Approximately Finitely Acting Operator Algebras

✍ Scribed by Stephen C. Power

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 345 KB
• Volume: 189
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2001.3858

✦ Synopsis

Let E be an operator algebra on a Hilbert space with finite-dimensional C g -algebra C g (E). A classification is given of the locally finite algebras A 0 = alg lim(A k , f k ) and the operator algebras A=K(A k , f 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 é K, E é 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.