zbMATH — the first resource for mathematics

K-theory for the reduced \(C^*\)-algebra of a semi-simple Lie group with real rank 1 and finite centre. (English) Zbl 0545.22006
The author discusses the Connes-Kasparov conjecture on the structure of K-groups of the reduced group \(C^*\)-algebra of a connected semisimple Lie group. This conjecture was proved by M. G. Penington and R. J. Plymen for complex semismple Lie groups [J. Funct. Anal. 53, 269- 286 (1983; Zbl 0542.22013)]. The author extends their argument to prove a weak form of the conjecture for semisimple Lie groups with finite center and real rank one, and the strong form for groups Spin(m,1). (The conjecture was recently proved by A. Wasserman for connected linear semisimple groups (unpublished), and (independently, but slightly later) by I. Mirković, J. L. Taylor and the reviewer for connected semisimple groups (unpublished)).
Reviewer: D.Miličić

22E20 General properties and structure of other Lie groups
22E46 Semisimple Lie groups and their representations
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
Full Text: DOI