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.

Definitions

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 $A$:

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

• 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 $ae_{\lambda }$ 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 $x1_{A}=x$ 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.

Examples

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 $E\oplus \ell ^{2}(A)\cong \ell ^{2}(A).$

See also

• Operator algebra

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