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Group algebras acting on -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 \ca{} 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.Comment: Version 1: 27 pages. Version 2: lots of minor corrections, and we got
rid of the second-countability assumption on the groups. 31 page
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