homology over traced *-algebras

Given a unital complex *-algebra , a tracial positive linear functional on that factors through a *-representation of on Hilbert space, and an -module possessing a resolution by finitely generated projective -modules, we construct homology spaces for . Each is a Hilbert space equipped with a *-representation of , independent (up to unitary equivalence) of the given resolution of . A short exact sequence of -modules gives rise to a long weakly exact sequence of homology spaces. There is a Künneth formula for tensor products. The von Neumann dimension which is defined for -invariant subspaces of gives well-behaved Betti numbers and an Euler characteristic for with respect to and .