IWOTA 2019

International Workshop on

Operator Theory and its Applications

IWOTA 2019

International Workshop on

Operator Theory and its Applications

July 22-26, 2019

Instituto Superior Técnico

Lisbon, Portugal

Instituto Superior Técnico

Lisbon, Portugal

- Dimitrios Andreou

National and Kapodistrian University of Athens, Greece ## Crossed products of dual operator spaces and the approximation property

We compare the Fubini crossed product and the spatial crossed product of a locally compact group $G$ acting on a dual operator space $X$, studied by Hamana and others. The two notions coincide when $X$ is a von Neumann algebra (Takesaki-Digernes). We show that they coincide if the dual action (of the group von Neumann algebra $L(G))$ on the Fubini crossed product satisfies a certain non-degeneracy property. This yields an alternative proof of the recent result of Crann and Neufang that the two notions coincide when $G$ has the approximation property of Haagerup-Kraus.

- Pere Ara

Universitat Autònoma de Barcelona, Spain ## Graded $K$-Theory, Filtered $K$-theory and the classification of graph algebras

In this talk, I will introduce the graded Grothendieck group $K^{gr}_0$ of a Leavitt path algebra and I will relate it to its algebraic filtered $K$-theory. Our main result is that an isomorphism of graded Grothendieck groups of two Leavitt path algebras induces an isomorphism of their algebraic filtered $K$-theory and consequently an isomorphism of filtered $K$-theory of their associated graph $C^*$-algebras. As an application, we show that, since for a finite graph $E$ with no sinks, $K^{gr}_0\left (L(E)\right )$ of the Leavitt path algebra $L(E)$ coincides with Krieger’s dimension group of its adjacency matrix $A_E$, our result relates the shift equivalence of graphs to the filtered $K$-theory and consequently gives that two arbitrary shift equivalent matrices give stably isomorphic graph $C^*$-algebras. This result was only known for irreducible graphs.

This is joint work with Roozbeh Hazrat and Huanhuan Li, both from Western Sydney University.

- Jonathan Brown

University of Dayton ## Dense subalgebras of purely infinite simple groupoid $C^*$-algebras

This work is joint with L. Orloff Clark and A. an Huef. A simple Steinberg algebra associated to an ample Hausdorff groupoid is algebraically purely infinite if and only if the characteristic functions of compact open subsets of the unit space are infinite idempotents. If a simple Steinberg algebra is algebraically purely infinite, then the reduced groupoid $C^*$-algebra $C^*_r(G)$ is simple and purely infinite. But the Steinberg algebra seems to small for the converse to hold. For this purpose we introduce an intermediate *-algebra $B(G)$ constructed using corners $1_U C^*_r(G) 1_U$ for all compact open subsets $U$ of the unit space of the groupoid. We then show that if $G$ is minimal and effective, then $B(G)$ is algebraically properly infinite if and only if the $C^*_r(G)$ is purely infinite simple. We apply our results to the algebras of higher-rank graphs.

- Alcides Buss

Universidade Federal de Santa Catarina ## Amenability of (partial) actions and Fell bundles

I will explain recent developments in the theory of amenable (partial) actions of groups on C*-algebras and Fell bundles based on joint papers with Siegfried Echterhoff, Rufus Willett, Fernando Abadie and Damián Ferraro.

- Lisa Orloff Clark

Victoria University of Wellington ## Ample groupoids and Kirchberg algebras

We will investigate the following theorem: If $A$ is a Kirchberg algebra, then $A$ is isomorphic to the reduced $C^*$-algebra of an ample groupoid. Many people have contributed to the proof of this theorem. The idea is as follows: build a groupoid whose reduce $C^*$-algebra is a Kirchberg algebra with a prescribed K-theory. Then apply the Kirchberg-Phillips classification theorem. In this talk we will survey some of the methods used to build the desired groupoids. We will also investigate exactly what groupoid properties are required to make the converse hold.

