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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.
Let $A$ be a unital operator algebra and let $alpha$ be an automorphism of $A$ that extends to a *-automorphism of its $ca$-envelope $cenv (A)$. In this paper we introduce the isometric semicrossed product $A times_{alpha}^{is} bbZ^+ $ and we show that $cenv(A times_{alpha}^{is} bbZ^+) simeq cenv (A) times_{alpha} bbZ$. In contrast, the $ca$-envelope of the familiar contractive semicrossed product $A times_{alpha} bbZ^+ $ may not equal $cenv (A) times_{alpha} bbZ$. Our main tool for calculating $ca$-envelopes for semicrossed products is the concept of a relative semicrossed product of an operator algebra, which we explore in the more general context of injective endomorphisms. As an application, we extend a recent result of Davidson and Katsoulis to tensor algebras of $ca$-correspondences. We show that if $T_{X}^{+}$ is the tensor algebra of a $ca$-correspondence $(X, fA)$ and $alpha$ a completely isometric automorphism of $T_{X}^{+}$ that fixes the diagonal elementwise, then the contractive semicrossed product satisfies $ cenv(T_{X}^{+} times_{alpha} bbZ^+)simeq O_{X} times_{alpha} bbZ$, where $O_{X}$ denotes the Cuntz-Pimsner algebra of $(X, fA)$.
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.
Given two correspondences X and Y, we show that (under mild hypotheses) the Cuntz-Pimsner algebra of the tensor product of X and Y embeds as a certain subalgebra of the tensor product of the Cuntz-Pimsner algebra of X and the Cuntz=Pimsner algebra of Y. Furthermore, this subalgebra can be described in a natural way in terms of the gauge actions on the Cuntz-Pimsner algebras. We explore implications for graph algebras, crossed products by the integers, and crossed products by completely positive maps. We also give a new proof of a result of Kaliszewski and Quigg related to coactions on correspondences.
We consider the construction of twisted tensor products in the category of C*-algebras equipped with orthogonal filtrations and under certain assumptions on the form of the twist compute the corresponding quantum symmetry group, which turns out to be the generalised Drinfeld double of the quantum symmetry groups of the original filtrations. We show how these results apply to a wide class of crossed products of C*-algebras by actions of discrete groups. We also discuss an example where the hypothesis of our main theorem is not satisfied and the quantum symmetry group is not a generalised Drinfeld double.
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