p-Adic Number Fields Acting On W^*-Probability Spaces

Ilwoo Cho^1,

¹Department of Mathematics, 421 Ambrose Hall, Saint Ambrose University, 518 W. Locust St., Davenport, Iowa, 52803, U. S. A.

Cite this paper:
Ilwoo Cho. p-Adic Number Fields Acting On W^*-Probability Spaces. Turkish Journal of Analysis and Number Theory. 2017; 5(2):31-56. doi: 10.12691/tjant-5-2-2

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.

Keywords:
Free Probability Free Probability Spaces p-Adic Number Fields Von Neumann Algebras W^*-Dynamical Systems Crossed Product W^*-Algebras.

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