Approximately finitely acting operator algebras

Let E be an operator algebra on a Hilbert space with finite-dimensional C*-algebra C*(E). A classification is given of the locally finite algebras A(0) = alg lim(A(k), phi(k)) and the operator algebras A = (lim) double under bar (A(k), phi(k)) obtained as limits of direct sums of matrix algebras over E with respect to star-extendible homomorphisms. The invariants in the algebraic case consist of an additive semigroup. with scale, which is a right module for the semiring V-E = Hom(u)(E circle times K, E circle times K) of unitary equivalence classes of star-extendible homomorphisms. This semigroup is referred to as the dimension module invariant. In the operator algebra case the invariants consist of a metrized additive semigroup with scale and a contractive right module V-E-action. Subcategories of algebras determined by restricted classes of embeddings, such as 1-decomposable embeddings between digraph algebras, are also classified in terms of simplified dimension modules. (C) 2002 Elsevier Science (USA).