| Generalized Dynamics preserving maps and application |
| کد مقاله : 1057-SHAA |
| نویسندگان: |
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میثم مصدق * گروه ریاضی دانشگاه آزاد اسلامی واحد لارستان |
| چکیده مقاله: |
| Let $M$ and $N$ be full Hilbert modules over $C^*$-algebras $A$ and $B,$ respectively. In this paper we demonstrate the concept of (inner) generalized dynamics preserving operators for automorphism groups of $C^*$-algebras and unitary morphisms groups of Hilbert $C^*$-modules and show that each generalized dynamics preserving operator on the group $U(M)$ of unitary operators on a Hilbert $A$-modules $M$ induces a natural generalized dynamics preserving operator on the quotient group of $U(M)$ over its kernel. We also, characterize the form of generalized dynamics preserving operators for unitary morphisms groups on the Cartesian product of two Hilbert $C^*$-modules. Let $psi:Ato B$ be an injective morphism of $C^*$-algebras and $theta:Mto N$ be a surjective $psi$-morphism. Introducing the notion of approximately inner generalized dynamical system on Hilbert $C^*$-modules, we show that for the inner $C^*$-dynamics preserving operator $Lambda_{psi}:Aut(A)to Aut(B)$ and the inner $Lambda_{psi}$-dynamics preserving operator $Phi_{theta}:U(M)to U(N),$ if ${alpha_t}_{tinmathbb{R}}$ is a (an approximately inner) generalized dynamical system on $M,$ then there exists a unique (approximately inner) $C^*$-dynamical system ${varphi_t}_{tinmathbb{R}}$ on $A$ such that ${Lambda_{psi}(varphi_t)}_{tinmathbb{R}}$ is a (an approximately inner) $C^*$-dynamical system on $B,$ and ${Phi_{theta}(alpha_t)}_{tinmathbb{R}}$ is a (an approximately inner) generalized dynamical system on $N$ whose associated $C^*$-dynamical system is exactly ${Lambda_{psi}(varphi_t)}_{tinmathbb{R}}.$ |
| کلیدواژه ها: |
| $C^*$-Dynamical systems, generalized derivation, Hilbert $C^*$-module, unitary operator |
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