Tannaka-Krein duality for Roelcke-precompact non-archimedean Polish groups

Abstract

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.