Hilbert module

Definition 1.

A (right) pre-Hilbert module over a C*-algebra A is a right A-module equipped with an A-valued inner productMathworldPlanetmath -,-:×A, i.e. a sesquilinear pairing satisfying

u,va = u,va (1)
u,v = v,u* (2)
v,v 0, with v,v=0 iff v=0, (3)

for all u,v and aA. Note, positive definiteness is well-defined due to the notion of positivity for C*-algebras. The norm of an element v is defined by v=v,v.

Definition 2.

A (right) Hilbert module over a C*-algebra A is a right pre-Hilbert module over A which is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath with respect to the norm.

Example 1 (Hilbert spaces)

A complex Hilbert spaceMathworldPlanetmath is a Hilbert C-module.

Example 2 (C*-algebras)

A C*-algebra A is a Hilbert A-module with inner product a,b=a*b.

Definition 3.

A Hilbert A-B-bimodule is a (right) Hilbert module over a C*-algebra B together with a *-homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath π from a C*-algebra A to End().

Title Hilbert module
Canonical name HilbertModule
Date of creation 2013-03-22 13:01:01
Last modified on 2013-03-22 13:01:01
Owner mhale (572)
Last modified by mhale (572)
Numerical id 8
Author mhale (572)
Entry type Definition
Classification msc 46C05
Synonym C*-module
Related topic HilbertSpace
Related topic FinitelyGeneratedProjectiveModule
Defines pre-Hilbert module