Hilbert C*-module

Hilbert C*-modules are mathematical objects that generalise the notion of Hilbert spaces (which are themselves generalisations of Euclidean space), in that they endow a linear space with an "inner product" that takes values in a C*-algebra. They were first introduced in the work of Irving Kaplansky in 1953, which developed the theory for commutative, unital algebras (though Kaplansky observed that the assumption of a unit element was not "vital"). In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke and Marc Rieffel, the latter in a paper that used Hilbert C*-modules to construct a theory of induced representations of C*-algebras. Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory, and provide the right framework to extend the notion of Morita equivalence to C*-algebras. They can be viewed as the generalization of vector bundles to noncommutative C*-algebras and as such play an important role in noncommutative geometry, notably in C*-algebraic quantum group theory, and groupoid C*-algebras.

Definitions

Inner-product C*-modules Let A {\displaystyle A} be a C*-algebra (not assumed to be commutative or unital), its involution denoted by

∗ {\displaystyle {}^{*}} . An inner-product A {\displaystyle A} -module (or pre-Hilbert A {\displaystyle A} -module) is a complex linear space E {\displaystyle E} equipped with a compatible right A {\displaystyle A} -module structure, together with a map

⟨ ⋅ , ⋅ ⟩ A : E × E → A {\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle _{A}:E\times E\rightarrow A}

that satisfies the following properties:

For all x {\displaystyle x} , y {\displaystyle y} , z {\displaystyle z} in E {\displaystyle E} , and α {\displaystyle \alpha } , β {\displaystyle \beta } in C {\displaystyle \mathbb {C} } :

⟨ x , y α + z β ⟩ A = ⟨ x , y ⟩ A α + ⟨ x , z ⟩ A β {\displaystyle \langle x,y\alpha +z\beta \rangle _{A}=\langle x,y\rangle _{A}\alpha +\langle x,z\rangle _{A}\beta }

(i.e. the inner product is C {\displaystyle \mathbb {C} } -linear in its second argument). For all x {\displaystyle x} , y {\displaystyle y} in E {\displaystyle E} , and a {\displaystyle a} in A {\displaystyle A} :

⟨ x , y a ⟩ A = ⟨ x , y ⟩ A a {\displaystyle \langle x,ya\rangle _{A}=\langle x,y\rangle _{A}a}

For all x {\displaystyle x} , y {\displaystyle y} in E {\displaystyle E} :

⟨ x , y ⟩ A = ⟨ y , x ⟩ A ∗ , {\displaystyle \langle x,y\rangle _{A}=\langle y,x\rangle _{A}^{*},}

from which it follows that the inner product is conjugate linear in its first argument (i.e. it is a sesquilinear form). For all x {\displaystyle x} in E {\displaystyle E} :

⟨ x , x ⟩ A ≥ 0 {\displaystyle \langle x,x\rangle _{A}\geq 0}

in the sense of being a positive element of A, and

⟨ x , x ⟩ A = 0 ⟺ x = 0. {\displaystyle \langle x,x\rangle _{A}=0\iff x=0.}

(An element of a C*-algebra A {\displaystyle A} is said to be positive if it is self-adjoint with non-negative spectrum.)

Hilbert C*-modules An analogue to the Cauchy–Schwarz inequality holds for an inner-product A {\displaystyle A} -module E {\displaystyle E} :

⟨ x , y ⟩ A ⟨ y , x ⟩ A ≤ ‖ ⟨ y , y ⟩ A ‖ ⟨ x , x ⟩ A {\displaystyle \langle x,y\rangle _{A}\langle y,x\rangle _{A}\leq \Vert \langle y,y\rangle _{A}\Vert \langle x,x\rangle _{A}}

for x {\displaystyle x} , y {\displaystyle y} in E {\displaystyle E} . On the pre-Hilbert module E {\displaystyle E} , define a norm by

‖ x ‖ = ‖ ⟨ x , x ⟩ A ‖ 1 2 . {\displaystyle \Vert x\Vert =\Vert \langle x,x\rangle _{A}\Vert ^{\frac {1}{2}}.}

The norm-completion of E {\displaystyle E} , still denoted by E {\displaystyle E} , is said to be a Hilbert A {\displaystyle A} -module or a Hilbert C*-module over the C*-algebra A {\displaystyle A} . The Cauchy–Schwarz inequality implies the inner product is jointly continuous in norm and can therefore be extended to the completion. The action of A {\displaystyle A} on E {\displaystyle E} is continuous: for all x {\displaystyle x} in E {\displaystyle E}

a λ → a ⇒ x a λ → x a . {\displaystyle a_{\lambda }\rightarrow a\Rightarrow xa_{\lambda }\rightarrow xa.}

