  Let E be an operator algebra on a Hilbert space with finite-dimensional
generated C*-algebra. A classification is given of the locally finite algebras
and the operator algebras 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 $V_E = Hom_u(E \otimes \sK, E \otimes \sK)$ 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 module
invariants.

Operator Algebras