An invariant for locally finite dimensional semisimple algebras.

Abstract

Complete invariants were found for the category of unital direct limits of finite dimensional semisimple complex algebras and the category of unital direct limits of finite dimensional semisimple real algebras by G. A. Elliott ( (E)) and by K. R. Goodearl and D. E. Handelman ( (GH)) respectively. We are naturally led to consider similar complete invariants for other algebras of this type. For other fields, the situation is much more complicated, since the set of division rings containing a field F that is neither real closed nor algebraically closed is infinite (even ignoring the noncommutative ones). So let $\Omega=\{D\sb{i}\}$ be a finite set of finite dimensional division algebras, we shall only study the categories of unital direct limits of finite direct products of matrix algebras involving just this set of division rings. The conjecture of (GH) concerning a proposed complete invariant for direct limit algebras is simplified, and we show that this invariant (essentially a diagram of ordered $K\sb0$-groups) is complete, establishing the conjecture.