Banach KK-Theory and the Baum-Connes Conjecture

Abstract

The report below describes the applications of Banach KK-theory to a conjecture of P. Baum and A. Connes about the K-theory of group C * -algebras, and a new proof of the classification by Harish-Chandra, the construction by Parthasarathy and the exhaustion by Atiyah and Schmid of the discrete series representations of connected semi-simple Lie groups. This report is intended to be very elementary. In the first part we outline the main results in Banach KK-theory and the applications to the Baum-Connes conjecture. In the second part we show how the Baum-Connes conjecture for connected semi-simple Lie groups can be applied to recover the classification of the discrete series representations.

Positive scalar curvature and Poincaré duality for proper actions

“…It can be verified that d H c v is a projection in the C * -algebra C * r (G) and hence defines an element in its K-theory. It is also known that the von Neumann trace τ G (defined above) of the class [d H c v ] ∈ K 0 (C * r (G)) is equal to (−1) d/2 times the formal degree (see [48]).…”

“…For a more extensive discussion we refer to [48]. It is shown there that the map taking (H, π) to [d H c v ] ∈ K 0 (C * r (G)) is an injection from the set of discrete series representations to a set of generating elements for K 0 (C * r (G)) [48].…”

“…We can relate the formula for the formal degree proved in [48] to the commutativity result Section 4, as follows. This gives an interpretation of a result of Atiyah and Schmid (p. 25 of [7]) in terms of equivariant index theory.…”

An equivariant index for proper actions III: The invariant and discrete series indices

“…Here K G 0 (M) is the (even) G-equivariant K-homology of M, and K 0 (C * r G) is the (even) K-theory of the reduced goup C * -algebra of G. Consider the inclusion map j : R ds (G) ֒→ K 0 (C * r G) given by j(π) = [d π c π ], where d π is the formal degree of π ∈Ĝ ds , and c π is the matrix coefficient of any unit vector in the representation space of π (see [21]). Let p ds : K 0 (C * r G) → K 0 (C * r G) be the projection onto the image of j.…”

Formal geometric quantisation for proper actions

“…In [7,8], it was shown that for compact N, quantisation commutes with induction, in the sense that Proof. The equality (11) implies that…”