Celebratio Mathematica

P. Baum and R. G. Douglas: “ $K$ homology and index theory,” pp. 117–173 in Operator algebras and applications (Kingston, ON, 14 July–2 August 1980), part 1. Edited by R. V. Kadison. Proceedings of Symposia in Pure Mathematics 38. American Mathematical Society (Providence, RI), 1982. MR 679698 Zbl 0532.55004 incollection

Ronald George Douglas	Related
Richard V. Kadison	Related

P. Baum: “On the index of equivariant elliptic operators,” pp. 41–49 in Operator algebras, quantization, and noncommutative geometry: A centennial celebration honoring John von Neumann and Marshall H. Stone. Edited by R. Doran and R. V. Kadison. Contemporary Mathematics 365. American Mathematical Society (Providence, RI), 2004. MR 2106816 Zbl 1081.46047 incollection

Robert Stuart Doran	Related
Marshall Harvey Stone	Related	Volume
Richard V. Kadison	Related
John von Neumann	Related

P. Baum, E. Guentner, and R. Willett: “Exactness and the Kadison–Kaplansky conjecture,” pp. 1–33 in Operator algebras and their applications: A tribute to Richard V. Kadison (San Antonio, TX, 10–11 January 2015). Edited by R. S. Doran and E. Park. Contemporary Mathematics 671. American Mathematical Society (Providence, RI), 2016. Dedicated to Richard Kadison on the occasion of his ninetieth birthday with affection and admiration. MR 3546676 Zbl 1366.46045 incollection

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.

Robert Stuart Doran	Related
Richard V. Kadison	Related
Rufus Willett	Related
Erik Paul Guentner	Related
Efton Park	Related

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