, k ∈ H and λ ∈ C.
Example 3. Every C ∗ -algebra A can be viewed as a (full) Hilbert
module over itself (the Hilbert A-module AA ), with
a · b := ab (the usual multiplication in A); and
ha, biA := a∗ b,
for a, b ∈ A. In this case, the norm k · kA on AA coincides with the
C ∗ -norm of A as we have
1/2
kakA = kha, aiA k1/2 = ka∗ ak1/2 = kak2
= kak.
Saeid Zahmatkesh
Hilbert C ∗ -modules
Definitions and Examples
Example 4. Let H be a Hilbert space and K(H) the C ∗ -algebra of
compact operators on H (as a C ∗ -subalgebra of bounded operators
B(H)). For x, y ∈ H, let x ⊗ y denote the rank-one operator
h 7→ hh|yix = x · hy, hiC
on H. Now H is a full left Hilbert K(H)-module with
T · x := T (x) and
Note that the norm
because
K(H) kxk
K(H) k
K(H) hx, yi
:= x ⊗ y.
· k of H coincides with the usual norm on H,
= kK(H) hx, xik1/2 = kx ⊗ xk1/2 = kxk2
1/2
= kxk.
To see that K(H) H is full, recall that the compact operators K(H) are
indeed spanned by finite-rank operators, more precisely
K(H) = span{x ⊗ y : x, y ∈ H} = span{K(H) hx, yi : x, y ∈ H}.
Saeid Zahmatkesh
Hilbert C ∗ -modules
Definitions and Examples
Example 5. Let T be a locally compact Hausdorff space, and H a
Hilbert space. Let
X = C0 (T, H) := {x : T → H : x is continuous and (t 7→ kx(t)k) ∈ C0 (T )}.
Then X is a Hilbert C0 (T )-module with
(x · f )(t) := f (t)x(t) and hx, yiC0 (T ) (t) := hy(t)|x(t)i,
where x, y ∈ X and f ∈ C0 (T ). We have
kxk2C0 (T ) = khx, xiC0 (T ) k
= supt∈T |hx, xiC0 (T ) (t)|
= supt∈T |hx(t)|x(t)i|
= supt∈T kx(t)k2
2
= supt∈T kx(t)k = kxk2∞ .
Therefore the norm k · kC0 (T ) agrees with the usual sup norm on
C0 (T, H). Moreover X is full, because...
Saeid Zahmatkesh
Hilbert C ∗ -modules
Definitions and Examples
the C ∗ -algebra C0 (T ), which can be viewed as a Hilbert module over
itself, admits an approximate unit, and therefore
C0 (T ) = span{hf, giC0 (T ) : f, g ∈ C0 (T )} = span{f g : f, g ∈ C0 (T )}.
But if we take x, y ∈ X = C0 (T, H) such that x(t) := f (t)h and
y(t) := g(t)h, where h ∈ H with khk = 1, then hx, yiC0 (T ) = f g.
Because
hx, yiC0 (T ) (t) = hg(t)h|f (t)hi = g(t)f (t)hh|hi = f (t)g(t) = (f g)(t).
Therefore C0 (T ) = span{hx, yiC0 (T ) : x, y ∈ X}.
Example 6. (Direct Sums of Hilbert modules) Let X and Y be Hilbert
A-modules. Then Z = X ⊕ Y := {(x, y) : x ∈ X, y ∈ Y } is a Hilbert
A-module with
(x, y) · a := (x · a, y · a) and;
h(x1 , y1 ), (x2 , y2 )iA := hx1 , x2 iA + hy1 , y2 iA .
Saeid Zahmatkesh
Hilbert C ∗ -modules
Definitions and Examples
Example 7. Let A be a C ∗ -algebra, and N := {0, 1, 2, 3, ...}. Then
`2 (N, A) = {f : N → A :
∞
X
f (n)∗ f (n) converges in A}
n=0
is a (full) Hilbert A-module with
(f · a)(n) := f (n)a and hf, giA :=
∞
X
f (n)∗ g(n).
n=0
Note that `2 (N, A) can be identified with the tensor product `2 (N) ⊗ A.
