Operator algebras and C*-correspondences: A survey
In this paper we survey our recent work on C*-correspondences and their associated operator algebras; in particular, on adding tails, the Shift Equivalence Problem and Hilbert bimodules.
math.OA↗
arXiv subjects
Evgenios T. A. Kakariadis
Publications and source records attributed to Evgenios T. A. Kakariadis.
Conjugate Dynamical Systems on C*-algebras
Let $(A, α)$ and $(B, β)$ be C*-dynamical systems where $α$ and $β$ are arbitrary *-endomorphisms. When $α$ is injective or surjective, we show that the semicrossed products $A \times_α\mathbb{Z}$ and $B \times_β\mathbb{Z}$ are isometrically isomorphic if and only if $(A, α)$ and $(B, β)$ are outer conjugate. This conclusion also holds in various other cases as well.
math.OA↗
Contributions to the theory of C*-correspondences with applications to multivariable dynamics
Motivated by the theory of tensor algebras and multivariable C*-dynamics, we revisit two fundamental techniques in the theory of C*-correspondences, the "addition of a tail" to a non-injective C*-correspondence and the dilation of an injective C*-correspondence to an essential Hilbert bimodule. We provide a very broad scheme for "adding a tail" to a non-injective C*-correspondence; our scheme includes the "tail" of Muhly and Tomforde as a special case. We illustrate the diversity and necessity of our tails with several examples from the theory of multivariable C*-dynamics. We also exhibit a transparent picture for the dilation of an injective C*-correspondence to an essential Hilbert bimodule. As an application of our constructs, we prove two results in the theory of multivariable dynamics that extend results of Davidson and Roydor and Peters. We also discuss the impact of our results on the description of the C*-envelope of a tensor algebra as the Cuntz-Pimsner algebra of the associated C*-correspondence.
math.OA↗
Semicrossed Products and Reflexivity
Given a w*-closed unital algebra $A$ acting on $H_0$ and a contractive w*-continuous endomorphism $β$ of $A$, there is a w*-closed (non-selfadjoint) unital algebra $\mathbb{Z}_+\bar{\times}_βA$ acting on $H_0\otimes\ell^2({\mathbb{Z}_+})$, called the w*-semicrossed product of $A$ with $β$. We prove that the w*-semicrossed product is a reflexive operator algebra provided $A$ is reflexive and $β$ is unitarily implemented, and that it has the bicommutant property if and only if so does $A$. Also, we show that the w*-semicrossed product generated by a commutative C*-algebra and a *-endomorphism is reflexive.
math.OA↗