Appears in collection : 2025 - T1 - Representation theory and noncommutative geometry

Part 7 - Kasparov’s gamma element for $SL(2,R)$

Practical person’s guide to the Baum-Connes conjecture from the point of view of Kasparov theory: Kasparov’s representation ring (as an abelian group), the gamma element, Kasparov’s theorem for connected Lie groups, reduction to $\gamma=1$. First examples where $\gamma$ is equal to 1. Examples when $\gamma$ is not equal to 1. Julg’s proposal regarding uniformly bounded representations. Kasparov’s homotopy for $SL(2,R)$.

Part 8 - Lafforgue’s Banach version of KK-theory

The C-algebra of a group, versus other convolution Banach algebras. Homotopies of unitary representations versus homotopies of Banach representations. Examples and action of the reduced C-algebra on $L^p$ spaces. Review of the role of compact operators in Kasparov theory. Dual pairs of Banach spaces and operators of compact type. Lafforgue’s version of Kasparov’s representation ring.

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