Strictly Positive Definite Functions on the Space of Infinite Diagonal Matrices

In this paper, using a Bochner-type theorem, we establish a characterization of strictly positive definite functions on the topological direct limit space $\mathcal D_{_\infty}$ of real diagonal matrices relatively to the action of the product group $K_{_\infty}=U_{_\infty}\times U_{_\infty}$, where $U_{_\infty}$ is the infinite dimensional unitary group.