Fredholm module

Fredholm modules represent abstract elliptic pseudo-differential operators.

Definition 1.

An odd Fredholm module $(\mathscr{H},F)$ over a $C^{*}$-algebra $A$ is given by an involutive representation $\pi$ of $A$ on a Hilbert space $\mathscr{H}$, together with an operator $F$ on $\mathscr{H}$ such that $F=F^{*}$, $F^2=1\mathrm{I}$ and $[F,\pi (a)]\in 𝕂(\mathscr{H})$ for all $a\in A$.

Definition 2.

An even Fredholm module $(\mathscr{H},F,\mathrm{\Gamma })$ is given by an odd Fredholm module $(\mathscr{H},F)$ together with a ${\mathbb{Z}}_2$-grading $\mathrm{\Gamma }$ on $\mathscr{H}$, $\mathrm{\Gamma }={\mathrm{\Gamma }}^{*}$, ${\mathrm{\Gamma }}^2=1\mathrm{I}$, such that $\mathrm{\Gamma }\pi (a)=\pi (a)\mathrm{\Gamma }$ and $\mathrm{\Gamma }F=-F\mathrm{\Gamma }$.

Definition 3.

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

Example 1 (Fredholm modules over $\mathrm{C}$)

An even Fredholm module $\mathrm{(}\mathrm{H}\mathrm{,}F\mathrm{,}\mathrm{\Gamma }\mathrm{)}$ over $\mathrm{C}$ is given by

$\mathscr{H}$ $=$

$F$ $=$

$\mathrm{\Gamma }$ $=$

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