  We study the operator-valued positive definite functions on a group using
positive block matrices. We give an alternative proof to Brehmer positivity for
doubly commuting contractions. We classify all commuting unitary
representations over a finite group. We show by examples that the power of a
positive-definite function may not be positive definite and also the power of a
unitary representation may not be a unitary representation. We also
characterize all unitary representations whose powers are also unitary
representations.
