Hilbert C*-module

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

:$\backslash langle\; \backslash ,\; \backslash cdot\; \backslash ,\; ,\; \backslash ,\; \backslash cdot\; \backslash ,\backslash rangle\_A\; :\; E\; \backslash times\; E\; \backslash rightarrow\; A$

that satisfies the following properties:

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

::$\backslash langle\; x\; ,y\; \backslash alpha\; +\; z\; \backslash beta\; \backslash rangle\_A\; =\; \backslash langle\; x,\; y\; \backslash rangle\_A\; \backslash alpha\; +\; \backslash langle\; x,\; z\; \backslash rangle\_A\; \backslash beta$

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

• For all $x$, $y$ in $E$, and $a$ in $A$:

::$\backslash langle\; x,\; y\; a\; \backslash rangle\_A\; =\; \backslash langle\; x,\; y\; \backslash rangle\_A\; a$

• For all $x$, $y$ in $E$:

::$\backslash langle\; x,\; y\; \backslash rangle\_A\; =\; \backslash langle\; y,\; x\; \backslash rangle\_A^*,$

: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 $E$:

::$\backslash langle\; x,\; x\; \backslash rangle\_A\; \backslash geq\; 0$

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

::$\backslash langle\; x,\; x\; \backslash rangle\_A\; =\; 0\; \backslash iff\; x\; =\; 0.$

:(An element of a C*-algebra $A$ is said to be positive if it is self-adjoint with non-negative spectrum.){{cite book| last = Arveson| first = William| authorlink = William Arveson|title = An Invitation to C*-Algebras| publisher = Springer-Verlag| year = 1976| page = 35}}In the case when $A$ is non-unital, the spectrum of an element is calculated in the C*-algebra generated by adjoining a unit to $A$.

An analogue to the Cauchy–Schwarz inequality holds for an inner-product $A$-module $E$:This result in fact holds for semi-inner-product $A$-modules, which may have non-zero elements $A$ such that $\backslash langle\; x\; ,\; x\; \backslash rangle\_A$

= 0, as the proof does not rely on the nondegeneracy property.

:$\backslash langle\; x,\; y\; \backslash rangle\_A\; \backslash langle\; y,\; x\; \backslash rangle\_A\; \backslash leq\; \backslash Vert\; \backslash langle\; y,\; y\; \backslash rangle\_A\; \backslash Vert\; \backslash langle\; x,\; x\; \backslash rangle\_A$

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

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

:$\backslash Vert\; x\; \backslash Vert\; =\; \backslash Vert\; \backslash langle\; x,\; x\; \backslash rangle\_A\; \backslash Vert^\backslash frac\{1\}\{2\}.$

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 $E$

:$a\_\{\backslash lambda\}\; \backslash rightarrow\; a\; \backslash Rightarrow\; xa\_\{\backslash lambda\}\; \backslash rightarrow\; xa.$

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

:$xe\_\backslash lambda\; \backslash rightarrow\; x.$

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

Let

:$\backslash langle\; E,\; E\; \backslash rangle\_A\; =\; \backslash operatorname\{span\}\; \backslash \{\; \backslash langle\; x,\; y\; \backslash rangle\_A\; \backslash mid\; x,\; y\; \backslash in\; E\; \backslash \},$

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

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

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

::$\backslash sigma\; f\; (x)\; =\; \backslash sigma(x)\; f(\backslash pi(x)),$

and the inner product is given by

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

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 $X$. {{cn|date=May 2023}}

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

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

:$A^n\; =\; \backslash bigoplus\_\{i=1\}^n\; A$

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

:$\backslash langle\; (a\_i),\; (b\_i)\; \backslash rangle\_A\; =\; \backslash sum\_\{i=1\}^n\; a\_i^*\; b\_i.$

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.

