Fredholm module

Fredholm modules represent abstract elliptic pseudo-differential operators.

Definition 1.

An odd Fredholm module (,F) over a C*-algebra A is given by an involutive representation π of A on a Hilbert spaceMathworldPlanetmath , together with an operator F on such that F=F*, F2=1I and [F,π(a)]𝕂() for all aA.

Definition 2.

An even Fredholm module (,F,Γ) is given by an odd Fredholm module (,F) together with a 2-grading Γ on , Γ=Γ*, Γ2=1I, such that Γπ(a)=π(a)Γ and ΓF=-FΓ.

Definition 3.

A Fredholm module is called degenerate if [F,π(a)]=0 for all aA. Degenerate Fredholm modules are homotopic to the 0-module.

Example 1 (Fredholm modules over C)

An even Fredholm module (H,F,Γ) over C is given by

= kkwith π(a)=(a1Ik000),
F = (01Ik1Ik0),
Γ = (1Ik00-1Ik).
Title Fredholm module
Canonical name FredholmModule
Date of creation 2013-03-22 12:57:43
Last modified on 2013-03-22 12:57:43
Owner mhale (572)
Last modified by mhale (572)
Numerical id 6
Author mhale (572)
Entry type Definition
Classification msc 19K33
Classification msc 46L87
Classification msc 47A53
Related topic KHomology