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. Hilbert C*-modules 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.

Inner-product C*-modules
Let $$A$$ be a C*-algebra (not assumed to be commutative or unital), its involution denoted by $${}^*$$. An inner-product $$A$$-module (or pre-Hilbert $$A$$-module) is a complex linear space $$E$$ equipped with a compatible right $A$-module structure, together with a map


 * $$ \langle \, \cdot \,, \, \cdot \,\rangle_A : E \times E \rightarrow A $$

that satisfies the following properties:


 * For all $$x$$, $$y$$, $$z$$ in $$E$$, and $$\alpha$$, $$\beta$$ in $$\mathbb{C}$$:


 * $$ \langle x ,y \alpha + z \beta \rangle_A =  \langle x, y \rangle_A \alpha + \langle x, z \rangle_A \beta $$


 * (i.e. the inner product is $$\mathbb{C}$$-linear in its second argument).


 * For all $$x$$, $$y$$ in $$E$$, and $$a$$ in $$A$$:
 * $$ \langle x, y a \rangle_A = \langle x, y \rangle_A a $$


 * For all $$x$$, $$y$$ in $$E$$:


 * $$ \langle x, y \rangle_A = \langle y, x \rangle_A^*,$$


 * from which it follows that the inner product is conjugate linear in its first argument (i.e. it is a sesquilinear form).


 * For all $$x$$ in $$E$$:


 * $$ \langle x, x \rangle_A \geq 0$$


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


 * $$ \langle x, x \rangle_A = 0 \iff x = 0.$$


 * (An element of a C*-algebra $$A$$ 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 $$A$$-module $$E$$:


 * $$\langle x, y \rangle_A \langle y, x \rangle_A \leq \Vert \langle y, y \rangle_A \Vert \langle x, x \rangle_A$$

for $$x$$, $$y$$ in $$E$$.

On the pre-Hilbert module $$E$$, define a norm by


 * $$\Vert x \Vert = \Vert \langle x, x \rangle_A \Vert^\frac{1}{2}.$$

The norm-completion of $$E$$, still denoted by $$E$$, is said to be a Hilbert $$A$$-module or a Hilbert C*-module over the C*-algebra $$A$$. The Cauchy–Schwarz inequality implies the inner product is jointly continuous in norm and can therefore be extended to the completion.

The action of $$A$$ on $$E$$ is continuous: for all $$x$$ in $$E$$


 * $$a_{\lambda} \rightarrow a \Rightarrow xa_{\lambda} \rightarrow xa.$$

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


 * $$ xe_\lambda \rightarrow x.$$

Whence it follows that $$EA$$ is dense in $$E$$, and $$x 1_A = x$$ when $$A$$ is unital. Let


 * $$ \langle E, E \rangle_A = \operatorname{span} \{ \langle x, y \rangle_A \mid x, y \in E \},$$

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

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

Vector bundles
If $$ X $$ is a locally compact Hausdorff space and $$ E $$ a vector bundle over $$ X $$ with projection $$\pi \colon E \to X$$ a Hermitian metric $$ g $$, then the space of continuous sections of $$ E $$ is a Hilbert $$ C(X) $$-module. Given sections $$\sigma, \rho$$ of $$ E $$ and $$ f \in C(X) $$ the right action is defined by
 * $$ \sigma f (x) = \sigma(x) f(\pi(x)),$$

and the inner product is given by
 * $$ \langle \sigma,\rho\rangle_{C(X)} (x):=g(\sigma(x),\rho(x)).$$

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

C*-algebras
Any C*-algebra $$ A $$ is a Hilbert $$ A $$-module with the action given by right multiplication in $$ A $$ and the inner product $$ \langle a, b \rangle = a^*b $$. By the C*-identity, the Hilbert module norm coincides with C*-norm on $$ A $$.

The (algebraic) direct sum of $$ n $$ copies of $$ A $$


 * $$ A^n = \bigoplus_{i=1}^n A$$

can be made into a Hilbert $$ A $$-module by defining


 * $$\langle (a_i), (b_i) \rangle_A = \sum_{i=1}^n a_i^* b_i.$$

If $$p$$ is a projection in the C*-algebra $$M_n(A)$$, then $$pA^n$$ is also a Hilbert $$A$$-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 $$ A $$


 * $$ \ell_2(A)= \mathcal{H}_A = \Big\{ (a_i) | \sum_{i=1}^{\infty} a_i^{*}a_i\text{ converges in }A \Big\}.$$

Endowed with the obvious inner product (analogous to that of $$ A^n $$), the resulting Hilbert $$ A $$-module is called the standard Hilbert module over $$ A $$.

The standard Hilbert module plays an important role in the proof of the Kasparov stabilization theorem which states that for any countably generated Hilbert $$A$$-module $$E$$ there is an isometric isomorphism $$E \oplus \ell^2(A) \cong \ell^2(A). $$