Just-infinite C-algebras and Their Invariants

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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.

Original languageEnglish
JournalInternational Mathematics Research Notices
Issue number12
Pages (from-to)3621-3645
Number of pages25
Publication statusPublished - 2019

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