HOMOLOGY FOR OPERATOR-ALGEBRAS .4. ON THE REGULAR CLASSIFICATION OF LIMITS OF 4-CYCLE ALGEBRAS

A 4-cycle algebra is a finite-dimensional digraph algebra (CSL algebra ) whose reduced digraph is a 4-cycle. A rigid embedding between such a lgebras is a direct sum of certain nondegenerate multiplicity one star -extendible embeddings. A complete classification is obtained for the regular isomorphism classes of direct systems A of 4-cycle algebras wi th rigid embeddings. The critical invariant is a binary relation in K( 0)A + H-1 A, generalising the scale of the K-0 group, called the joint scale. The joint scale encapsulates other invariants and compatibilit y conditions of regular isomorphism. These include the scale of H(1)A, the scale of H(0)A + H(1)A, sign compatibility, congruence compatibil ity and H0N1 coupling classes. These invariants are also important for lifting K-0 + H-1 isomorphisms to algebra isomorphisms; we resolve th is lifting problem for various classes of 4-cycle algebra direct syste ms. (C) 1997 Academic Press.