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Article page (unfinished but already has a sensible story) Reconstruction of Groups.
I have added links to that page from “Tannaka duality” and “Tannaka-Krein theorem”
Notice an interesting discussion at the beginning of the paper on the relation between the functional analysis picture and algebraic geometry:
Note that the Tannaka–Krein duality reflects the fact that any compact group is the set of real points of an affine pro-algebraic group (‘pro-’ stands for projective limit). This observation had lead to an extension of Tannaka–Krein duality to pro-algebraic groups ([D], [DM], [R1], [S]) and Lie algebras ([H-C], [R1]).
Continuous finite-dimensional representations of compact groups are rigid in the following sense: every subcategory of the category $f. Rep(G)$ of finite-dimensional representations of a compact group $G$, which separates elements of $G$ and is closed under the tensor product and complex conjugation, is naturally equivalent to the category $f. Rep(G)$ itself. This is a consequence of a similar rigidity property of regular representations of affine pro-algebraic groups. It follows from the results of this paper that infinite-dimensional representations of an arbitrary locally compact or Lie group do not possess the rigidity property.
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