# Fredholm module

Fredholm modules represent abstract elliptic pseudo-differential operators.

###### Definition 1.

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

###### Definition 2.

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

###### 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 $\mathbb{C}$)

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

 $\displaystyle\mathord{\mathcal{H}}$ $\displaystyle=$ $\displaystyle\mathbb{C}^{k}\oplus\mathbb{C}^{k}\quad\mbox{with\ }\pi(a)=\left(% \begin{array}[]{cc}a\mathord{\mathrm{1\!\!\!\>I}}_{k}&0\\ 0&0\end{array}\right),$ $\displaystyle F$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}0&\mathord{\mathrm{1\!\!\!\>I}}_{k}\\ \mathord{\mathrm{1\!\!\!\>I}}_{k}&0\end{array}\right),$ $\displaystyle\Gamma$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}\mathord{\mathrm{1\!\!\!\>I}}_{k}&0\\ 0&-\mathord{\mathrm{1\!\!\!\>I}}_{k}\end{array}\right).$
Title Fredholm module FredholmModule 2013-03-22 12:57:43 2013-03-22 12:57:43 mhale (572) mhale (572) 6 mhale (572) Definition msc 19K33 msc 46L87 msc 47A53 KHomology