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

:$\backslash ell\_2(A)=\; \backslash mathcal\{H\}\_A\; =\; \backslash Big\backslash \{\; (a\_i)\; |\; \backslash sum\_\{i=1\}^\{\backslash infty\}\; a\_i^\{*\}a\_i\backslash text\{\; converges\; in\; \}A\; \backslash Big\backslash \}.$

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 $E\; \backslash oplus\; \backslash ell^2(A)\; \backslash cong\; \backslash ell^2(A).$ {{cite journal| last = Kasparov| first = G. G.| title = Hilbert C*-modules: Theorems of Stinespring and Voiculescu| journal = Journal of Operator Theory| volume = 4| pages = 133–150| publisher = ThetaFoundation| year = 1980}}

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 $\backslash mathcal\{L\}(E,F)$,

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 $\backslash mathbb\{C\}$ these reduce to

bounded and compact operators on Hilbert spaces respectively.

A map (not necessarily linear)

$T\; \backslash colon\; E\; \backslash to\; F$ is defined to be adjointable if there

is another map $T^*\; \backslash colon\; F\; \backslash to\; E$, known as the adjoint

of $T$, such that for every

$e\; \backslash in\; E$ and $f\; \backslash in\; F$,

:$\backslash langle\; f,\; Te\; \backslash rangle\; =\; \backslash langle\; T^*\; f,\; e\; \backslash rangle.$

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 $S\; \backslash colon\; F\; \backslash to\; G$

is a second adjointable map, $ST$ is adjointable with adjoint

$S^*\; T^*$.

The adjointable operators $E\; \backslash to\; F$ form a subspace $\backslash mathbb\{B\}(E,F)$

of $\backslash mathcal\{L\}(E,F)$, which is complete in the operator norm.

In the case $F\; =\; E$, the space $\backslash mathbb\{B\}(E,E)$ of

adjointable operators from $E$ to itself is denoted $\backslash mathbb\{B\}(E)$, and is a

C*-algebra.Wegge-Olsen 1993, pp. 240-241.

Given $e\; \backslash in\; E$ and $f\; \backslash in\; F$, the map

$|\; f\; \backslash rangle\; \backslash langle\; e\; |\; \backslash colon\; E\; \backslash to\; F$ is defined, analogously to the

rank one operators of Hilbert spaces, to be

:$g\; \backslash mapsto\; f\; \backslash langle\; e,\; g\; \backslash rangle.$

This is adjointable with adjoint $|\; e\; \backslash rangle\; \backslash langle\; f\; |$.

The compact adjointable operators $\backslash mathbb\{K\}(E,F)$ are defined to be the closed span

of

:$\backslash \{\; |\; f\; \backslash rangle\; \backslash langle\; e\; |\; \backslash mid\; e\; \backslash in\; E,\; \backslash ;\; f\; \backslash in\; F\; \backslash \}$

in $\backslash mathbb\{B\}(E,F)$.

As with the bounded operators, $\backslash mathbb\{K\}(E,E)$ is denoted

$\backslash mathbb\{K\}(E)$. This is a

(closed, two-sided) ideal of

$\backslash mathbb\{B\}(E)$.Wegge-Olsen 1993, pp. 242-243.

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,Brown, Ozawa 2008, section 4.6.

and can be employed to put the structure of a bicategory on the collection of C*-algebras.Buss, Meyer, Zhu, 2013, section 2.2.

If $E$ is an $(A,B)$ and $F$ a $(B,C)$ correspondence,

the algebraic tensor product $E\; \backslash odot\; F$ 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

:$\backslash langle\; e\; \backslash odot\; f,\; e\text{'}\; \backslash odot\; f\text{'}\; \backslash rangle\_C\; :=\; \backslash langle\; f,\; \backslash langle\; e,\; e\text{'}\; \backslash rangle\_B\; f\; \backslash rangle\_C.$

This is positive semidefinite, and the Hausdorff completion of $E\; \backslash odot\; F$

in the resulting seminorm is denoted $E\; \backslash otimes\_B\; F$. The left- and right-actions of

$A$ and $C$ extend to make this an $(A,C)$ correspondence.Brown, Ozawa 2008, pp. 138-139.

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 $B\; \backslash to\; A$, and isomorphisms of correspondences (bijective module maps that preserve

inner products) as 2-arrows.Buss, Meyer, Zhu 2013, section 2.2.

Given a C*-algebra $A$, and an $(A,A)$ correspondence $E$,

its Toeplitz algebra $\backslash mathcal\{T\}(E)$ 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.Brown, Ozawa, 2008, section 4.6.