- Raphael Clouatre

University of Manitoba ## Residual finite-dimensionality for general operator algebras

Finite-dimensional approximation properties have proven to be a fruitful tool in the realm of $C^\ast$-algebras. It is thus natural to hope that similar ideas can elucidate the structure of general (not necessarily self-adjoint) operator algebras. In this talk we will study residual finite-dimensionality from that perspective. The departure from the self-adjoint world involves some interesting subtleties. For instance, it is well known that finite-dimensional operator algebras cannot necessarily be represented completely isometrically inside of an algebra of matrices, in contrast with the situation for $C^\ast$-algebras. As such, it it not immediately obvious what the "natural" definition of this more general notion of residual finite-dimensionality should be. After clarifying this issue, we will explore the extent to which the residual finite-dimensionality of an operator algebra carries over to its $C^\ast$-envelope or its maximal $C^\ast$-cover. This is joint work with Christopher Ramsey.

- Shirly Geffen

Ben-Gurion University of the Negev ## Classification of reversible and irreversible Pimsner operator algebras

In this talk we will show that two intensely studied topics, classification of non-self adjoint algebras, and classification of $C^\ast$-algebras with additional structure, are intimately connected.

One of our main motivations from the non-self adjoint world is the work of Solel and the work of Katsoulis and Kribs. They showed that non-self adjoint graph tensor algebras contain all the information about the graphs themselves. Namely, for any two graphs $G$ and $G'$, $\mathcal{T}_+(G)\cong \mathcal{T}_+(G')$ if and only if $G\cong G'$.

On the other hand, from the self-adjoint world, a recent result by Brownlowe, Laca, Robertson and Sims shows that for any two finite directed graphs $G$ and $G'$, $\mathcal{T}(G)\cong \mathcal{T}(G')$ by a base-preserving, gauge-invariant *-isomorphism if and only if $G\cong G'$. We extend this result to arbitrary graphs.

We will see that the result for graphs is an application of a broader $C^\ast$-rigidity problem. Taking $E$ and $F$ to be $C^\ast$-correspondences over two $C^\ast$-algebras, we ask when an isomorphism between the Toeplitz algebras, $\mathcal{T}(E)\cong \mathcal{T}(F)$ (or, the tensor algebras, $\mathcal{T}_+(E)\cong \mathcal{T}_+(F)$) gives an isomorphism between the $C^\ast$-correspondences. We give answers under conditions on the isomorphisms and the base algebras.- Rachid El Harti

University Hassan 1st, Morocco ## The amenability to algebraic and analytical perspective.

In this talk, we investigate the amenability to the point of algebraic and analytical view and its relationship with the semisimplicity in the case of operator algebras and crossed product Banach algebras associated with a class of $C^\ast$-dynamical systems.

- Michael Hartz

Fernuniversitaet in Hagen ## Dilations in finite dimensions and matrix convexity

Dilation theorems make it possible to represent fairly general operators on Hilbert space as pieces of better understood operators on a larger Hilbert space. However, in classical dilation results such as Sz.-Nagy's dilation theorem, the dilation typically acts on an infinite dimensional space, even if the original operator lives in finite dimensions. To remedy this drawback, finite dimensional versions of classical dilation theorems have been established by several authors.

I will talk about an abstract dilation result for completely positive maps on finite dimensional operator systems. This result shows when an infinite dimensional dilation theorem has a finite dimensional cousin. Moreover, I will explain how these questions are related to matrix convexity. This is joint work with Martino Lupini.

- Evgenios Kakariadis

Newcastle University ## Using entropy to compute phase transitions for Pimsner algebras

KMS-states of Pimsner algebras have been under thorough study in the past years. Their phase transitions can give an invariant for gauge-preserving isomorphisms. In this talk I will present how the notion of entropy mixes with their parametrization and recovers past results. Time permitting we will discuss how this works for graph algebras and Nica-Pimsner algebras.

- Aristides Katavolos

National and Kapodistrian University of Athens, Department of Mathematics ## Ideals, bimodules, synthesis, and boundaries.

We study a correspondence between closed (left) ideals of the algebra $L^1(G)$ (resp. the Fourier algebra A(G)) and bimodules of $B(L^2(G))$ over the dual algebras. A duality relation is established and used to explore spectral vs operator synthesis (resp. non-commutative Poisson boundaries).

This is a survey of joint work with M. Anoussis (Univ. of the Aegean) and I. G. Todorov (QUB Belfast).

