Matrix coefficients of unitary representations and associated compactifications

Abstract

We study, for a locally compact group $G$, the compactifications $(\pi,G^\pi)$ associated with unitary representations $\pi$, which we call {\it $\pi$-Eberlein compactifications}. We also study the Gelfand 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. We illustrate our results with examples including various abelian and compact groups, and the $ax+b$-group. In particular, we witness algebras $\fA(\pi)$, for certain non-self-conjugate $\pi$, as being generalised algebras of analytic functions.