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 important results of Godement on square integrable positive definite functions to matrix valued square integrable positive definite functions. We show that a matrix-valued continuous $L^2$ positive definite function can always be written as a convolution of a $L^2$ positive definite function with itself. We also prove that, given two $L^2$ matrix valued positive definite functions $\Phi$ and $\Psi$, $\int_G Trace(\Phi(g) \bar{\Psi(g)}^t) d g \geq 0$. In addition this integral equals zero if and only if $\Phi * \Psi=0$. Our proofs are operator-theoretic and independent of the group.