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Paul S. Muhly

Publications and source records attributed to Paul S. Muhly.

At least 37 records · Page 2Linked to original sources
arXiv2007

Groupoid and Inverse Semigroup Presentations of Ultragraph $C^{*}$-Algebras

Inspired by the work of Paterson on $C^{\ast}$-algebras of directed graphs, we show how to associate a groupoid $\mathfrak{G}_{\mathcal{G}}$ to an ultragraph $\mathcal{G}$ in such a way that the $C^*$-algebra of $\mathfrak{G}_{\mathcal{G}}$ is canonically isomorphic to Tomforde's $C^*$-algebra $C^*(\mathcal{G})$. The groupoid $\mathfrak{G}_{\mathcal{G}}$ is built from an inverse semigroup $S_{\mathcal{G}}$ naturally associated to $\mathcal{G}$.

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arXiv2007

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 $\mathbb{D}((E^σ)^*)$, 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 $\mathbb{D}((E^σ)^*)$. We also connect our Poission kernel to various "point evaluations" and to the idea of curvature.

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arXiv2007

Schur Class Operator Functions and Automorphisms of Hardy Algebras

Let $E$ be a $W^{\ast}$-correspondence over a von Neumann algebra $M$ and let $H^{\infty}(E)$ be the associated Hardy algebra. If $σ$ is a faithful normal representation of $M$ on a Hilbert space $H$, then one may form the dual correspondence $E^σ$ and represent elements in $H^{\infty}(E)$ as $B(H)$-valued functions on the unit ball $\mathbb{D}(E^σ)^{\ast}$. The functions that one obtains are called Schur class functions and may be characterized in terms of certain Pick-like kernels. We study these functions and relate them to system matrices and transfer functions from systems theory. We use the information gained to describe the automorphism group of $H^{\infty}(E)$ in terms of special Möbius transformations on $\mathbb{D}(E^σ)$. Particular attention is devoted to the $H^{\infty}% $-algebras that are associated to graphs.

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arXiv2006

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.

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arXiv2005

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

Let $A$ be a unital $C^*$-algebra and $α$ be an injective, unital endomorphism of $A$. A covariant representation of $(A,α)$ is a pair $(π,T)$ consisting of a $C^*$-representation $π$ of $A$ on a Hilbert space $H$ and a contraction $T$ in $B(H)$ satisfying $Tπ(α(a))=π(a)T$. It follows from more general results of ours that such a covariant representation can be extended to a covariant representation $(ρ,V)$ (on a larger space $K$) such that $V$ is a coisometry and it can be dilated to a covariant representation $(σ,U)$ (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.

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arXiv2005

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.

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arXiv2004

Cuntz-Pimsner Algebras, Completely Positive Maps and Morita Equivalence

Let $P$ be a completely positive map on $M_n(\mathbb{C})$ and let $E_P$ be the associated \emph{GNS}-$C^*$-correspondence. We prove a result that implies, in particular, that the Cuntz-Pimsner algebra of $E_P$, $\mathcal{O}(E_P)$, is strongly Morita equivalent to the Cuntz algebra $\mathcal{O}_{d(P)}$, where $d(P)$ is the index of $P$.

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arXiv2004

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.

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arXiv2004

Duality of $W^*$-correspondences and applications

This is an exposition of our work on the canonical models of representations of Hardy algebras associated with $W^*$-correspondences. It generalizes the model theory of Sz.-Nagy and Foias and the work of Popescu on models of row contractions. We also emphasize the concept of duality for $W^*$-correspondences and its relation to Morita equivalence of correspondenses.

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arXiv2004

Hardy algebras associated with $W^*$-correspondences (point evaluation and Schur class functions)

This is primarily an exposition of our work on Hardy algebras associated with $W^*$-correspondences with an emphasis on interpolation results (a generalized Nevanlinna-Pick theorem) and the concepts of Schur class operator functions (and realizations). We describe in detail the construction of the Hardy algebras, its representations and the important concepts of duality and point evaluation. Many examples are provided.

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arXiv2004

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.

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arXiv2003

Hardy Algebras, W*-Correspondences and Interpolation Theory

Given a von Neumann algebra $M$ and a $W^{\ast}$-correspondence $E$ over $M$, we construct an algebra $H^{\infty}(E)$ that we call the Hardy algebra of $E$. When $M=\mathbb{C}=E$, then $H^{\infty}(E)$ is the classical Hardy space $H^{\infty}(\mathbb{T})$ of bounded analytic functions on the unit disc. We show that given any faithful normal representation $σ$ of $M$ on a Hilbert space $H$ there is a natural correspondence $E^σ$ over the commutant $σ(M)^{\prime}$, called the $σ$-dual of $E$, and that $H^{\infty}(E)$ can be realized in terms of ($B(H)$-valued) functions on the open unit ball $\mathbb{D}((E^σ)^{\ast})$ in the space of adjoints of elements in $E^σ$. We prove analogues of the Nevanlinna-Pick theorem in this setting and discover other aspects of the value ``distribution theory'' for elements in $H^{\infty}(E)$. We also analyze the ``boundary behavior'' of elements in $H^{\infty}(E)$ and obtain generalizations of the Sz.-Nagy--Foia\c {s} functional calculus. The correspondence $E^σ$ has a dual that is naturally isomorphic to $E$ and the commutants of certain, so-called induced representations of $H^{\infty}(E)$ can be viewed as induced representations of $H^{\infty}(E^σ)$. For these induced representations a double commutant theorem is proved.

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arXiv2002

The Dixmier-Douady Class of Groupoid Crossed Products

We give a formula for the Dixmier-Douady class of a continuous-trace groupoid crossed product that arises from an action of a locally trivial, proper, principal groupoid on a bundle of elementary $C^*$-algebras that satisfies Fell's condition.

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arXiv2002

Continuous-Trace Groupoid Crossed Products

Let $G$ be a second countable, locally compact groupoid with Haar system, and let $\mathcal{A}$ be a bundle of $C^{\ast}$-algebras defined over the unit space of $G$ on which $G$ acts continuously. We determine conditions under which the associated crossed product $C^{\ast}(G;\mathcal{A})$ is a continuous trace $C^{\ast}$-algebra.

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