2011/12/20 by Nico Spronk, Spronk, Nico, Ross Stokke +1
Mathematics · #43A07 #43A25 #43A30 #43A65 #46E25 #46J10 #47D03 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:43A07 #msc:43A25 #msc:43A30 #msc:43A65 #msc:46E25 #msc:46J10 #msc:47D03
paper · pdf · doi:10.48550/arxiv.1112.4878
40 pages, some theorems improved
arxiv created 2012/07/11 · arxiv updated 2012/07/12
We study, for a locally compact group G, the compactifications (π,Gπ) associated with unitary representations π, which we call \it π-Eberlein compactifications. We also study the Gelfand spectra ΦA(π) of the uniformly closed algebras A(π) generated by matrix coefficients of such π. We note that ΦA(π)∪\0\ is itself a semigroup and show that the Šilov boundary of A(π) is Gπ. We study containment relations of various uniformly closed algebras generated by matrix coefficients, and give a new characterisation of amenability: the constant function 1 can be uniformly approximated by matrix coefficients of representations weakly contained in the left regular representation if and only if G is amenable. We show that for the universal representation ω, the compactification (ω,Gω) has a certain universality property: it is universal amongst all compactifications of G which may be embedded as contractions on a Hilbert space, a fact which was also recently proved by Megrelishvili. We illustrate our results with examples including various abelian and compact groups, and the ax+b-group. In particular, we witness algebras \fA(π), for certain non-self-conjugate π, as being generalised algebras of analytic functions.