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

Let $G$ be a Roelcke-precompact non-archimedean Polish group, $\mathcal {A}_G$ the algebra generated by indicator maps of cosets of open subgroups in $G$. Then $\mathcal{A}_G$ is dense in the algebra of matrix coefficients of $G$. We prove that multiplicative linear functionals on $\mathcal{A}_G$ 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 $\mathbf{H}(G)$ of $G$. One is the space $\mathbf {P}(\mathcal{M}_G)$ of partial elementary maps with algebraically closed domain on $\mathcal{M}_G$, 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 $\aleph _0$-categorical structures. The other is $\mathbf{T}(G)$, 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.