### Electronic Research Archive

2020, Issue 2: 739-776. doi: 10.3934/era.2020038

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

• Received: 01 February 2020 Revised: 01 March 2020 Published: 01 June 2020
• 05E15, 11G15, 11R47, 11R56, 46L10, 46L54, 47L30, 47L55

• In this paper, we study the Banach $*$-probability space $(A\otimes_{\Bbb{C}}\Bbb{LS},$ $\tau_{A}^{0})$ generated by a fixed unital $C^{*}$-probability space $(A,$ $\varphi_{A}),$ and the semicircular elements $\Theta_{p,j}$ induced by $p$-adic number fields $\Bbb{Q}_{p},$ for all $p$ $\in$ $\mathcal{P},$ $j$ $\in$ $\Bbb{Z},$ where $\mathcal{P}$ is the set of all primes, and $\Bbb{Z}$ is the set of all integers. In particular, from the order-preserving shifts $g\times h_{\pm }$ on $\mathcal{P}$ $\times$ $\Bbb{Z},$ and $*$-homomorphisms $\theta$ on $A,$ we define the corresponding $*$-homomorphisms $\sigma_{(\pm ,1)}^{1:\theta }$ on $A\otimes_{\Bbb{C}}\Bbb{LS},$ and consider free-distributional data affected by them.

Citation: Ilwoo Cho. Certain $*$-homomorphisms acting on unital $C^{*}$-probability spaces and semicircular elements induced by $p$-adic number fields over primes $p$[J]. Electronic Research Archive, 2020, 28(2): 739-776. doi: 10.3934/era.2020038

### Related Papers:

• In this paper, we study the Banach $*$-probability space $(A\otimes_{\Bbb{C}}\Bbb{LS},$ $\tau_{A}^{0})$ generated by a fixed unital $C^{*}$-probability space $(A,$ $\varphi_{A}),$ and the semicircular elements $\Theta_{p,j}$ induced by $p$-adic number fields $\Bbb{Q}_{p},$ for all $p$ $\in$ $\mathcal{P},$ $j$ $\in$ $\Bbb{Z},$ where $\mathcal{P}$ is the set of all primes, and $\Bbb{Z}$ is the set of all integers. In particular, from the order-preserving shifts $g\times h_{\pm }$ on $\mathcal{P}$ $\times$ $\Bbb{Z},$ and $*$-homomorphisms $\theta$ on $A,$ we define the corresponding $*$-homomorphisms $\sigma_{(\pm ,1)}^{1:\theta }$ on $A\otimes_{\Bbb{C}}\Bbb{LS},$ and consider free-distributional data affected by them.

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