Homology for Operator Algebras III Partial Isometry Homotopy and Triangular Algebras.pdf

New York Journal of Mathematics New York J Math    Homology for Operator Algebras III Partial Isometry Homotopy and Triangular Algebras S C Power Abstract The partial isometry homology groups Hn dened in Power  and a related chain complex homology CH are calculated for various triangu lar operator algebras including the disc algebra These invariants are closely connected with Ktheory Simplicial homotopy reductions are used to iden tify both Hn and CHn for the lexicographic products AG  A with AG a digraph algebra and A a triangular subalgebra of the Cuntz algebra Om Specically HnAG  A  HnG ZKC A and CHnAG  A is the simplicial homology group HnGKCA with coecients in KCA Contents  The Partial Isometry Homology HnA C   Vanishing Homology  The Proof of Theorem   Kregular Inclusion and Homotopy  The Cuntz Algebras and TOm  The Proof of Theorem    The Partial Isometry Chain Complex Homology  References  Taking the perspective that equivalence classes of projections in the stable al gebra of a nonselfadjoint algebra A may be viewed as simplexes one can often identify the resulting homology group HA as KC A Analogously viewing partial isometries in the stable algebra as simplexes one can similarly formulate higher order homology group invariants for A This was done recently in Power  with the intention of extending the limit homology groups introduced by Davidson and Power  for regular limit algebras to subalgebras of general Calgebras In the present paper we develop further these homology invariants together with a related chain complex homology CH also derived from partial isometries in the stable algebra Our main purpose is to indicate methods for calculation mainly in Received January   Mathematics Subject Classication D  K  Key words and phrases operator algebra homology group nonselfadjoint Cunt