Just-infinite C* - algebras (Mikael Rordam, Copenhagen)

23.03.2017 16:15

There is a well-established notion of just infinite groups, i.e., infinite groups for which all proper quotients are finite. The residually finite just infinite groups are particularly interesting. They are either branch groups (e.g., Grigorchuk's group of intermediate growth) or hereditarily just infinite groups (eg. Z, the infinite dihedral group, and SLn(Z)). It is natural to consider the analogous notion for C*-algebras, whereby a C*-algebra is just-infinite if it is infinite dimensional and all its proper quotients are finite dimensional. The study of these C*-algebras was motivated by a question of Grigorchuk if the group C*-algebra associated with his group might have this property. The just-infinite residually finite dimensional C*-algebras turn out to be a new class of C*-algebras not studied before - even the existence of such algebras is not obvious. We discuss how these C*-algebras arise, both from a point of view of AF-algebras and from a point of view of groups, and we will also discuss the question of Grigorchuk. This is joint work with R. Grigorchuk and M. Musat.

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