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

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  • 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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