Just-infinite C*-algebras
Rostislav Grigorchuk, Magdalena Musat, Mikael R{\o}rdam

TL;DR
This paper introduces and classifies just-infinite C*-algebras, exploring their properties, examples, and connections to group algebras, revealing new instances and open problems in the field.
Contribution
It systematically studies just-infinite C*-algebras, providing a classification, explicit examples, and analyzing their relation to group C*-algebras and representations.
Findings
Classification of just-infinite C*-algebras into a trichotomy.
Existence of residually finite dimensional just-infinite AF-algebras.
Identification of a just-infinite algebra from a group action.
Abstract
By analogy with the well-established notions of just-infinite groups and just-infinite (abstract) algebras, we initiate a systematic study of just-infinite C*-algebras, i.e., infinite dimensional C*-algebras for which all proper quotients are finite dimensional. We give a classification of such C*-algebras in terms of their primitive ideal space that leads to a trichotomy. We show that just-infinite, residually finite dimensional C*-algebras do exist by giving an explicit example of (the Bratteli diagram of) an AF-algebra with these properties. Further, we discuss when C*-algebras and *-algebras associated with a discrete group are just-infinite. If is the Burnside-type group of intermediate growth discovered by the first named author, which is known to be just-infinite, then its group algebra and its group C*-algebra are not just-infinite. Furthermore, we show…
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