On matrix valued square integrable positive definite functions

In this paper, we study matrix valued positive definite functions on a unimodular group. We generalize two theorems of Godement on positive definite functions. We show that a matrix-valued continuous positive definite function can always be written as the convolution of a matrix-valued positive definite function with itself. We also prove that, given two matrix valued positive definite functions and , . In addition this integral equals zero if and only if . Our proofs are operator-theoretic and independent of the group.