Generalized Dynamics preserving maps and application
کد مقاله : 1057-SHAA
نویسندگان
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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