Certain $*$-homomorphisms acting on unital $C^{*}$-probability spaces and semicircular elements induced by $p$-adic number fields over primes $p$

Abstract

In this paper, we study the Banach * -probability space (A ⊗ C LS, τ 0 A ) generated by a fixed unital C * -probability space (A, ϕ A ), and the semicircular elements Θ p,j induced by p-adic number fields Qp, for all p ∈ P, j ∈ Z, where P is the set of all primes, and Z is the set of all integers. In particular, from the order-preserving shifts g × h ± on P × Z, and * -homomorphisms θ on A, we define the corresponding * -homomorphisms σ 1:θ (±,1) on A ⊗ C LS, and consider free-distributional data affected by them.