Invariants for Approximately Finitely Acting Operator Algebras
by
Stephen Power
Lancaster University

I'll survey very recent variants and generalizations of Elliott's K₀ classification of AF C*-algebras to the setting of nonself-adjoint operator algebras.

There are two strategies which I'll whimsically coin ``augmentation" and ``recreation".

In the first mindset, for a particular class of algebras one augments the K₀ invariants (which detemine the maximal self-adjoint subalgebra) by extra invariants appropriate to obtain the necessary uniqueness and existence properties for classification. This approach is proving succesful for certain limits of cycle algebras, where the augmentation is by a partial isometry homology group H₁ and associated scales in K₀ \oplusH₁. (Joint work with Allan Donsig.) Another augmentation which is effective for limits of digraph algebras which are ``nearly" self-adjoint comes via ``enlarged" Bratteli diagrams whose nodes correspond to digraph edges (rather than vertices) of the reduced digraph of the building block subalgebras.

With ``recreation" one defines Grothendieck group invariants afresh: for limits A of direct sums of operator algebras of the form

E \otimesMn1 \oplus... \oplusE \otimesMnk,

with E fixed and ``finitely acting" (contained in M_m), the approximately inner conjugacy classes of morphisms E --> A \otimesK give an abelian semigroup Hom_a(E, A\otimesK) with a natural metric. Moreover if the morphisms defining A are constrained one may similarly restrict morphisms to define the appropriate metrized abelian semigroup of morphism classes. Such an approach, with the inclusion of scale and a natural right action from Hom_a(E, E \otimesK) leads to classifying invariants.

With all such strategies the question of the well-definedness of the invariants is an interesting issue.

Date received: April 9, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacw-20.