In particular, graph algebras , crossed products by $\backslash mathbb\{Z\}$ , and the

Cuntz algebras are all quotients of specific Toeplitz algebras.

A Toeplitz representationFowler, Raeburn, 1999, section 1. of $E$ in a C*-algebra $D$

is a pair $(S,\backslash phi)$

of a linear map $S\; \backslash colon\; E\; \backslash to\; D$ and a homomorphism

$\backslash phi\; \backslash colon\; A\; \backslash to\; D$ such that

• $S$ is "isometric":

:$S(e)^*\; S(f)\; =\; \backslash phi(\backslash langle\; e,\; f\; \backslash rangle)$ for all $e,f\; \backslash in\; E$,

• $S$ resembles a bimodule map:

:$S(a\; e)\; =\; \backslash phi(a)\; S(e)$ and $S(ea)\; =\; S(e)\; \backslash phi(a)$ for $e\; \backslash in\; E$ and $a\; \backslash in\; A$.

The Toeplitz algebra $\backslash mathcal\{T\}(E)$ is the universal Toeplitz representation.

That is, there is a Toeplitz representation $(T,\; \backslash iota)$ of $E$

in $\backslash mathcal\{T\}(E)$ such that if $(S,\backslash phi)$ is any Toeplitz representation

of $E$ (in an arbitrary algebra $D$) there is a unique *-homomorphism

$\backslash Phi\; \backslash colon\; \backslash mathcal\{T\}(E)\; \backslash to\; D$ such that $S\; =\; \backslash Phi\; \backslash circ\; T$

and $\backslash phi\; =\; \backslash Phi\; \backslash circ\; \backslash iota$.Fowler, Raeburn, 1999, Proposition 1.3.

If $A$ is taken to be the algebra of complex numbers, and $E$

the vector space $\backslash mathbb\{C\}^n$, endowed with the natural

$(\backslash mathbb\{C\},\backslash mathbb\{C\})$-bimodule structure, the corresponding Toeplitz algebra

is the universal algebra generated by $n$ isometries with mutually orthogonal

range projections.Brown, Ozawa, 2008, Example 4.6.10.

In particular, $\backslash mathcal\{T\}(\backslash mathbb\{C\})$ is the universal algebra generated by

a single isometry, which is the classical Toeplitz algebra.

• Operator algebra

{{Reflist}}

• {{cite book |last=Lance |first=E. Christopher|title=Hilbert C*-modules: A toolkit for operator algebraists |series=London Mathematical Society Lecture Note Series|year=1995 |publisher=Cambridge University Press |location=Cambridge, England}}
• {{cite book| last = Wegge-Olsen | first = N. E.| title = K-Theory and C*-Algebras | publisher = Oxford University Press | year = 1993}}
• {{cite book

| last1 = Brown

| first1 = Nathanial P.

| last2 = Ozawa

| first2 = Narutaka

| date = 2008

| title = C*-Algebras and Finite-Dimensional Approximations

| url = https://bookstore.ams.org/gsm-88

| publisher = American Mathematical Society

}}

• {{cite journal

| last1 = Buss

| first1 = Alcides

| last2 = Meyer

| first2 = Ralf

| last3 = Zhu

| first3 = Chenchang

| date = 2013

| title = A higher category approach to twisted actions on c* -algebras

| url =

| journal = Proceedings of the Edinburgh Mathematical Society

| volume = 56

| issue = 2

| pages = 387–426

| doi = 10.1017/S0013091512000259

| arxiv = 0908.0455

}}

• {{cite journal

| last1 = Fowler

| first1 = Neal J.

| last2 = Raeburn

| first2 = Iain

| date = 1999

| title = The Toeplitz algebra of a Hilbert bimodule

| url = https://www.jstor.org/stable/24900141

| journal = Indiana University Mathematics Journal

| volume = 48

| issue = 1

| pages = 155–181

| doi = 10.1512/iumj.1999.48.1639

| jstor = 24900141

| arxiv = math/9806093

}}

• {{MathWorld |title=Hilbert C*-Module |urlname=HilbertC-Star-Module}}
• [http://www.imn.htwk-leipzig.de/~mfrank/hilmod.html Hilbert C*-Modules Home Page], a literature list

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Category:C*-algebras

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