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Matrix coefficients of unitary representations and associated compactifications

We study, for a locally compact group $G$, the compactifications $(π,G^π)$ associated with unitary representations $π$, which we call {\it $π$-Eberlein compactifications}. We also study the Gelfand spectra $Φ_{\mathcal{A}}(π)}$ of the uniformly closed algebras $\mathcal{A}(π)$ generated by matrix coefficients of such $π$. We note that $Φ_{\mathcal{A}(π)}\cup\{0\}$ is itself a semigroup and show that the Šilov boundary of $\mathcal{A}(π)$ is $G^π$. 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 $(ω,G^ω)$ 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.