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

Abstract

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