Measurable selector in Kadison's carpenter's theorem

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.