Cohomologie des groupes localement compacts et produits tensoriels continus de représentations

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 $\text{L}{}^1\text{(Kβ}\text{G}\text{K}\text{)}$ 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).