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").^[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 C*-algebraic quantum group theory,^[6][7] and groupoid C*-algebras.

Definitions

Inner-product C*-modules

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

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

that satisfies the following properties:

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

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

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

• For all ${\displaystyle x}$, ${\displaystyle y}$ in ${\displaystyle E}$, and ${\displaystyle a}$ in ${\displaystyle A}$:

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

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

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

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

• For all ${\displaystyle x}$ in ${\displaystyle E}$:

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

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

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

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

Hilbert C*-modules

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

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

for ${\displaystyle x}$, ${\displaystyle y}$ in ${\displaystyle E}$.

On the pre-Hilbert module ${\displaystyle E}$, define a norm by

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

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

The action of ${\displaystyle A}$ on ${\displaystyle E}$ is continuous: for all ${\displaystyle x}$ in ${\displaystyle E}$

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

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

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

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

Let

${\displaystyle \langle E,E\rangle _{A}=\operatorname {span} \{\langle x,y\rangle _{A}\mid x,y\in E\},}$

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

Examples

Hilbert spaces

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

Vector bundles

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

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

and the inner product is given by

${\displaystyle \langle \sigma ,\rho \rangle _{C(X)}(x):=g(\sigma (x),\rho (x)).}$

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

C*-algebras

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

The (algebraic) direct sum of ${\displaystyle n}$ copies of ${\displaystyle A}$

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

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

${\displaystyle \langle (a_{i}),(b_{i})\rangle _{A}=\sum _{i=1}^{n}a_{i}^{*}b_{i}.}$

If ${\displaystyle p}$ is a projection in the C*-algebra ${\displaystyle M_{n}(A)}$, then ${\displaystyle pA^{n))$ is also a Hilbert ${\displaystyle A}$-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 ${\displaystyle A}$

${\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 ${\displaystyle A^{n))$), the resulting Hilbert ${\displaystyle A}$-module is called the standard Hilbert module over ${\displaystyle A}$.

The standard Hilbert module plays an important role in the proof of the Kasparov stabilization theorem which states that for any countably generated Hilbert ${\displaystyle A}$-module ${\displaystyle E}$ there is an isometric isomorphism ${\displaystyle E\oplus \ell ^{2}(A)\cong \ell ^{2}(A).}$ ^[11]