Stationary stochastic processes in a group system

Abstract

In this paper, we discuss positive definite functions of locally compact groups into the predual of von Neumann algebras or C*-algebras. For a predual-valued or a C*-algebra valued weakly continuous covariant positive definite function, the corresponding covariant representation in a covariant group system is constructed. We consider some generalization of the time domain of stochastic processes to a locally compact group, and we use covariant representations in a group system to obtain invariant stationary stochastic processes induced from predual-valued covariant positive definite functions and covariant stationary stochastic processes induced from C*-algebra valued covariant positive definite functions.