Group Algebras Acting on Lp-Spaces

For we study representations of a locally compact group on -spaces and -spaces. The universal completions and of with respect to these classes of representations (which were first considered by Phillips and Runde, respectively), can be regarded as analogs of the full group -algebra of (which is the case ). We study these completions of in relation to the algebra of -pseudofunctions. We prove a characterization of group amenability in terms of certain canonical maps between these universal Banach algebras. In particular, is amenable if and only if . One of our main results is that for , there is a canonical map which is contractive and has dense range. When is amenable, is injective, and it is never surjective unless is finite. We use the maps to show that when is discrete, all (or one) of the universal completions of are amenable as a Banach algebras if and only if is amenable. Finally, we exhibit a family of examples showing that the characterizations of group amenability mentioned above cannot be extended to -operator crossed products of topological spaces.