- Elias Katsoulis

East Carolina University, USA ## The non-selfadjoint approach to the Hao-Ng isomorphism

We survey recent work with Chris Ramsey and also with Adam Dor-On regarding the Hao-Ng isomorphism problem and the way it relates to non-selfadjoint operator algebras.

- Arundhathi Krishnan

University College Cork ## Markovianity and Representations of the Thompson monoid $F^+$

In the process of identifying a suitable distributional symmetry to describe Markovianity, it has been conjectured by C. Köstler that there is a certain correspondence between unilateral Markov shifts and representations of the Thompson monoid $F^+$.

After having illustrated this correspondence in the context of tensor products of $W^*$-algebraic probability spaces, I will present the following two general results. A representation of the Thompson monoid $F^+$ in the endomorphisms of a $W^*$-algebraic probability space yields a noncommutative Markov process (in the sense of K\"ummerer). Conversely, such a representation is obtained from a noncommutative Markov process which is given as coupling to a so-called spreadable noncommutative Bernoulli shift.

(with B. V. R. Bhat, C. Köstler, V. Kumar, S. Wills)

- Bartosz Kwasniewski

University of Bialystok ## Noncommutative Cartan $C^\ast$-subalgebras and aperiodicity

Celebrated Renault’s result states that all regular maximal abelian $C^\ast$-subalgebras in separable $C^\ast$-algebras with faithful conditional expectation correspond to twisted groupoids. Recently, Ruy Exel generalized this result to noncommutative subalgebras, by replacing commutants with virtual commutants and groupoids with Fell bundles over inverse semigroups.

In this talk we extend this result in several ways. We cover non-separable $C^\ast$-algebras, we identify the inverse semigroup actions that give rise to Exel's noncommutative Cartan subaglebras, and characterise such subalgebras in terms of uniqueness of conditional expectation. We also introduce a notion of aperiodicity for $C^\ast$-inclusions, that gives natural simplicity and pure infiniteness criteria for a number of $C^\ast$-algebraic constructions. We discuss its relationship with noncommutative Cartan subalgebras.

Based on joint work with Ralf Meyer.

- Marcelo Laca

University of Victoria, Canada ## Ground states of groupoid $C^*$-algebras, phase transitions, and Hecke algebras.

We obtain a characterization of the ground states of a groupoid $C^*$-algebra under the time evolution determined by a cocycle, refining earlier work of Renault, and we apply our results to the study of equilibrium on Hecke $C^*$-algebras and Bost-Connes type systems arising from algebraic number fields. (this is joint work with N.S. Larsen and S. Neshveyev)

- Daniel Markiewicz

Ben-Gurion University of the Negev ## $E_0$-semigroups arising from boundary weight maps

An $E_0$-semigroup of $B(H)$ is a one parameter strongly continuous semigroup of $*$-endomorphisms of $B(H)$ that preserve the identity. The classification of $E_0$-semigroups up to cocycle conjugacy remains an intriguing problem. In this talk we will discuss it in a slightly different guise: the search for a rich class where classification is possible. Every $E_0$-semigroup that possesses a strongly continuous intertwining semigroup of isometries is cocycle conjugate to an $E_0$-semigroup obtained by the Bhat dilation of a $CP$-flow over a separable Hilbert space $K$. And Robert T. Powers showed how to construct $CP$-flows from boundary weight maps over $K$. In this talk we show how to construct and classify all $E_0$-semigroups (up to cocycle conjugacy) arising from boundary weight maps over finite-dimensional spaces that are $q$-pure in the following sense. We say an $E_0$-semigroup $α$ is $q$-pure if the $CP$-subordinates $\beta$ of norm one (i.e.$\|\beta t(I)\|=1$ and $\alpha t−\beta t$ is completely positive for all $t \geq 0$) are totally ordered in the sense that if $\beta$ and $γ$ are two $CP$-subordinates of $\alpha \circ f$ norm one, then $\beta \geq γ$ or $γ\geq \beta$.