Similarly, if ( e λ ) {\displaystyle (e_{\lambda })} is an approximate unit for A {\displaystyle A} (a net of self-adjoint elements of A {\displaystyle A} for which a e λ {\displaystyle ae_{\lambda }} and e λ a {\displaystyle e_{\lambda }a} tend to a {\displaystyle a} for each a {\displaystyle a} in A {\displaystyle A} ), then for x {\displaystyle x} in E {\displaystyle E}

x e λ → x . {\displaystyle xe_{\lambda }\rightarrow x.}

Whence it follows that E A {\displaystyle EA} is dense in E {\displaystyle E} , and x 1 A = x {\displaystyle x1_{A}=x} when A {\displaystyle A} is unital. Let

⟨ E , E ⟩ A = span ⁡ { ⟨ x , y ⟩ A ∣ x , y ∈ E } , {\displaystyle \langle E,E\rangle _{A}=\operatorname {span} \{\langle x,y\rangle _{A}\mid x,y\in E\},}

then the closure of ⟨ E , E ⟩ A {\displaystyle \langle E,E\rangle _{A}} is a two-sided ideal in A {\displaystyle A} . Two-sided ideals are C*-subalgebras and therefore possess approximate units. One can verify that E ⟨ E , E ⟩ A {\displaystyle E\langle E,E\rangle _{A}} is dense in E {\displaystyle E} . In the case when ⟨ E , E ⟩ A {\displaystyle \langle E,E\rangle _{A}} is dense in A {\displaystyle A} , E {\displaystyle E} is said to be full. This does not generally hold.

Examples

Hilbert spaces Since the complex numbers C {\displaystyle \mathbb {C} } are a C*-algebra with an involution given by complex conjugation, a complex Hilbert space H {\displaystyle {\mathcal {H}}} is a Hilbert C {\displaystyle \mathbb {C} } -module under scalar multipliation by complex numbers and its inner product.

Vector bundles If X {\displaystyle X} is a locally compact Hausdorff space and E {\displaystyle E} a vector bundle over X {\displaystyle X} with projection π : E → X {\displaystyle \pi \colon E\to X} a Hermitian metric g {\displaystyle g} , then the space of continuous sections of E {\displaystyle E} is a Hilbert C ( X ) {\displaystyle C(X)} -module. Given sections σ , ρ {\displaystyle \sigma ,\rho } of E {\displaystyle E} and f ∈ C ( X ) {\displaystyle f\in C(X)} the right action is defined by

σ f ( x ) = σ ( x ) f ( π ( x ) ) , {\displaystyle \sigma f(x)=\sigma (x)f(\pi (x)),}

and the inner product is given by

⟨ σ , ρ ⟩ C ( X ) ( x ) := g ( σ ( x ) , ρ ( x ) ) . {\displaystyle \langle \sigma ,\rho \rangle _{C(X)}(x):=g(\sigma (x),\rho (x)).}

The converse holds as well: Every countably generated Hilbert C*-module over a commutative unital C*-algebra A = C ( X ) {\displaystyle A=C(X)} is isomorphic to the space of sections vanishing at infinity of a continuous field of Hilbert spaces over X {\displaystyle X} .

C*-algebras Any C*-algebra A {\displaystyle A} is a Hilbert A {\displaystyle A} -module with the action given by right multiplication in A {\displaystyle A} and the inner product ⟨ a , b ⟩ = a ∗ b {\displaystyle \langle a,b\rangle =a^{*}b} . By the C*-identity, the Hilbert module norm coincides with C*-norm on A {\displaystyle A} . The (algebraic) direct sum of n {\displaystyle n} copies of A {\displaystyle A}

A n = ⨁ i = 1 n A {\displaystyle A^{n}=\bigoplus _{i=1}^{n}A}

can be made into a Hilbert A {\displaystyle A} -module by defining

⟨ ( a i ) , ( b i ) ⟩ A = ∑ i = 1 n a i ∗ b i . {\displaystyle \langle (a_{i}),(b_{i})\rangle _{A}=\sum _{i=1}^{n}a_{i}^{*}b_{i}.}

If p {\displaystyle p} is a projection in the C*-algebra M n ( A ) {\displaystyle M_{n}(A)} , then p A n {\displaystyle pA^{n}} is also a Hilbert A {\displaystyle A} -module with the same inner product as the direct sum.

