Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures

TL;DR

This paper investigates the conditions under which matrix-valued measures can be reduced to smaller blocks, establishing that all reductions are unitary if and only if a certain algebraic subspace is $*$-invariant, with applications to group-related polynomials.

Contribution

It proves that the subspace of matrices commuting with a measure is $*$-invariant if and only if all reductions are unitarily performed, clarifying the structure of reducibility for matrix-valued measures.

Findings

Reductions of matrix-valued measures are unitary if and only if a specific subspace is $*$-invariant.

The commutant algebra for certain group-related measures is two-dimensional, leading to a $2 imes 2$ block diagonal form.

No non-unitary reductions exist beyond the established unitary block diagonalization.

Abstract

A matrix-valued measure $\Theta$ reduces to measures of smaller size if there exists a constant invertible matrix $M$ such that $M\Theta M^*$ is block diagonal. Equivalently, the real vector space ${\mathscr A}$ of all matrices $T$ such that $T\Theta(X)=\Theta(X) T^*$ for any Borel set $X$ is non-trivial. If the subspace $A_h$ of self-adjoints elements in the commutant algebra $A$ of $\Theta$ is non-trivial, then $\Theta$ is reducible via a unitary matrix. In this paper we prove that ${\mathscr A}$ is $*$-invariant if and only if $A_h={\mathscr A}$, i.e., every reduction of $\Theta$ can be performed via a unitary matrix. The motivation for this paper comes from families of matrix-valued polynomials related to the group ${\rm SU}(2)\times {\rm SU}(2)$ and its quantum analogue. In both cases the commutant algebra $A=A_h\oplus iA_h$ is of dimension two and the matrix-valued measures reduce unitarily into a $2\times 2$ block diagonal matrix. Here we show that there is no further non-unitary reduction.