# Just-infinite $C_{∗}$-algebras

### Rostislav Grigorchuk

Texas A&M University, College Station, USA### Magdalena Musat

University of Copenhagen, Denmark### Mikael Rørdam

University of Copenhagen, Denmark

## 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 $G$ 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 $C[G]$ and its group $C_{∗}$-algebra $C_{∗}(G)$ are not just-infinite. Furthermore, we show that the algebra $B=π(C[G])$ under the Koopman representation $π$ of $G$ associated with its canonical action on a binary rooted tree is just-infinite. It remains an open problem whether the residually finite dimensional $C_{∗}$-algebra $C_{π}(G)$ is just-infinite.

## Cite this article

Rostislav Grigorchuk, Magdalena Musat, Mikael Rørdam, Just-infinite $C_{∗}$-algebras. Comment. Math. Helv. 93 (2018), no. 1, pp. 157–201

DOI 10.4171/CMH/432