2016/04/29 by Rostislav Grigorchuk, Grigorchuk, Rostislav, Magdalena Musat +3
Mathematics · #20C07 #37A55 #46L05 #46L36 #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #math.GR #math.OA #msc:20C07 #msc:37A55 #msc:46L05 #msc:46L36
paper · pdf · doi:10.48550/arxiv.1604.08774
36 pages. To appear in Commentarii Math. Helvetici
arxiv created 2017/04/03 · arxiv updated 2017/04/04
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.