Английская Википедия:Hilbert C*-module

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

<math> \langle \, \cdot \, , \, \cdot \,\rangle_A : E \times E \rightarrow A </math>

that satisfies the following properties:

• For all <math>x</math>, <math>y</math>, <math>z</math> in <math>E</math>, and <math>\alpha</math>, <math>\beta</math> in <math>\mathbb{C}</math>:

<math> \langle x ,y \alpha + z \beta \rangle_A = \langle x, y \rangle_A \alpha + \langle x, z \rangle_A \beta </math>

(i.e. the inner product is <math>\mathbb{C}</math>-linear in its second argument).

• For all <math>x</math>, <math>y</math> in <math>E</math>, and <math>a</math> in <math>A</math>:

<math> \langle x, y a \rangle_A = \langle x, y \rangle_A a </math>

• For all <math>x</math>, <math>y</math> in <math>E</math>:

<math> \langle x, y \rangle_A = \langle y, x \rangle_A^*,</math>

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

• For all <math>x</math> in <math>E</math>:

<math> \langle x, x \rangle_A \geq 0</math>

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

<math> \langle x, x \rangle_A = 0 \iff x = 0.</math>

(An element of a C*-algebra <math>A</math> 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 <math>A</math>-module <math>E</math>:

<math>\langle x, y \rangle_A \langle y, x \rangle_A \leq \Vert \langle y, y \rangle_A \Vert \langle x, x \rangle_A</math>

for <math>x</math>, <math>y</math> in <math>E</math>.

On the pre-Hilbert module <math>E</math>, define a norm by

<math>\Vert x \Vert = \Vert \langle x, x \rangle_A \Vert^\frac{1}{2}.</math>

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

The action of <math>A</math> on <math>E</math> is continuous: for all <math>x</math> in <math>E</math>

<math>a_{\lambda} \rightarrow a \Rightarrow xa_{\lambda} \rightarrow xa.</math>

Similarly, if <math>(e_\lambda)</math> is an approximate unit for <math>A</math> (a net of self-adjoint elements of <math>A</math> for which <math>a e_\lambda</math> and <math>e_\lambda a</math> tend to <math>a</math> for each <math>a</math> in <math>A</math>), then for <math>x</math> in <math>E</math>

<math> xe_\lambda \rightarrow x.</math>

Whence it follows that <math>EA</math> is dense in <math>E</math>, and <math>x 1_A = x</math> when <math>A</math> is unital.

Let

<math> \langle E, E \rangle_A = \operatorname{span} \{ \langle x, y \rangle_A \mid x, y \in E \},</math>

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

Examples

Hilbert spaces

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

Vector bundles

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

<math> \sigma f (x) = \sigma(x) f(\pi(x)),</math>

and the inner product is given by

<math> \langle \sigma,\rho\rangle_{C(X)} (x):=g(\sigma(x),\rho(x)).</math>

The converse holds as well: Every countably generated Hilbert C*-module over a commutative unital C*-algebra <math>A = C(X)</math> is isomorphic to the space of sections vanishing at infinity of a continuous field of Hilbert spaces over <math> X </math>. Шаблон:Cn

C*-algebras

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

The (algebraic) direct sum of <math> n </math> copies of <math> A </math>

<math> A^n = \bigoplus_{i=1}^n A</math>

can be made into a Hilbert <math> A </math>-module by defining

<math>\langle (a_i), (b_i) \rangle_A = \sum_{i=1}^n a_i^* b_i.</math>

If <math>p</math> is a projection in the C*-algebra <math>M_n(A)</math>, then <math>pA^n</math> is also a Hilbert <math>A</math>-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 <math> A </math>

<math> \ell_2(A)= \mathcal{H}_A = \Big\{ (a_i) | \sum_{i=1}^{\infty} a_i^{*}a_i\text{ converges in }A \Big\}.</math>

Endowed with the obvious inner product (analogous to that of <math> A^n </math>), the resulting Hilbert <math> A </math>-module is called the standard Hilbert module over <math> A </math>.

The standard Hilbert module plays an important role in the proof of the Kasparov stabilization theorem which states that for any countably generated Hilbert <math>A</math>-module <math>E</math> there is an isometric isomorphism <math>E \oplus \ell^2(A) \cong \ell^2(A). </math>

See also

• Operator algebra

References

• Шаблон:Cite book

External links

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