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

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.