Extensions and Dilations for C∗C^*-dynamical Systems

Let $A$ be a unital $C^*$-algebra and $\alpha$ be an injective, unital endomorphism of $A$. A covariant representation of $(A,\alpha)$ is a pair $(\pi,T)$ consisting of a $C^*$-representation $\pi$ of $A$ on a Hilbert space $H$ and a contraction $T$ in $B(H)$ satisfying $T\pi(\alpha(a))=\pi(a)T$. It follows from more general results of ours that such a covariant representation can be extended to a covariant representation $(\rho,V)$ (on a larger space $K$) such that $V$ is a coisometry and it can be dilated to a covariant representation $(\sigma,U)$ (on a larger space $K_1$) with $U$ unitary. Our objective here is to give self-contained, elementary proofs of these results which avoid the technology of $C^*$-correspondences. We also discuss the non uniqueness of the extension.Comment: 11 page

Quantum Markov Semigroups (Product Systems and Subordination)

We show that if a product system comes from a quantum Markov semigroup, then it carries a natural Borel structure with respect to which the semigroup may be realized in terms of a measurable representation. We show, too, that the dual product system of a Borel product system also carries a natural Borel structure. We apply our analysis to study the order interval consisting of all quantum Markov semigroups that are subordinate to a given one.Comment: Revised according to the referee's comments and suggestions; to appear in International Journal of Mathematic

Groupoid Methods in Wavelet Analysis

We describe how the Deaconu-Renault groupoid may be used in the study of wavelets and fractals.Comment: To appear in "Group representations, ergodic theory, and mathematical physics: A tribute to George W. Mackey", the proceedings of the AMS special session dedicated to the memory of George W. Mackey at the January 2007 AMS meetin

Quantum Markov Processes (Correspondences and Dilations)

We study the structure of quantum Markov Processes from the point of view of product systems and their representations.Comment: 44 pages, Late

Morita Transforms of Tensor Algebras

We show that if $M$ and $N$ are $C^{*}$-algebras and if $E$ (resp. $F$) is a $C^{*}$-correspondence over $M$ (resp. $N$), then a Morita equivalence between $(E,M)$ and $(F,N)$ implements a isometric functor between the categories of Hilbert modules over the tensor algebras of $\mathcal{T}_{+}(E)$ and $\mathcal{T}_{+}(F)$. We show that this functor maps absolutely continuous Hilbert modules to absolutely continuous Hilbert modules and provides a new interpretation of Popescu's reconstruction operator

Adding tails to C*-correspondences

We describe a method of adding tails to C*-correspondences which generalizes the process used in the study of graph C*-algebras. We show how this technique can be used to extend results for augmented Cuntz-Pimsner algebras to C*-algebras associated to general C*-correspondences, and as an application we prove a gauge-invariant uniqueness theorem for these algebras. We also define a notion of relative graph C*-algebras and show that properties of these C*-algebras can provide insight and motivation for results about relative Cuntz-Pimsner algebras.Comment: 24 pages, uses XY-pic. A few typos corrected, final version for publicatio

Topological Quivers

Topological quivers are generalizations of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to such a topological quiver Q is a C*-correspondence, and from this correspondence one may construct a Cuntz-Pimsner algebra C*(Q). In this paper we develop the general theory of topological quiver C*-algebras and show how certain C*-algebras found in the literature may be viewed from this general perspective. In particular, we show that C*-algebras of topological quivers generalize the well-studied class of graph C*-algebras and in analogy with that theory much of the operator algebra structure of C*(Q) can be determined from Q. We also show that many fundamental results from the theory of graph C*-algebras have natural analogues in the context of topological quivers (often with more involved proofs). These include the Gauge-Invariant Uniqueness theorem, the Cuntz-Krieger Uniqueness theorem, descriptions of the ideal structure, and conditions for simplicity.Comment: 55 pages, uses XY-pic. A few typos corrected. This is the version that will be publishe

The Poisson Kernel for Hardy Algebras

This note contributes to a circle of ideas that we have been developing recently in which we view certain abstract operator algebras $H^{\infty}(E)$, which we call Hardy algebras, and which are noncommutative generalizations of classical $H^{\infty}$, as spaces of functions defined on their spaces of representations. We define a generalization of the Poisson kernel, which ``reproduces'' the values, on $\mathbb{D}((E^{\sigma})^*)$, of the ``functions'' coming from $H^{\infty}(E)$. We present results that are natural generalizations of the Poisson integral formuala. They also are easily seen to be generalizations of formulas that Popescu developed. We relate our Poisson kernel to the idea of a characteristic operator function and show how the Poisson kernel identifies the ``model space'' for the canonical model that can be attached to a point in the disc $\mathbb{D}((E^{\sigma})^*)$. We also connect our Poission kernel to various "point evaluations" and to the idea of curvature

C*-algebras associated with branched coverings

In this note we analyze the C*-algebra associated with a branched covering both as a groupoid C*-algebra and as a Cuntz-Pimsner algebra. We determine conditions when the algebra is simple and purely infinite. We indicate how to compute the K-theory of several examples, including one related to rational maps on the Riemann sphere

Canonical Models for Representations of Hardy Algebras

In this paper we study canonical models for representations of the Hardy algebras that generalize the model theory of Sz.-Nagy and Foias for contraction operators. The Hardy algebras are non selfadjoint operator algebras associated with $W^*$-correspondences. This class includes the classical $H^{\infty}$ algebra, free semigroup algebras, quiver algebras and other classes of algebras studied in the literature