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

Further, we discuss when -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 -algebra are not just-infinite. Furthermore, we show that the algebra under the Koopman representation of associated with its canonical action on a binary rooted tree is just-infinite. It remains an open problem whether the residually finite dimensional -algebra is just-infinite.

Cite this article

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

DOI 10.4171/CMH/432