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 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 A is *-invariant if and only if A(h) = 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 SU(2) x SU(2) and its quantum analogue. In both cases the commutant algebra A = A(h) circle plus iA(h) is of dimension two and the matrix-valued measures reduce unitarily into a 2 x 2 block diagonal matrix. Here we show that there is no further non-unitary reduction.