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On the index of equivariant elliptic operators. (English) Zbl 1081.46047
Doran, Robert S. (ed.) et al., Operator algebras, quantization, and noncommutative geometry. A centennial celebration honoring John von Neumann and Marshall H. Stone. Proceedings of the AMS special session, Baltimore, MD, USA, January 15–16, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3402-9/pbk). Contemporary Mathematics 365, 41-49 (2004).
This survey covers the link between the Baum-Connes (BC) conjecture and the trace conjecture, following a result of W. Lück [Invent. Math. 149, No. 1, 123–152 (2002; Zbl 1035.19003)]. Let \(\Gamma\) be a discrete countable (or finite) group. The group algebra acts by convolution on \(L^2(\Gamma)\). Its completion \(C^*_r\Gamma\) with respect to the operator norm is called the reduced \(C^*\)-algebra of \(\Gamma\). The trace functional on \(C^*_r\Gamma\) is simply evaluation at the identity element of \(\Gamma\). Informally, the BC conjecture asserts that the \(K\)-theory of \(C^*_r\Gamma\) can be realized by \(\Gamma\)-equivariant \(\text{Spin}^c\) Dirac operators, uniquely up to some elementary moves. The trace conjecture asserts that the values of the trace on the \(K_0\) group of \(C^*_r\Gamma\) lie inside a certain subring of the field of rational numbers related to finite subgroups of \(\Gamma\). Lück’s result shows that BC implies the trace conjecture. Note that a priori it is not even clear that the trace takes rational values. Several variants of the BC conjecture are also discussed.
For the entire collection see [Zbl 1056.58001].
46L80 \(K\)-theory and operator algebras (including cyclic theory)
19K33 Ext and \(K\)-homology
19K35 Kasparov theory (\(KK\)-theory)
19K56 Index theory
58J20 Index theory and related fixed-point theorems on manifolds