"Piecewise Interpretable Hilbert spaces"
We introduce piecewise Hilbert spaces in continuous logic. We say that a Hilbert space is piecewise interpretable when it is a direct limit of imaginary sorts of a model M of a theory T. Piecewise interpretable Hilbert spaces arise in many interesting contexts in model theory where they can code various kinds of information about T or the particular model M. They also provide a point of contact between model theory and the theory of unitary group representations. Using the stability of Hilbert spaces and the standard tools of local stability theory in continuous logic, we are able to prove decomposition theorems for piecewise interpretable Hilbert spaces under some relatively weak assumptions. These decomposition theorems generalise the classification of unitary representations of oligomorphic groups due to Tsankov.