This talk is based on the paper: C. Jankowski, D. Markiewicz and R.T. Powers,

*Classification of $q$-pure $q$-weight maps over finite dimensional Hilbert spaces*, to appear in the Journal of Functional Analysis, and preprint arXiv:1807.09824.- David Milan

University of Texas at Tyler ## On non-Hausdorff étale groupoids

Recent work by Exel and Pitts investigates the role of free points (units with trivial isotropy) in the unit space of a non-Hausdorff étale groupoid. In the topologically free case the free points are dense and the essential $C^*$-algebra of the groupoid is defined.

We investigate a larger subset of the unit space of a second countable étale groupoid which we call the Hausdorff points. We show that the Hausdorff points are dense and we consider the quotient of the $C^*$-algebra of the groupoid by the ideal of functions vanishing at these points. We relate our quotient to the essential $C^*$-algebra of the groupoid. Applications to ample groupoids and inverse semigroups are also considered.

This preliminary report is based on joint work with Allan Donsig and Ruy Exel.

- David Pitts

University of Nebraska, Lincoln, USA. ## Weak Cartan Inclusions

An inclusion is a pair of $C^*$-algebras $(\mathcal{C},\mathcal{D})$ with $\mathcal{D}\subseteq \mathcal{C}$, $\mathcal{D}$ abelian, and $\mathcal{D}$ containing an approximate unit of $\mathcal{C}$; the inclusion is regular when the set $\mathcal{N}(\mathcal{C},\mathcal{D}):=\{v\in \mathcal{C}: v^*\mathcal{D} v\cup v\mathcal{D} v^*\subseteq \mathcal{D}\}$ has dense linear span. Renault showed that when $(\mathcal{C},\mathcal{D})$ is a Cartan inclusion with $\mathcal{C}$ separable, there is a pair $(\Sigma, G)$ consisting of a Hausdorff topologically free étale groupoid $G$ and a twist $\Sigma$ over $G$ such that the pair $(\mathcal{C},\mathcal{D})$ is isomorphic to the Cartan inclusion $(C^*_{{\rm red}}(\Sigma, G), C_0(G^{(0)}))$.

In this talk, I will describe recent joint work with Ruy Exel in which we give a variant of Renault's result for

*weak Cartan inclusions*. Here the class of twisted groupoids $(\Sigma, G)$ which arise have $G$ topologically free and étale, but $G$ is not in general Hausdorff. Given such $(\Sigma, G)$, the*essential*groupoid $C^*$-algebra $C^*_{{\rm ess}}(\Sigma, G)$, is the completion of $C_c(\Sigma, G)$ with respect to a $C^*$-seminorm minimial among all $C^*$-seminorms on $C_c(\Sigma, G)$ which agree with the norm on $C_c(G^{(0)})$. The main result is that if $(\mathcal{C},\mathcal{D})$ is a weak Cartan inclusion, then there is a twisted topologically free étale groupoid $(\Sigma, G)$ such that $(\mathcal{C},\mathcal{D})$ is isomorphic to the weak Cartan inclusion $(C^*_{{\rm ess}}(\Sigma, G), C_0(G^{(0)}))$.I will conclude by discussing some interesting examples of weak Cartan inclusions which are not Cartan inclusions.

- Ahmed Al Rawashdeh

UAE University ## On the Extension of Unitary Group Isomorphisms of $C^*$-Algebras

The unitary groups are considered as a tool for the classifications of unital $C^*$-algebras. Dye showed that an isomorphism between the unitary groups in two factors not of type $I_{n}$ is implemented by a linear (or a conjugate linear) $*$-isomorphism of the factors. Al-Rawashdeh, Booth and Giordano proved that if the unitary groups of two simple unital AH-algebras with real rank zero are isomorphic as abstract groups, then their $K_0$-ordered groups are isomorphic. Also, using Dadarlat-Gong's classification theorem, we prove that such $C^*$-algebras are isomorphic if and only if their unitary groups are topological isomorphic. In the case of simple, unital purely infinite $C^*$-algebras, we show that two unital Kirchberg algebras are $*$-isomorphic if and only if their discrete unitary groups are isomorphic. We also discuss the extension problem, and following Dye's approach, if $\varphi$ is an isomorphism between the unitary groups of two unital $C^*$-algebras, we prove that there exists a $*$-isomorphism between the algebras which extends $\varphi$ on a certain subgroup of the unitaries. Indeed, if $\varphi$ is a continuous automorphism of the unitary group of a UHF-algebra $A$, then $\varphi$ is implemented by a linear or a conjugate linear $*$-automorphism of $A$. We also discuss whether $\varphi$ is fully $*$-extendable by an isomorphism between the $*$-algebras.

