Cohomologie des groupes localement compacts et produits tensoriels continus de représentations

A. Guichardet

Journal of Multivariate Analysis, 1976, vol. 6, issue 1, 138-158

Abstract: The following is an expository paper, containing few and sometimes incomplete proofs, on continuous tensor products of Hilbert spaces and of group representations, and on the irreducibility of the latter; the principal results in the last direction are due to Verchik, Gelfand, and Graiev. The theory of continuous tensor products of Hilbert spaces, based on a fundamental theorem of Araki and Woods, is closely related to that of conditionally positive definite functions; it relies on the technique of symmetric Hilbert spaces, which also can be used to give a new proof of the classical Lévy-Khinchin formula (see A. Guichardet, (1973). J. Multiv.3 249-261.). Another basic tool for what follows is the 1-cohomology of unitary representations of locally compact groups; here, the main results are due to P. Delorme; let us mention, for instance, his results for the case of a group G containing a compact subgroup K such that L1(K[beta]G/K) is commutative, using a Lévy-Khinchin's type formula for K-invariant functions due to Gangolli, Faraut, and Harzallah. We add that the results exposed in that paper should have interesting connections with the central limit theorems à la Parthasarathy-Schmidt (see K. Parthasarathy, (1974). J. Multiv. Anal.4 123-149).

Keywords: groups; representations; positive; definite; functions; Lévy-Khinchin; formula; cohomology; of; groups; symmetric; Hilbert; spaces; central; limit; theorem (search for similar items in EconPapers)
Date: 1976
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