Homology for Operator 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 algebras is a direct sum of certain nondegenerate multiplicity one star-extendible embeddings. A complete classification is obtained for the regular isomorphism classes of direct systemsAof 4-cycle algebras with rigid embeddings. The critical invariant is a binary relation inK0A⊕H1A, generalising the scale of theK0group, called the joint scale. The joint scale encapsulates other invariants and compatibility conditions of regular isomorphism. These include the scale ofH1A, the scale ofH0A⊕H1A, sign compatibility, congruence compatibility andH0H1coupling classes. These invariants are also important for liftingK0⊕H1isomorphisms to algebra isomorphisms; we resolve this lifting problem for various classes of 4-cycle algebra direct systems