Fredholm module

Remarks

Example: let $A = C(\mathbb{T})$ act on $H = L^2(\mathbb{T})$ by multiplication, and let $F = 2P - 1$, where $P$ is the orthogonal projection onto the Hardy space $H^2$ spanned by $z^n$, $n \ge 0$. Then $F$ is a self-adjoint unitary, and $[P, M_f]$ is compact for continuous $f$ (it is of finite rank for trigonometric polynomials), so this is an odd Fredholm module. Non-example: with $A = C(\mathbb{T})$ on $L^2(\mathbb{T})$, let $F = J$ be the flip $(J f)(z) = f(\bar{z})$. Then $F$ is a self-adjoint unitary, but $[F, M_z] = (M_{\bar{z}} - M_z) J$, and this is not compact, because multiplication by a nonzero continuous function on $L^2(\mathbb{T})$ is never compact. The module is degenerate if the three operators above are zero for all $a$; degenerate modules represent zero in K-homology.