Hilbert C*-module

Hilbert C*-modules are mathematical objects that generalise the notion of a Hilbert space (which itself is a generalisation 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").^[1] In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke^[2] and Marc Rieffel, the latter in a paper that used Hilbert C*-modules to construct a theory of induced representations of C*-algebras.^[3] Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory,^[4] and provide the right framework to extend the notion of Morita equivalence to C*-algebras.^[5] 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 ,^[6][7] and groupoid C*-algebras.

Definitions[]

Inner-product A-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

${\displaystyle \langle \cdot ,\cdot \rangle :E\times E\rightarrow A}$

that satisfies the following properties:

• For all x, y, z in E, and α, β in C:

${\displaystyle \langle x,y\alpha +z\beta \rangle =\langle x,y\rangle \alpha +\langle x,z\rangle \beta }$

(i.e. the inner product is linear in its second argument).

• For all x, y in E, and a in A:

${\displaystyle \langle x,ya\rangle =\langle x,y\rangle a}$

• For all x, y in E:

${\displaystyle \langle x,y\rangle =\langle y,x\rangle ^{*},}$

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:

${\displaystyle \langle x,x\rangle \geq 0}$

and

${\displaystyle \langle x,x\rangle =0\iff x=0.}$

(An element of a C*-algebra A is said to be positive if it is self-adjoint with non-negative spectrum.)^[8][9]

Hilbert A-modules[]

An analogue to the Cauchy–Schwarz inequality holds for an inner-product A-module E:^[10]

${\displaystyle \langle x,y\rangle \langle y,x\rangle \leq \Vert \langle y,y\rangle \Vert \langle x,x\rangle }$

for x, y in E.

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

${\displaystyle \Vert x\Vert =\Vert \langle x,x\rangle \Vert ^{\frac {1}{2}}.}$

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

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

Similarly, if {e_λ} is an approximate unit for A (a net of self-adjoint elements of A for which ae_λ and e_λa tend to a for each a in A), then for x in E

${\displaystyle xe_{\lambda }\rightarrow x}$

whence it follows that EA is dense in E, and x1 = x when A is unital.

Let

${\displaystyle \langle E,E\rangle =\operatorname {span} \{\langle x,y\rangle |x,y\in E\},}$

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

Examples[]

Hilbert spaces[]

A complex Hilbert space H is a Hilbert C-module under its inner product, the complex numbers being a C*-algebra with an involution given by complex conjugation.

Vector bundles[]

If X is a locally compact Hausdorff space and E a vector bundle over X with a Riemannian metric g, then the space of continuous sections of E is a Hilbert C(X)-module. The inner product is given by

${\displaystyle \langle f,h\rangle (x):=g(f(x),h(x)).}$

The converse holds as well: Every countably generated Hilbert C*-module over a commutative 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 under the inner product <a,b> = 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

${\displaystyle A^{n}=\oplus _{1}^{n}A}$

can be made into a Hilbert A-module by defining

${\displaystyle \langle (a_{i}),(b_{i})\rangle =\sum a_{i}^{*}b_{i}.}$

One may also consider the following subspace of elements in the countable direct product of A

${\displaystyle \ell _{2}(A)={\mathcal {H}}_{A}=\{(a_{i})|\sum a_{i}^{*}a_{i}{\text{ converges in }}A\}.}$

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

See also[]

• Operator algebra

Notes[]

References[]

• Lance, E. Christopher (1995). Hilbert C*-modules: A toolkit for operator algebraists. London Mathematical Society Lecture Note Series. Cambridge, England: Cambridge University Press.

External links[]

• Weisstein, Eric W. "Hilbert C*-Module". MathWorld.
• Hilbert C*-Modules Home Page, a literature list