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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 is NOT adjointable.) 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 cannot be adjointable. 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