- Jean Renault

Université d’Orléans ## Groupoid extensions, group bundles and weights

We give a groupoid version of Mackey normal subgroup analysis in a $C^*$-algebraic framework. More precisely, we describe the $C^*$-algebra of a groupoid which is an extension by a group bundle. When the group bundle is abelian, one obtains a twisted groupoid $C^*$-algebra. $C^*$-bundles and the canonical weight of a group bundle play a crucial role in this study. This is joint work with M. Ionescu, A. Kumjian, A. Sims and D. Williams.

- Pedro Resende

CAMGSD, Instituto Superior Técnico ## Generalized $C^*$-diagonals via maps of quantales

A general construction of quantale maps from Fell bundles has been studied in [1], where a dictionary relating properties of the bundles to those of the maps is provided. In ongoing work with J. P. Santos, the reverse direction is being pursued, namely addressing how maps of quantales $p:\operatorname{Max} A \to O(G)$, where $A$ is a $C^*$-algebra and $G$ is a locally compact Hausdorff étale groupoid, determine Fell bundles on $G$. The inverse image homomorphism $p^*$ of such a map sends $G_0$ to an abelian subalgebra of $A$, and in certain circumstances this subalgebra contains all the information that is needed. This happens, in particular, if $G$ is topologically principal.

### Reference

- P. Resende, Quantales and Fell bundles, Adv. Math. 325 (2018) 312-374

- Baruch Solel

Technion, Israel Institute of Technology ## Homomorphisms of noncommutative Hardy algebras

I will discuss completely contractive homomorphisms among Hardy algebras that are associated with $W^*$-correspondences. I will present interpolation ("Nevanlinna-Pick type") results and discuss the properties of the maps (on the representation spaces) that induce such homomorphisms.

This is a joint work with Paul Muhly.

- Vrej Zarikian

United States Naval Academy ## Unique Extension Properties for $C^\ast$-Inclusions

Let $A\subseteq B$ be a $C^\ast$-inclusion}, i.e., an inclusion of unital $C^\ast$-algebras with the same unit. Structural properties of the inclusion are often reflected by the fact that certain families of UCP (unital completely positive) maps on $A$ extend uniquely to UCP maps on $B$. In particular, depending on the structure of $A\subseteq B$, it could be the case that

- every pure state on $\mathcal{A}$ extends uniquely to a pure state on $\mathcal{B}$ (i.e., $\mathcal{A} \subseteq \mathcal{B}$ has the
**pure extension property**); - a weak* dense set of pure states on $\mathcal{A}$ extend uniquely to pure states on $\mathcal{B}$ (i.e., $\mathcal{A} \subseteq \mathcal{B}$ has the
**almost extension property**); - the identity map $\operatorname{id}:\mathcal{A} \to \mathcal{A}$ extends uniquely to a UCP map $E:\mathcal{B} \to \mathcal{A}$ (i.e., $\mathcal{A} \subseteq \mathcal{B}$ has a unique
**conditional expectation**); - the identity map $\operatorname{id}:\mathcal{A} \to \mathcal{A}$ extends uniquely to a UCP map $\theta:\mathcal{B} \to I(\mathcal{A})$, where $I(\mathcal{A})$ is the injective envelope of $\mathcal{A}$ (i.e., $\mathcal{A} \subseteq \mathcal{B}$ has a unique
**pseudo-expectation**).

In this talk, we explore properties (i)-(iv) above, with a special emphasis on abelian inclusions $C(X)\subseteq C(Y)$ and inclusions $A\subseteq A\rtimes_r G$ arising from actions of discrete groups. Applications to determining the simplicity of reduced crossed products are provided.

- every pure state on $\mathcal{A}$ extends uniquely to a pure state on $\mathcal{B}$ (i.e., $\mathcal{A} \subseteq \mathcal{B}$ has the