Piecewise Interpretable Hilbert Spaces

TL;DR

This paper introduces a new framework for analyzing Hilbert spaces within first-order theories, demonstrating they can be decomposed into interpretable, asymptotically free components under a scatteredness condition, with applications across various mathematical structures.

Contribution

It establishes a novel decomposition theorem for Hilbert spaces in model theory using local stability, extending previous results and providing new methods for understanding their structure.

Findings

Hilbert spaces decompose into asymptotically free components under scatteredness.

Applications include $L^2$-spaces, Galois groups, and unitary representations.

Main theorem generalizes Tsankov's result for automorphism groups.

Abstract

We study Hilbert spaces $H$ interpreted, in an appropriate sense, in a first-order theory. Under a new finiteness hypothesis that we call {\em scatteredness} we prove that $H$ is a direct sum of {\em asymptotically free} components, where short-range interactions are controlled by algebraic closure and long-range interactions vanish. Examples include $L^2$-spaces relative to Macpherson-Steinhorn definable measures; $L^2$ spaces relative to the Haar measure of the absolute Galois groups; irreducible unitary representations of $p$-adic Lie groups; and unitary representations of the automorphism group of an $\omega$-categorical theory. In the last case, our main result specialises to a theorem of Tsankov. New methods are required, making essential use of local stability theory in continuous logic.