The standard Hilbert module One may also consider the following subspace of elements in the countable direct product of A {\displaystyle A}

ℓ 2 ( A ) = H A = { ( a i ) | ∑ i = 1 ∞ a i ∗ a i converges in A } . {\displaystyle \ell _{2}(A)={\mathcal {H}}_{A}={\Big \{}(a_{i})|\sum _{i=1}^{\infty }a_{i}^{*}a_{i}{\text{ converges in }}A{\Big \}}.}

Endowed with the obvious inner product (analogous to that of A n {\displaystyle A^{n}} ), the resulting Hilbert A {\displaystyle A} -module is called the standard Hilbert module over A {\displaystyle A} . The fact that there is a unique separable Hilbert space has a generalization to Hilbert modules in the form of the Kasparov stabilization theorem, which states that if E {\displaystyle E} is a countably generated Hilbert A {\displaystyle A} -module, there is an isometric isomorphism E ⊕ ℓ 2 ( A ) ≅ ℓ 2 ( A ) . {\displaystyle E\oplus \ell _{2}(A)\cong \ell _{2}(A).}

Maps between Hilbert modules Let E {\displaystyle E} and F {\displaystyle F} be two Hilbert modules over the same C*-algebra A {\displaystyle A} . These are then Banach spaces, so it is possible to speak of the Banach space of bounded linear maps L ( E , F ) {\displaystyle {\mathcal {L}}(E,F)} , normed by the operator norm. The adjointable and compact adjointable operators are subspaces of this Banach space defined using the inner product structures on E {\displaystyle E} and F {\displaystyle F} . In the special case where A {\displaystyle A} is C {\displaystyle \mathbb {C} } these reduce to bounded and compact operators on Hilbert spaces respectively.

Adjointable maps A map (not necessarily linear)

T : E → F {\displaystyle T\colon E\to F} is defined to be adjointable if there is another map T ∗ : F → E {\displaystyle T^{*}\colon F\to E} , known as the adjoint of T {\displaystyle T} , such that for every

e ∈ E {\displaystyle e\in E} and f ∈ F {\displaystyle f\in F} ,

⟨ f , T e ⟩ = ⟨ T ∗ f , e ⟩ . {\displaystyle \langle f,Te\rangle =\langle T^{*}f,e\rangle .}

Both T {\displaystyle T} and T ∗ {\displaystyle T^{*}} are then automatically linear and also A {\displaystyle A} -module maps. The closed graph theorem can be used to show that they are also bounded. Analogously to the adjoint of operators on Hilbert spaces, T ∗ {\displaystyle T^{*}} is unique (if it exists) and itself adjointable with adjoint T {\displaystyle T} . If S : F → G {\displaystyle S\colon F\to G}

is a second adjointable map, S T {\displaystyle ST} is adjointable with adjoint

S ∗ T ∗ {\displaystyle S^{*}T^{*}} . The adjointable operators E → F {\displaystyle E\to F} form a subspace B ( E , F ) {\displaystyle \mathbb {B} (E,F)}

of L ( E , F ) {\displaystyle {\mathcal {L}}(E,F)} , which is complete in the operator norm. In the case F = E {\displaystyle F=E} , the space B ( E , E ) {\displaystyle \mathbb {B} (E,E)} of adjointable operators from E {\displaystyle E} to itself is denoted B ( E ) {\displaystyle \mathbb {B} (E)} , and is a C*-algebra.

Compact adjointable maps Given e ∈ E {\displaystyle e\in E} and f ∈ F {\displaystyle f\in F} , the map

| f ⟩ ⟨ e | : E → F {\displaystyle |f\rangle \langle e|\colon E\to F} is defined, analogously to the rank one operators of Hilbert spaces, to be

g ↦ f ⟨ e , g ⟩ . {\displaystyle g\mapsto f\langle e,g\rangle .}

This is adjointable with adjoint | e ⟩ ⟨ f | {\displaystyle |e\rangle \langle f|} . The compact adjointable operators K ( E , F ) {\displaystyle \mathbb {K} (E,F)} are defined to be the closed span of

{ | f ⟩ ⟨ e | ∣ e ∈ E , f ∈ F } {\displaystyle \{|f\rangle \langle e|\mid e\in E,\;f\in F\}}

in B ( E , F ) {\displaystyle \mathbb {B} (E,F)} . As with the bounded operators, K ( E , E ) {\displaystyle \mathbb {K} (E,E)} is denoted

K ( E ) {\displaystyle \mathbb {K} (E)} . This is a (closed, two-sided) ideal of

B ( E ) {\displaystyle \mathbb {B} (E)} .

C*-correspondences If A {\displaystyle A} and B {\displaystyle B} are C*-algebras, an ( A , B ) {\displaystyle (A,B)} C*-correspondence is a Hilbert B {\displaystyle B} -module equipped with a left action of A {\displaystyle A} by adjointable maps that is faithful. (NB: Some authors require the left action to be non-degenerate instead.) These objects are used in the formulation of Morita equivalence for C*-algebras, see applications in the construction of Toeplitz and Cuntz-Pimsner algebras, and can be employed to put the structure of a bicategory on the collection of C*-algebras.

