Just-infinite C^∗-algebras and Their Invariants

Just-infinite C^∗-algebras, that is, infinite dimensional C^∗-algebras, whose proper quotients are finite dimensional, were investigated in [3]. One particular example of a just-infinite residually finite dimensional AF-algebras was constructed in [3]. In this article, 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^∗-algebra. The trace simplex of any unital residually finite dimensional C^∗-algebra is hence realized by a just-infinite one. We determine the trace simplex of the particular residually finite dimensional AF-algebras constructed in [3], and we show that it has precisely one extremal trace of type II₁. 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.