Just-infinite $C^{\ast }$-algebras

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 $C^{\ast }$-algebras, i.e. infinite dimensional $C^{\ast }$-algebras for which all proper quotients are finite dimensional. We give a classification of such $C^{\ast }$-algebras in terms of their primitive ideal space, that leads to a trichotomy. We show that just-infinite, residually finite dimensional $C^{\ast }$-algebras do exist by giving an explicit example of (the Bratteli diagram of) an AF-algebra with these properties.

Further, we discuss when $C^{\ast }$-algebras and ${}^{\ast }$-algebras associated with a discrete group are just-infinite. If $\mathcal{G}$ 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 $\mathbb{C}[\mathcal{G}]$ and its group $C^{\ast }$-algebra $C^{\ast }(\mathcal{G})$ are not just-infinite. Furthermore, we show that the algebra $B=\pi (\mathbb{C}[\mathcal{G}])$ under the Koopman representation $\pi$ of $\mathcal{G}$ associated with its canonical action on a binary rooted tree is just-infinite. It remains an open problem whether the residually finite dimensional $C^{\ast }$-algebra $C_{\pi }^{\ast }(\mathcal{G})$ is just-infinite.