Measurable selector in Kadison's carpenter's theorem

TL;DR

This paper proves the existence of a measurable selector in Kadison's Carpenter's Theorem, enabling a characterization of spectral functions of shift-invariant subspaces and extending the theorem to type I$_inf$ von Neumann algebras.

Contribution

It establishes a measurable selector in Kadison's Theorem, solving a previously posed problem and broadening the theorem's applicability to new algebraic contexts.

Findings

Existence of a measurable selector in Kadison's Carpenter's Theorem.

Characterization of spectral functions of shift-invariant subspaces.

Extension of Carpenter's Theorem to type I$_inf$ von Neumann algebras.

Abstract

We show the existence of a measurable selector in Carpenter's Theorem due to Kadison. This solves a problem posed by Jasper and the first author. As an application we obtain a characterization of all possible spectral functions of shift-invariant subspaces of $L^2(\mathbb R^d)$ and Carpenter's Theorem for type I$_\infty$ von Neumann algebras.