nLab Fredholm module

Idea

For $A$ a C-star algebra, an odd Fredholm module over $A$ is a representation $\pi : A \to \mathcal{B}(\mathcal{H})$ of $A$ on a Hilbert space $\mathcal{H}$ together with a Fredholm operator $F$ on $\mathcal{H}$ such that $[F,\pi(a)] \in \mathcal{K}(\mathcal{H})$ for all $a \in A$.

(…)

(Here $\mathcal{B}(-)$ denotes bounded operators and $\mathcal{K}(-)$ denotes compact operators).

A Fredholm module defines a class in K-homology, hence in KK-theory.

Related concepts

• unbounded Fredholm module (spectral triple)

• K-homology

• KK-theory

• Dirac operator