Matrix coefficients of unitary representations and associated compactifications Nico SpronkRoss Stokke Primary 43A3047D0346J1043A07Secondary 43A2543A6546E25.Fourier-Stieltjes algebrassemitopological compactification\\v{s}ilov boundaryamenable group. 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. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.4825 10.1512/iumj.2013.62.4825 en Indiana Univ. Math. J. 62 (2013) 99 - 148 state-of-the-art mathematics http://iumj.org/access/