Hilbert C*-module

Template:Short description 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

$⟨\cdot ,\cdot {⟩}_A:E\times E\to A$

that satisfies the following properties:

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

$⟨x,y\alpha +z\beta {⟩}_A=⟨x,y{⟩}_A\alpha +⟨x,z{⟩}_A\beta$

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

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

$⟨x,ya{⟩}_A=⟨x,y{⟩}_{A}a$

• For all $x$, $y$ in $E$:

$⟨x,y{⟩}_A=⟨y,x{⟩}_A^{*},$

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 $E$:

$⟨x,x{⟩}_A\ge 0$

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

$⟨x,x{⟩}_A=0⟺x=0.$

(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 $E$:

$⟨x,y{⟩}_A⟨y,x{⟩}_A\le \Vert ⟨y,y{⟩}_A\Vert ⟨x,x{⟩}_A$

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

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

$\Vert x\Vert =\Vert ⟨x,x{⟩}_A{\Vert }^{\frac{1}{2}}.$

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 $E$

$a_{\lambda }\to a\Rightarrow x{a}_{\lambda }\to xa.$

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

$x{e}_{\lambda }\to x.$

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

Let

$⟨E,E{⟩}_A=\mathrm{span}\{⟨x,y{⟩}_A\mid x,y\in E\},$

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

Examples

Hilbert spaces

Since the complex numbers $ℂ$ are a C*-algebra with an involution given by complex conjugation, a complex Hilbert space $\mathscr{H}$ is a Hilbert $ℂ$-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 $\pi :E\to X$ 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

$⟨\sigma ,\rho {⟩}_{C(X)}(x):=g(\sigma (x),\rho (x)).$

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 $X$. Template:Cn

C*-algebras

Any C*-algebra $A$ is a Hilbert $A$-module with the action given by right multiplication in $A$ and 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 $A$

$A^n=\underset{i=1}{\overset{n}{⨁}}A$

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

$⟨(a_i),(b_i){⟩}_A=\sum _{i=1}^n{a}_i^{*}{b}_i.$

If $p$ is a projection in the C*-algebra $M_n(A)$, then $p{A}^n$ 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 $A$

${\ell }_2(A)={\mathscr{H}}_A=\{(a_i)|\sum _{i=1}^{\mathrm{\infty }}{a}_i^{*}{a}_i\text{ converges in }A\}.$

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 $ℒ(E,F)$, 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 $ℂ$ these reduce to bounded and compact operators on Hilbert spaces respectively.

Adjointable maps

A map (not necessarily linear) $T:E\to F$ is defined to be adjointable if there is another map $T^{*}:F\to E$, known as the adjoint of $T$, such that for every $e\in E$ and $f\in F$,

$⟨f,Te⟩=⟨T^{*}f,e⟩.$

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 $S:F\to G$ is a second adjointable map, $ST$ is adjointable with adjoint $S^{*}{T}^{*}$.

The adjointable operators $E\to F$ form a subspace $𝔹(E,F)$ of $ℒ(E,F)$, which is complete in the operator norm.

In the case $F=E$, the space $𝔹(E,E)$ of adjointable operators from $E$ to itself is denoted $𝔹(E)$, and is a C*-algebra.

Compact adjointable maps

Given $e\in E$ and $f\in F$, the map $|f⟩⟨e|:E\to F$ is defined, analogously to the rank one operators of Hilbert spaces, to be

$g\mapsto f⟨e,g⟩.$

This is adjointable with adjoint $|e⟩⟨f|$.

The compact adjointable operators $𝕂(E,F)$ are defined to be the closed span of

$\{|f⟩⟨e|\mid e\in E,f\in F\}$

in $𝔹(E,F)$.

As with the bounded operators, $𝕂(E,E)$ is denoted $𝕂(E)$. This is a (closed, two-sided) ideal of $𝔹(E)$.

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

$⟨e\odot f,e^{\prime }\odot f^{\prime }{⟩}_C:=⟨f,⟨e,e^{\prime }{⟩}_{B}f{⟩}_C.$

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 $𝒯(E)$ 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 $(S,\varphi )$ of a linear map $S:E\to D$ and a homomorphism $\varphi :A\to D$ such that

• $S$ is "isometric":

$S(e{)}^{*}S(f)=\varphi (⟨e,f⟩)$ for all $e,f\in E$,

• $S$ resembles a bimodule map:

$S(ae)=\varphi (a)S(e)$ and $S(ea)=S(e)\varphi (a)$ for $e\in E$ and $a\in A$.

Toeplitz algebra

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

Examples

If $A$ is taken to be the algebra of complex numbers, and $E$ the vector space ${ℂ}^n$, endowed with the natural $(ℂ,ℂ)$-bimodule structure, the corresponding Toeplitz algebra is the universal algebra generated by $n$ isometries with mutually orthogonal range projections.

In particular, $𝒯(ℂ)$ is the universal algebra generated by a single isometry, which is the classical Toeplitz algebra.

See also

• Operator algebra

References

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External links

• Template:MathWorld
• Hilbert C*-Modules Home Page, a literature list