Submission date: 11 October 2021

Abstract:

We define and study piecewise interpretable Hilbert spaces in continuous logic. These are Hilbert spaces which arise as direct limits of imaginary sorts of a model M of a theory T. We introduce natural examples of piecewise interpretable Hilbert spaces in a wide variety of contexts. We show that piecewise interpretable Hilbert spaces can be seen as encoding interesting model theoretic information about M or T, such as properties of definable measures or Galois-theoretic information. We also show that piecewise interpretable Hilbert spaces encode unitary group representations in various ways. We carry out a systematic structural analysis of piecewise interpretable Hilbert spaces with scattered subsets. As an application of this work, we recover the classification of the unitary representations of oligomorphic groups first discovered by Tsankov (2012). Our main tool is local stability theory in continuous logic.

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Full text arXiv 2110.05142: pdf, ps.