HOMOLOGY FOR OPERATOR-ALGEBRAS .1. SPECTRAL HOMOLOGY FOR REFLEXIVE ALGEBRAS

A stable homology theory is defined for completely distributive CSL al gebras in terms of the point-neighbourhood homology of the partially o rdered set of meet-irreducible elements of the invariant projection la ttice. This specialises to the simplicial homology of the underlying s implicial complex in the case of a digraph algebra. These groups are c omputable and useful. In particular it is shown that if the first spec tral homology group is trivial then Schur automorphisms are automatica lly quasispatial. This motivates the introduction of essential Hochsch ild cohomology which we define by using the point weak star closure of coboundaries in place of the usual coboundaries. (C) 1995 Academic Pr ess.