Abstract
In this paper, we study relations between free probability on crossed product W * -algebras with a von Neumann algebra over p-adic number fields ℚp (for primes p), and free probability on the subalgebra Φ, generated by the Euler totient function ϕ, of the arithmetic algebra A , consisting of all arithmetic functions. In particular, we apply such free probability to consider operator-theoretic and operator-algebraic properties of W * -dynamical systems induced by ℚp under free-probabilistic (and hence, spectral-theoretic) techniques.
Highlights
While in standard probability spaces, the random variables are functions, and their analysis entails only abelian algebras of functions.By contrast, in free probability, one studies noncommutative random variables in terms of fixed linear functionals
The main purpose of this paper is to show the free probability on W ∗ -dynamical systems induced by
We address-and-summarize the main theorems, (i) in a given free probability space, either global, or one of the prime factors, how do we identify mutually free sub-systems? See, for example, Theorem 8.6; and (ii) how do our global systems factor in terms of the prime free probability spaces? See especially
Summary
While in standard probability spaces, the random variables are functions (measurable with respect to a prescribed σ-algebra), and their analysis entails only abelian algebras of functions. One can understand arithmetic functions as Krein-space operators, via certain representations (See [11,12]) These studies are all motivated by well-known number-theoretic results (e.g., [13,14,15,16,17]) with help of free probability techniques (e.g., [8,11,12]). In [21], we considered von Neumann algebras L∞ (Qp ) induced by p-adic number fields Qp , and realized the connection between non-Archimedean calculus on L∞ (Qp ) and free probability on (A, gp ), liked via Euler totient function φ.
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