P-Adic Number Fields Acting On W*-Probability Spaces

Abstract

In this paper, we study how a p -adic number field acts on an arbitrarily fixed W * -algebra, and how it affects the original free-probabilistic information on the W * -algebra, for each prime p . In particular, by understanding the σ-algebra of as a semigroup equipped with the setintersection, we act on a unital tracial W * -probability space ( M , tr ), creating the corresponding semigroup W * -dynamical system. From such a dynamical system, construct the crossed product W * -algebra equipped with a suitable linear functional. We study free probability on such W * -dynamical operator-algebraic structures determined by primes, and those on corresponding free products of such structures over primes. As application, we study cases where given W * -probability spaces are generated by countable discrete groups.