IUMJ

Title: Matrix coefficients of unitary representations and associated compactifications

Authors: Nico Spronk and Ross Stokke

Issue: Volume 62 (2013), Issue 1, 99-148

Abstract:

We study, for a locally compact group $G$, the compactifications $(\pi,G^{\pi})$ associated with unitary representations $\pi$, which we call $\pi$-\emph{Eberlein compactifications}. We also study the Gel\-fand spectra $\Phi_{\mathcal{A}(\pi)}$ of the uniformly closed algebras $\mathcal{A}(\pi)$ generated by matrix coefficients of such $\pi$. We note that $\Phi_{\mathcal{A}(\pi)}\cup\{0\}$ is itself a semigroup, and show that the \v{S}ilov boundary of $\mathcal{A}(\pi)$ is $G^{\pi}$. 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 $\omega$, the compactification $(\omega,G^{\omega})$ 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 [M. Megrelishvili, \textit{Reflexively representable but not Hilbert representable compact flows and semitopological semigroups}, Colloq. Math. \textbf{110} (2008), no. 2, 383--407]. We illustrate our results with examples including various abelian and compact groups, and the $ax+b$-group. In particular, we witness algebras $\mathcal{A}(\pi)$, for certain non-self--conjugate $\pi$, as being generalised algebras of analytic functions.