Homology for Operator Algebras III: Partial Isometry Homotopy and Triangular Algebras

Abstract. The partial isometry homology groups Hn defined in Power [17] and a related chain complex homology CH ∗ are calculated for various triangular operator algebras, including the disc algebra. These invariants are closely connected with K-theory. Simplicial homotopy reductions are used to identify both Hn and CHn for the lexicographic products A(G) ⋆A with A(G) a digraph algebra and A a triangular subalgebra of the Cuntz algebra Om. Specifically Hn(A(G) ⋆A)=Hn(∆(G)) ⊗Z K0(C ∗ (A)) and CHn(A(G) ⋆A) is the simplicial homology group Hn(∆(G); K0(C ∗ (A))) with coefficients i