Tannaka–Kreĭn duality for Roelcke-precompact non-archimedean Polish groups

Let G be a Roelcke-precompact non-archimedean Polish group, AG the algebra generated by indicator maps of cosets of open subgroups in G. Then AG is dense in the algebra of matrix coefficients of G. We prove that multiplicative linear functionals on AG are automatically continuous, an analogue of a result of Kreĭn for finite-dimensional representations of topological groups. We deduce two abstract realizations of the Hilbert compactification H(G) of G. One is the space P(MG) of partial elementary maps with algebraically closed domain on MG, the countable set of open cosets of G seen as a homogeneous first-order logical structure. This can be seen as a reformulation of a similar identification by Ben Yaacov, Ibarlucía and Tsankov for ℵ0-categorical structures. The other is T(G), the Tannaka monoid of G. The group can be recovered from these constructions, generalizing Tannaka’s and Kreĭn’s duality theories to this context. Finally, we show that the natural functor that sends G to the category of its representations is full and faithful.