Tensor products and the bicategory of correspondences If E {\displaystyle E} is an ( A , B ) {\displaystyle (A,B)} and F {\displaystyle F} a ( B , C ) {\displaystyle (B,C)} correspondence, the algebraic tensor product E ⊙ F {\displaystyle E\odot F} of E {\displaystyle E} and F {\displaystyle F}

as vector spaces inherits left and right A {\displaystyle A} - and C {\displaystyle C} -module structures respectively. It can also be endowed with the C {\displaystyle C} -valued sesquilinear form defined on pure tensors by

⟨ e ⊙ f , e ′ ⊙ f ′ ⟩ C := ⟨ f , ⟨ e , e ′ ⟩ B f ⟩ C . {\displaystyle \langle e\odot f,e'\odot f'\rangle _{C}:=\langle f,\langle e,e'\rangle _{B}f\rangle _{C}.}

This is positive semidefinite, and the Hausdorff completion of E ⊙ F {\displaystyle E\odot F}

in the resulting seminorm is denoted E ⊗ B F {\displaystyle E\otimes _{B}F} . The left- and right-actions of

A {\displaystyle A} and C {\displaystyle C} extend to make this an ( A , C ) {\displaystyle (A,C)} correspondence. The collection of C*-algebras can then be endowed with the structure of a bicategory, with C*-algebras as objects, ( A , B ) {\displaystyle (A,B)} correspondences as arrows B → A {\displaystyle B\to A} , and isomorphisms of correspondences (bijective module maps that preserve inner products) as 2-arrows.

Toeplitz algebra of a correspondence Given a C*-algebra A {\displaystyle A} , and an ( A , A ) {\displaystyle (A,A)} correspondence E {\displaystyle E} , its Toeplitz algebra T ( E ) {\displaystyle {\mathcal {T}}(E)} is defined as the universal algebra for Toeplitz representations (defined below). The classical Toeplitz algebra can be recovered as a special case, and the Cuntz-Pimsner algebras are defined as particular quotients of Toeplitz algebras. In particular, graph algebras , crossed products by Z {\displaystyle \mathbb {Z} } , and the Cuntz algebras are all quotients of specific Toeplitz algebras.

Toeplitz representations A Toeplitz representation of E {\displaystyle E} in a C*-algebra D {\displaystyle D} is a pair ( S , ϕ ) {\displaystyle (S,\phi )} of a linear map S : E → D {\displaystyle S\colon E\to D} and a homomorphism

ϕ : A → D {\displaystyle \phi \colon A\to D} such that

S {\displaystyle S} is "isometric":

S ( e ) ∗ S ( f ) = ϕ ( ⟨ e , f ⟩ ) {\displaystyle S(e)^{*}S(f)=\phi (\langle e,f\rangle )} for all e , f ∈ E {\displaystyle e,f\in E} ,

S {\displaystyle S} resembles a bimodule map:

S ( a e ) = ϕ ( a ) S ( e ) {\displaystyle S(ae)=\phi (a)S(e)} and S ( e a ) = S ( e ) ϕ ( a ) {\displaystyle S(ea)=S(e)\phi (a)} for e ∈ E {\displaystyle e\in E} and a ∈ A {\displaystyle a\in A} .

Toeplitz algebra The Toeplitz algebra T ( E ) {\displaystyle {\mathcal {T}}(E)} is the universal Toeplitz representation. That is, there is a Toeplitz representation ( T , ι ) {\displaystyle (T,\iota )} of E {\displaystyle E} in T ( E ) {\displaystyle {\mathcal {T}}(E)} such that if ( S , ϕ ) {\displaystyle (S,\phi )} is any Toeplitz representation of E {\displaystyle E} (in an arbitrary algebra D {\displaystyle D} ) there is a unique *-homomorphism

Φ : T ( E ) → D {\displaystyle \Phi \colon {\mathcal {T}}(E)\to D} such that S = Φ ∘ T {\displaystyle S=\Phi \circ T}

and ϕ = Φ ∘ ι {\displaystyle \phi =\Phi \circ \iota } .

Examples If A {\displaystyle A} is taken to be the algebra of complex numbers, and E {\displaystyle E}

the vector space C n {\displaystyle \mathbb {C} ^{n}} , endowed with the natural

( C , C ) {\displaystyle (\mathbb {C} ,\mathbb {C} )} -bimodule structure, the corresponding Toeplitz algebra is the universal algebra generated by n {\displaystyle n} isometries with mutually orthogonal range projections. In particular, T ( C ) {\displaystyle {\mathcal {T}}(\mathbb {C} )} is the universal algebra generated by a single isometry, which is the classical Toeplitz algebra.

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