Given an ample groupoid, we construct a spectral sequence with groupoid homology integer coefficients on the second sheet, converging to K-groups of (reduced) C*-algebra, provided has torsion-free stabilizers and satisfies strong form Baum-Connes conjecture. The construction is based triangulated category approach conjecture developed by Meyer Nest. We also present few applications topological ...

By a paramedial groupoid we mean a groupoid satisfying the equation xy · zu = uy · zx. As it is easy to see, the equational theory of paramedial groupoids, as well as the equational theory based on any balanced equation, is decidable. In this paper we are going to investigate the equational theory of paramedial cancellation groupoids; by this we mean the set of all equations satisfied by parame...

Taking advantage of the quantale-theoretic description \'etale groupoids we study principal bundles, Hilsum-Skandalis maps, and Morita equivalence in terms modules on inverse quantal frames. The Hilbert module quantale sheaves leads naturally to a formulation bimodules that resemble imprimitivity C*-algebras.

We prove that Matui's AH conjecture holds for graph groupoids of infinite graphs. This is a which relates the topological full group an ample groupoid with homology groupoid. Our main result complements in finite case, makes true all covered by assumptions said conjecture. Furthermore, we observe arbitrary graphs, coincides $K$-theory its $C^*$-algebra.