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 byan 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 }$$=$