Grothendieck group invariants for partly self-adjoint operator algebras

Abstract

Various partially ordered Grothendieck group invariants are introduced for general operator algebras and these are used in the classification of direct systems and direct limits of finite-dimensional complex incidence algebras with common reduced digraph H (systems of H-algebras). In particular the dimension distribution group G(A; C), defined for an operator algebra A and a self-adjoint subalgebra C, generalises both the K0 group of a sigma unital C*-algebra B and the spectrum (fundamental relation) R(A) of a regular limit A of triangular digraph algebras. This invariant is more economical and computable than the so called regular Grothendieck group which nevertheless forms the basis for a complete classification of regular systems of H-algebras.