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

E. KOELINK; P. ROMÁN

SYMMETRY, INTEGRABILITY AND GEOMETRY

NATL ACAD SCI UKRAINE

Año: 2016 vol. 12 p. 1 - 9

1815-0659

A matrix-valued measure $Theta$ reduces to measures of smaller size if there exists a constant invertible matrix $M$ such that $MTheta M^*$ is block diagonal. Equivalently, the real vector space $cA$ of all matrices $T$ such that $TTheta(X)=Theta(X) T^*$ 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 $cA$ is $*$-invariant if and only if $A_h=cA$, 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 $SU(2)imes SU(2)$ and its quantum analogue. In both cases the commutant algebra $A=A_hoplus iA_h$ is of dimension two and the matrix-valued measures reduce unitarily into a $2imes 2$ block diagonal matrix. Here we show that there is no further non-unitary reduction.