Kernels of representations and group extensions

Abstract

We first determine the kernels of those unitary representations of a locally compact group which are obtained by integrating, inducing and tensoring. We then describe a broad class of group extensions for which the previous results can be used to find (1) the kernel of any representation and (2) necessary and sufficient conditions for these extensions to be maximally almost periodic. Introduction. The primary purpose of this paper is to determine the kernel of any (unitary) representation of a reasonably general locally compact group extension. (See ?4 for a specific description of the exten- sion.) Roughly speaking, the representations of such an extension are obtained by successively tensoring, inducing and integrating (at least in the separable case) representations of certain subgroups. In ?1 we deter- mine the kernel of a direct integral. (This is the only place where we find it necessary to introduce separability assumptions.) This determination has the effect of reducing our problem to the consideration of irreducible representations only. In ?2 we describe the kernel of an induced repre- sentation, thus extending Lemma 2.1 of (11) to the nonseparable case. Regarding the tensor product of representations, it is well known (see (9) and (6, ?17)) that the most useful kernel to determine is that of the tensor product of projective representations or, more accurately, of representa- tions of corresponding central group extensions of the circle. We do this in ?3. Finally, in ?4 we describe the type of extension for which the preceding results solve the problem stated at the beginning. This section is essentially a combination and augmentation of R. J. Blattner's work in (3) and the relevant results of J. M. G. Fell in ?17 of (6). We conclude by indicating how the previous results can be used to find necessary and sufficient conditions for the extension to be maximally almost periodic. Throughout this paper, G will be a locally compact group and all representations will be unitary. If h is a group homomorphism then