Certain $*$-homomorphisms acting on unital $C^{*}$-probability spaces and semicircular elements induced by $p$-adic number fields over primes $p$
Authors: Cho Ilwoo 
Published: April 2020

<abstract><p>In this paper, we study the Banach <inline-formula><tex-math id="M5">\begin{document}$ * $\end{document}</tex-math></inline-formula>-probability space <inline-formula><tex-math id="M6">\begin{document}$ (A\otimes_{\Bbb{C}}\Bbb{LS}, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M7">\begin{document}$ \tau_{A}^{0}) $\end{document}</tex-math></inline-formula> generated by a fixed unital <inline-formula><tex-math id="M8">\begin{document}$ C^{*} $\end{document}</tex-math></inline-formula>-probability space <inline-formula><tex-math id="M9">\begin{document}$ (A, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M10">\begin{document}$ \varphi_{A}), $\end{document}</tex-math></inline-formula> and the semicircular elements <inline-formula><tex-math id="M11">\begin{document}$ \Theta_{p,j} $\end{document}</tex-math></inline-formula> induced by <inline-formula><tex-math id="M12">\begin{document}$ p $\end{document}</tex-math></inline-formula>-adic number fields <inline-formula><tex-math id="M13">\begin{document}$ \Bbb{Q}_{p}, $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M14">\begin{document}$ p $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M15">\begin{document}$ \in $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M16">\begin{document}$ \mathcal{P}, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M17">\begin{document}$ j $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M18">\begin{document}$ \in $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M19">\begin{document}$ \Bbb{Z}, $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M20">\begin{document}$ \mathcal{P} $\end{document}</tex-math></inline-formula> is the set of all primes, and <inline-formula><tex-math id="M21">\begin{document}$ \Bbb{Z} $\end{document}</tex-math></inline-formula> is the set of all integers. In particular, from the order-preserving shifts <inline-formula><tex-math id="M22">\begin{document}$ g\times h_{\pm } $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M23">\begin{document}$ \mathcal{P} $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M24">\begin{document}$ \times $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M25">\begin{document}$ \Bbb{Z}, $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M26">\begin{document}$ * $\end{document}</tex-math></inline-formula>-homomorphisms <inline-formula><tex-math id="M27">\begin{document}$ \theta $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M28">\begin{document}$ A, $\end{document}</tex-math></inline-formula> we define the corresponding <inline-formula><tex-math id="M29">\begin{document}$ * $\end{document}</tex-math></inline-formula>-homomorphisms <inline-formula><tex-math id="M30">\begin{document}$ \sigma_{(\pm ,1)}^{1:\theta } $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M31">\begin{document}$ A\otimes_{\Bbb{C}}\Bbb{LS}, $\end{document}</tex-math></inline-formula> and consider free-distributional data affected by them.</p></abstract>