$K$-theory and $K$-homology relative to a $\textrm {II}_{\infty }$-factor

Abstract:

Let X be a compact space and M be a factor of type $\mathrm {II}_\infty$ acting on a separable Hilbert space. Let ${K_M}(X)$ denote the Grothendieck group generated by the semigroup of isomorphism classes of M-vector bundles over X, and, if X is also metric, let $\operatorname {Ext}^M(X)$ denote the group of equivalence classes of extensions of $C(X)$ relative to M. We show that ${K_M}(X)$ is the direct sum of the even-dimensional Čech cohomology groups of X, and that $\operatorname {Ext}^M(X)$ is the direct product of the odd-dimensional Čech homology groups of X.