# 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 $A$ 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 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 $E$ is a countably generated Hilbert $A$-module, there is an isometric isomorphism $E\oplus \ell ^{2}(A)\cong \ell ^{2}(A).$

## Maps between Hilbert modules

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

### Adjointable maps

A map (not necessarily linear) $T\colon E\to F$ is defined to be adjointable if there is another map $T^{*}\colon F\to E$, known as the adjoint of $T$, such that for every $e\in E$ and $f\in F$,
    $\langle f,Te\rangle =\langle T^{*}f,e\rangle .$
Both $T$ and $T^{*}$ are then automatically linear and also $A$-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, $T^{*}$ is unique (if it exists) and itself adjointable with adjoint $T$. If $S\colon F\to G$ is a second adjointable map, $ST$ is adjointable with adjoint $S^{*}T^{*}$.
The adjointable operators $E\to F$ form a subspace $\mathbb {B} (E,F)$ of ${\mathcal {L}}(E,F)$, which is complete in the operator norm.
In the case $F=E$, the space $\mathbb {B} (E,E)$ of adjointable operators from $E$ to itself is denoted $\mathbb {B} (E)$, and is a C*-algebra.

### Compact adjointable maps

Given $e\in E$ and $f\in F$, the map $|f\rangle \langle e|\colon E\to F$ is defined, analogously to the rank one operators of Hilbert spaces, to be
    $g\mapsto f\langle e,g\rangle .$
This is adjointable with adjoint $|e\rangle \langle f|$.
The compact adjointable operators $\mathbb {K} (E,F)$ are defined to be the closed span of
    $\{|f\rangle \langle e|\mid e\in E,\;f\in F\}$
in $\mathbb {B} (E,F)$.
As with the bounded operators, $\mathbb {K} (E,E)$ is denoted $\mathbb {K} (E)$. This is a (closed, two-sided) ideal of $\mathbb {B} (E)$.

## C*-correspondences

If $A$ and $B$ are C*-algebras, an $(A,B)$ C*-correspondence is a Hilbert $B$-module equipped with a left action of $A$ 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 $E$ is an $(A,B)$ and $F$ a $(B,C)$ correspondence, the algebraic tensor product $E\odot F$ of $E$ and $F$ as vector spaces inherits left and right $A$- and $C$-module structures respectively.
It can also be endowed with the $C$-valued sesquilinear form defined on pure tensors by
    $\langle e\odot f,e'\odot f'\rangle _{C}:=\langle f,\langle e,e'\rangle _{B}f\rangle _{C}.$
This is positive semidefinite, and the Hausdorff completion of $E\odot F$ in the resulting seminorm is denoted $E\otimes _{B}F$. The left- and right-actions of $A$ and $C$ extend to make this an $(A,C)$ correspondence.
The collection of C*-algebras can then be endowed with the structure of a bicategory, with C*-algebras as objects, $(A,B)$ correspondences as arrows $B\to A$, and isomorphisms of correspondences (bijective module maps that preserve inner products) as 2-arrows.

### Toeplitz algebra of a correspondence

Given a C*-algebra $A$, and an $(A,A)$ correspondence $E$, its Toeplitz algebra ${\mathcal {T}}(E)$ 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 $\mathbb {Z}$ , and the Cuntz algebras are all quotients of specific Toeplitz algebras.

#### Toeplitz representations

A Toeplitz representation of $E$ in a C*-algebra $D$ is a pair $(S,\phi )$ of a linear map $S\colon E\to D$ and a homomorphism $\phi \colon A\to D$ such that
* $S$ is "isometric":
    $S(e)^{*}S(f)=\phi (\langle e,f\rangle )$ for all $e,f\in E$,
* $S$ resembles a bimodule map:
    $S(ae)=\phi (a)S(e)$ and $S(ea)=S(e)\phi (a)$ for $e\in E$ and $a\in A$.

#### Toeplitz algebra

The Toeplitz algebra ${\mathcal {T}}(E)$ is the universal Toeplitz representation. That is, there is a Toeplitz representation $(T,\iota )$ of $E$ in ${\mathcal {T}}(E)$ such that if $(S,\phi )$ is any Toeplitz representation of $E$ (in an arbitrary algebra $D$) there is a unique *-homomorphism $\Phi \colon {\mathcal {T}}(E)\to D$ such that $S=\Phi \circ T$ and $\phi =\Phi \circ \iota$.

#### Examples

If $A$ is taken to be the algebra of complex numbers, and $E$ the vector space $\mathbb {C} ^{n}$, endowed with the natural $(\mathbb {C} ,\mathbb {C} )$-bimodule structure, the corresponding Toeplitz algebra is the universal algebra generated by $n$ isometries with mutually orthogonal range projections.
In particular, ${\mathcal {T}}(\mathbb {C} )$ is the universal algebra generated by a single isometry, which is the classical Toeplitz algebra.