Hilbert C*-module

Hilbert C * -modules are considered in the mathematical branch of functional analysis. They play an important role in the construction of KK - theory, the elements of the groups occurring there are those modules with some additional structure. Hilbert -C * -modules are defined analogously to the Hilbert space with the inner product assumes values ​​in a C * algebra. They were introduced in 1953 by Irving Kaplansky in the case of commutative C * - algebras and 1973 by William Paschke for the general case.

Definition

It is a C * - algebra. A pre- Hilbert - module is a right B- module together with a picture, so that

The apparent analogy with the definition of a Hilbert space can be extended further. It shows the Cauchy- Schwarz inequality

And so receives a seminorm

, which is a standard for the condition 5, as shown. If the pre- Hilbert - Module respect to this norm complete, so it is called a Hilbert - module. The main difference between Hilbert spaces is that one can prove no set of projection, that is, there are sub - Hilbert - modules that are not constantly be projected.

Examples

• A C * -algebra with the definition of a Hilbert - module. Meanwhile, sub - Hilbert B -modules are closed right ideals. is a Hilbert - exactly countably generated module, ie there is a countable subset, so that the smallest amount of this sub-module is comprehensive if it is σ - unital.
• A Hilbert space is a Hilbert - module.
• For a C *-algebra is the space of all sequences converges for. With the definition becomes a Hilbert - module. Apparently, the separable sequence space of the square sum consequences.

Constructions

Direct sums

Direct sums of Hilbert - modules are the definition apparently back Hilbert - modules.

Algebras of operators

For two Hilbert - modules and be the set of all operators for which there is an operator such that for all and. We show that such operators - linear and form a closed subspace of continuous linear operators. is a C * - algebra with the operator norm and the involution. In the special case is isomorphic to the algebra of multipliers.

Certain operators can set out as follows in analogy to the one-dimensional operators defined on Hilbert spaces. If and so was. It is easy to confirm the formula and thus. The generated by these operators, closed subspace is denoted by and is called its elements, the compact operators from to, even if it is in general not to compact operators in the sense of Banach space theory. Easily one confirmed for one, from which it follows, and quite similar too. This is a two-sided ideal. The ideal of compact operators Apparently on.

These constructions are related as follows: For every C * - algebra and every Hilbert - module is isomorphic to the multiplier algebra. In particular, is a * - isomorphism, the maps on.

Unitary equivalence

Two Hilbert - modules and are called unitarily equivalent, in symbols, if a bijective linear mapping are with all.

Interior tensor

Let be a Hilbert - module, Hilbert - module and a * - homomorphism. By the formula to a left - module and can therefore form the algebraic tensor that by the definition of a right - module. By the formula

We obtain by means of linear expansion a sesquilinear form on which all rules from the definition of the pre- Hilbert - met on module to any point 5, ie there may be vectors of length 0. Out by dividing the space of vectors of length, that is, the space factor in accordance with transitions, and then completed to obtain a Hilbert - module, which is referred to and is called the inner tensor of the Hilbert -C * -modules.

Exterior tensor

There were again a Hilbert - Hilbert module and a - module. Then the algebraic tensor product is defined by

A law - module and is obtained from by means of linear expansion

A sesquilinear form. If the spatial tensor product of C *-algebras, one constructed by dividing out by vectors of length 0 and by expanding on the completions of a Hilbert - module, which is referred to and called the outer tensor product of Hilbert C * -modules.

Pushout

Is a Hilbert - module and a surjective * - homomorphism, so defining. If the quotient map, then by the definitions

Is to show the well- definedness of a Hilbert - module is called the pushout of respect. One can show that by conceives as a subalgebra of.

Graduates of Hilbert C * -modules

Especially for the KK - theory of Hilbert C *-algebras with additional structure, a so-called graduation, or more precisely a graduation be used. It should be a Graduate C * - algebra with Graduierungsautomorphismus, which means that it is

Then the direct sum decomposition is to graduation. A graduated Hilbert - module is a Hilbert - module together with a linear bijection such that

Again, we obtain a direct sum decomposition, where

And it follows that

Then obtained by the automorphism and also a graduation.

Graduates of Hilbert C * -modules are called unitarily equivalent if they as Hilbert C * -modules are unitarily equivalent to a unitary operator who receives graduation.

This generalizes the notions introduced above, without graduation, for every C * - algebra can be graduation means trivial and just as any Hilbert - module means.

In order to graduate well, you have two options, namely and. We therefore define

The above constructions can be defined also for graduate Hilbert C * -modules, where the tensor product is graduated to take and all occurring morphisms with the graduations must be compatible. The connected therewith details are very technical and are omitted here.

Stabilization set of Kasparov

For the KK - theory, the so-called stabilization set of Kasparov proves to be important. This rate applies for graduate and non- graduate Hilbert C * -modules, he says that already contains all countably generated Hilbert C * -modules as direct summands, and analogously for postgraduate modules. It is even something more:

• Is a countably generated Hilbert - module over a C *-algebra, then.
• Is a countably generated, graded Hilbert - module over a graded C * - algebra, so is.