Just-infinite $ C^*$ -algebras and their invariants

Abstract

Just-infinite C* s, i.e., infinite dimensional C* s, whose proper quotients are finite dimensional, were investigated in \cite{GMR:JI}. One particular example of a just-infinite residually finite dimensional AF-algebras was constructed in \cite{GMR:JI}. In this paper we extend that construction by showing that each infinite dimensional metrizable Choquet simplex is affinely homeomorphic to the trace simplex of a just-infinite residually finite dimensional C*. The trace simplex of any unital residually finite dimensional C*{} is hence realized by a just-infinite one. We determine the trace simplex of the particular residually finite dimensional AF-algebras constructed in \cite{GMR:JI}, and we show that it has precisely one extremal trace of type II$ _1$ . We give a complete description of the Bratteli diagrams corresponding to residually finite dimensional AF-algebras. We show that a modification of any such Bratteli diagram, similar to the modification that makes an arbitrary Bratteli diagram simple, will yield a just-infinite residually finite dimensional AF-algebra.