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Exactness and the Kadison-Kaplansky conjecture. (English) Zbl 1366.46045
Doran, Robert S. (ed.) et al., Operator algebras and their applications. A tribute to Richard V. Kadison. AMS special session on operator algebras and their applications: a tribute to Richard V. Kadison, San Antonio, TX, USA, January 10–11, 2015. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-1948-6/pbk; 978-1-4704-3500-4/ebook). Contemporary Mathematics 671, 1-33 (2016).
Summary: We survey results connecting exactness in the sense of \(C^*\)-algebra theory, coarse geometry, geometric group theory, and expander graphs. We summarize the construction of the (in)famous non-exact monster groups whose Cayley graphs contain expanders, following Gromov, Arzhantseva, Delzant, Sapir, and Osajda. We explain how failures of exactness for expanders and these monsters lead to counterexamples to Baum-Connes type conjectures: the recent work of Osajda allows us to give a more streamlined approach than currently exists elsewhere in the literature.
We then summarize our work on reformulating the Baum-Connes conjecture using exotic crossed products, and show that many counterexamples to the old conjecture give confirming examples to the reformulated one; our results in this direction are a little stronger than those in our earlier work. Finally, we give an application of the reformulated Baum-Connes conjecture to a version of the Kadison-Kaplansky conjecture on idempotents in group algebras.
For the entire collection see [Zbl 1347.46001].

46L35 Classifications of \(C^*\)-algebras
20F65 Geometric group theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C40 Connectivity
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