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تسجيل مستخدم جديدOperator synthesis and tensor products
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We show that Kraus property $S_{sigma}$ is preserved under taking weak* closed sums with masa-bimodules of finite width, and establish an intersection formula for weak* closed spans of tensor products, one of whose terms is a masa-bimodule of finite width. We initiate the study of the question of when operator synthesis is preserved under the formation of products and prove that the union of finitely many sets of the form $kappa times lambda$, where $kappa$ is a set of finite width, while $lambda$ is operator synthetic, is, under a necessary restriction on the sets $lambda$, again operator synthetic. We show that property $S_{sigma}$ is preserved under spatial Morita subordinance. En route, we prove that non-atomic ternary masa-bimodules possess property $S_{sigma}$ hereditarily.
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We study crossed products of arbitrary operator algebras by locally compact groups of completely isometric automorphisms. We develop an abstract theory that allows for generalizations of many of the fundamental results from the selfadjoint theory to
our context. We complement our generic results with the detailed study of many important special cases. In particular we study crossed products of tensor algebras, triangular AF algebras and various associated C*-algebras. We make contributions to the study of C*-envelopes, semisimplicity, the semi-Dirichlet property, Takai duality and the Hao-Ng isomorphism problem. We also answer questions from the pertinent literature.
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Let $(mathcal G, Sigma)$ be an ordered abelian group with Haar measure $mu$, let $(mathcal A, mathcal G, alpha)$ be a dynamical system and let $mathcal Artimes_{alpha} Sigma $ be the associated semicrossed product. Using Takai duality we establish a
stable isomorphism [ mathcal Artimes_{alpha} Sigma sim_{s} big(mathcal A otimes mathcal K(mathcal G, Sigma, mu)big)rtimes_{alphaotimes {rm Ad}: rho} mathcal G, ] where $mathcal K(mathcal G, Sigma, mu)$ denotes the compact operators in the CSL algebra ${rm Alg}:mathcal L(mathcal G, Sigma, mu)$ and $rho$ denotes the right regular representation of $mathcal G$. We also show that there exists a complete lattice isomorphism between the $hat{alpha}$-invariant ideals of $mathcal Artimes_{alpha} Sigma$ and the $(alphaotimes {rm Ad}: rho)$-invariant ideals of $mathcal A otimes mathcal K(mathcal G, Sigma, mu)$. Using Takai duality we also continue our study of the Radical for the crossed product of an operator algebra and we solve open problems stemming from the earlier work of the authors. Among others we show that the crossed product of a radical operator algebra by a compact abelian group is a radical operator algebra. We also show that the permanence of semisimplicity fails for crossed products by $mathbb R$. A final section of the paper is devoted to the study of radically tight dynamical systems, i.e., dynamical systems $(mathcal A, mathcal G, alpha)$ for which the identity ${rm Rad}(mathcal A rtimes_alpha mathcal G)=({rm Rad}:mathcal A) rtimes_alpha mathcal G$ persists. A broad class of such dynamical systems is identified.
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