Proposition (The Cauchy-Schwarz inequality)
If X is an inner product A-module, then
hx, yi∗A hx, yiA ≤ khx, xiA khy, yiA for all x, y ∈ X,
Saeid Zahmatkesh
Hilbert C ∗ -modules
Adjointable operators on Hilbert C ∗ -modules
Recall that the bounded operators on Hilbert spaces are automatically
adjointable. But we are going to see that this is not true for operators on
Hilbert C ∗ -modules. More precisely, Adjointable operators on Hilbert
C ∗ -modules are bounded but a bounded operator on a Hilbert
C ∗ -module may not be adjointable.
Definition
Let X and Y be Hilbert A-modules, where A is a C ∗ -algebra. A map
T : X → Y is adjointable if there is a map T ∗ : Y → X such that
hT (x), yiA = hx, T ∗ (y)iA for all x ∈ X, y ∈ Y.
Theorem
If a map T : X → Y is adjointable, then it is a bounded linear A-module
map from X to Y .
Note that A-module map means that T is A-linear, which preserves the
module action:
T (x · a) = T (x) · a for all x ∈ X, a ∈ A.
Saeid Zahmatkesh
Hilbert C ∗ -modules
Adjointable operators on Hilbert C ∗ -modules
Example 8. (A bounded A-linear operator on a Hilbert A-module which
Suppose A = C([0, 1]), and let J = {f ∈ A : f (0) = 0}. Then A and J
are Hilbert A-modules. Now take X := A ⊕ J, and define T : X → X by
T (f, g) = (g, 0). Then one can see that T is bounded such that kT k = 1
and A-linear. Assume that T has an adjoint T ∗ satisfying
hT (x), yiA = hx, T ∗ (y)iA . If (f, g) := T ∗ (1, 0), then for all (h, k) ∈ X
k = h(k, 0), (1, 0)iA
=
=
=
=
hT (h, k), (1, 0)iA
h(h, k), T ∗ (1, 0)iA
h(h, k), (f, g)iA
hf + kg.
So we must have f ≡ 0 and g ≡ 1, which contradicts g(0) = 0. Thus T
Saeid Zahmatkesh
Hilbert C ∗ -modules
Adjointable operators on Hilbert C ∗ -modules
Definition
If X and Y are Hilbert A-modules, then the set of all adjointable
operators from X to Y is denoted by L(X, Y ). For L(X, X), we simply
write L(X).
One can see that if T ∈ L(X), then T ∗ is unique, T ∗ ∈ L(X), and
(T ∗ )∗ = T .
Also L(X) is in fact a subalgebra of the Banach algebra B(X) of
bounded operators on X, and T 7→ T ∗ is an involution on L(X). We
indeed have:
Theorem
If X is a Hilbert A-module, then L(X) is a C ∗ -algebra with respect to
the operator norm.
Saeid Zahmatkesh
Hilbert C ∗ -modules
Adjointable operators on Hilbert C ∗ -modules
Example 9. Consider L(`2 (N, A)), the C ∗ -algebra of adjointable
operators on the Hilbert A-modules `2 (N, A). Then define the map
S : `2 (N, A) → `2 (N, A) by
S(f (0), f (1), f (2), ...) := (0, f (0), f (1), f (2), ...) f ∈ `2 (N, A)
Then S ∈ L(`2 (N, A)) such that
S ∗ (f (0), f (1), f (2), ...) = (f (1), f (2), ...).
Also it is easy to that S ∗ S = I but SS ∗ 6= I, which means S is a
nonunitary isometry.
Saeid Zahmatkesh
Hilbert C ∗ -modules
Adjointable operators on Hilbert C ∗ -modules
Moreover if α is an endomorphism of A, then the map
πα : A → L(`2 (N, A)) defined by
πα (a)(f (0), f (1), f (2), ...)
= (α0 (a)f (0), α1 (a)f (1), α2 (a)f (2), ...)
= (af (0), α(a)f (1), α2 (a)f (2), ...)
is an injective ∗-homomorphism of C ∗ -algebra. So in fact, πα is a faithful
representation of A on the Hilbert A-modules `2 (N, A).
Saeid Zahmatkesh
Hilbert C ∗ -modules
Thank you
Saeid Zahmatkesh
Hilbert C ∗ -modules
```
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