  Let G be a Roelcke-precompact non-archimedean Polish group, B(G) the algebra
of matrix coefficients of G arising from its continuous unitary
representations. The Gel'fand spectrum H(G) of the norm closure of B(G) is
known as the Hilbert compactification of G. Let A be the dense subalgebra of
B(G) generated by indicator maps of open cosets in G. We prove that
multiplicative linear functionals on A are automatically continuous,
generalizing a result of Krein for finite dimensional representations of
topological groups. We deduce two abstract realizations of H(G). One is the
space P(MG) of partial isomorphisms with algebraically closed domain of MG, the
countable set of open cosets of G seen as a homogeneous first order logical
structure. The other is T(G) the Tannaka monoid of G. We also obtain that the
natural functor that sends G to the category of its representations is full and
faithful.
