# The 2017 Great Plains Operator Theory Symposium

## Schedule and Abstracts

All talks will take place in Lecture Halls 1-4 of the Sid W. Richardson building, which is less than 10 minutes from the dormitories. See the campus map.

Photo of the Lecture Hall 1. (All lecture halls have a similar setup.) The wall area between the speakers is used instead of a screen.

### Abstracts

#### Plenary Speakers

• Nate Brown, Penn State University.
Nuclear $$C^*$$-algebras and analogies
Abstract: Understanding the structure and classification of injective factors was one of the great $$W^*$$-achievements of the last century. Over the last decade, progress in nuclear $$C^*$$-algebra theory revealed deep and powerful analogies with that work. These analogies were crucial beacons for progress on both the Toms-Winter Conjecture and Elliott's Classification Program.
On the geometric side, analogies between nuclear $$C^*$$-algebras and coarse geometry are also emerging. In this case, we are hopeful said analogies will lead to $$K$$-theory results for certain nuclear $$C^*$$-algebras (e.g. Kunneth Theorem or Baum-Connes type results). With Rufus Willett and Guoliang Yu, we have formalized this into the "K-computability Project."
In this talk, I will review the analytic analogies described in the first paragraph and discuss the geometric analogies which lead to the $$K$$-computability Project.
• George Elliott, University of Toronto
The classification question for nuclear $$C^*$$-algebras
Abstract: Since the beginning of the subject of operator algebras, classification questions have risen to the surface.
Amazingly, the amenable von Neumann algebras and $$C^*$$-algebras have been largely amenable to this purpose!—even to the extent of allowing actions of amenable groups to be studied as well. In the case of $$C^*$$-algebras, this co-operation has been somewhat grudging. (And the analysis of amenable inclusions has been so far almost entirely restricted to von Neumann algebras.)
This lecture will survey progress on the classification of amenable (= nuclear) $$C^*$$-algebras, which has recently attained heights in the finite case comparable to those achieved in the infinite case by Kirchberg and Phillips twenty years ago.
• Thierry Giordano, University of Ottowa
A model of Cantor minimal $$\mathbb{Z}^2$$-systems
Joint work with Ian F. Putnam and Christian F. Skau.
In 1992, Herman, Putnam and Skau used ideas from operator algebras to present a complete model for minimal actions of the group $$\mathbb{Z}$$ on the Cantor set, i.e. a compact, totally disconnected, metrizable space with no isolated points. The data (a Bratteli diagram, with extra structure) is basically combinatorial and the two great features of the model were that it contained, in a reasonably accessible form, the orbit structure of the resulting dynamical system and also cohomological data provided either from the K-theory of the associated C*-algebra or more directly from the dynamics via group cohomology. This led to a complete classification of such systems up to orbit equivalence. This classification was extended to include minimal actions of $$\mathbb{Z}^2$$ and then to minimal actions of finitely generated abelian groups. However, what was not extended was the original model and this has handicapped the general understanding of these actions.
In this talk I will indicate how we can associate to any dense subgroup $$H$$ of $$\mathbb{R}^2$$ containing $$\mathbb{Z}^2$$ a minimal action of $$\mathbb{Z}^2$$ on the Cantor set, such that its first cohomology group is isomorphic to $$H$$.
• Adrian Ioana, UC San Diego
Prime II$$_1$$ factors arising from irreducible lattices in products of simple Lie groups of rank one
Abstract: A II$$_1$$ factor is called prime if it cannot be decomposed as a tensor product of II$$_1$$ factors. In this talk, I will present joint work with Daniel Drimbe and Daniel Hoff in which we show that II$$_1$$ factors associated to icc irreducible lattices in products of simple Lie groups of rank one are prime. This provides the first examples of prime II$$_1$$ factors arising from lattices in higher rank semisimple Lie groups.
Operator $$*$$-correspondences: Representations and pairings with unbounded $$KK$$-theory
Abstract: In this talk I will describe a very general class of hermitian bimodules called operator $$*$$-correspondences. This kind of bimodules typically arises as the domain of a metric connection acting on a $$C^*$$-correspondence. Relying on the representation theory of completely bounded multilinear maps we shall then see how operator $$*$$-correspondences can be represented as bounded operators on a Hilbert space. As a further application and motivation for introducing operator $$*$$-correspondences I will describe how they (under an extra compactness assumption) admit an explicit pairing with a suitable abelian monoid of twisted unbounded Kasparov modules. The talk is partly based on joint work with David Blecher and Bram Mesland.
• Mehrdad Kalantar, University of Houston
• Stationary C*-dynamical systems
Abstract: We introduce the notion of stationary actions in the context of C*-algebras. As an application of this concept we prove a new characterization of C*-simplicity in terms of unique stationarity. This ergodic theoretical characterization provides an intrinsic and conceptual understanding of why C*-simplicity is stronger than the unique trace property. In addition it allows us to conclude C*-simplicity of new classes of examples, including recurrent subgroups of C*-simple groups.
This is joint work with Yair Hartman.
• David Kerr, Texas A&M
Almost finiteness and $$\mathcal{Z}$$-stability Abstract: I will introduce a notion of almost finiteness for group actions on compact spaces as an analogue of both hyperfiniteness in the measure-preserving setting and of $$\mathcal{Z}$$-stability in the C*-algebraic setting. This generalizes Matui's concept of the same name from the zero-dimensional context and is related to dynamical comparison in the same way that $$\mathcal{Z}$$-stability is related to strict comparison in the context of the Toms-Winter conjecture. Moreover, for free minimal actions of countably infinite groups on compact metrizable spaces the property of almost finiteness implies that the crossed product is $$\mathcal{Z}$$-stable, which leads to new examples of classifiable crossed products.
• Nadia Larsen, University of Oslo
Cuntz-Pimsner algebras from subgroup $$C^*$$-correspondences
Abstract: Rieffel-induction can be described by means of a $$C^*$$-corres\-pondence with right and left actions encoding how representations of a locally compact group and a fixed closed subgroup are related via induction and restriction. Viewing the $$C^*$$-correspondence over the group $$C^*$$-algebra of the subgroup, it seems natural to study its associated Cuntz-Pimsner algebra. It turns out that the resulting objects can be characterized in terms of familiar $$C^*$$-algebras, and there is a variety of outcomes depending on whether the subgroup is open, discrete or compact. Some intriguing connections with other constructions surface along the way. This is joint work with Steve Kaliszewski and John Quigg.
• Terry Loring, University of New Mexico Multivariate Pseudospectrum and $$K$$-theory
Abstract: We will discuss a common generalization of the Taylor spectrum and the pseudospectrum, called the multivariate pseudospectrum. Applied to the case of three or more hermitian matrices, this variation on the spectrum leads to spaces with interesting topology that can be computed by reliable numerical methods. Connections with $$D$$-branes and topologically protected states of matter will be discussed, but only briefly. Emphasis will be placed on connections with recent advances in real $$K$$-theory.
• John McCarthy, Washington University
Functional Calculus for Noncommuting Operators
Abstract: The Riesz-Dunford functional calculus lets you make sense of $$f(T)$$ when $$f$$ is a holomorphic function on some open set $$U \subseteq \mathbb{C}$$ and $$T$$ is an operator with spectrum in $$U$$. The Taylor functional calculus generalizes this to when $$U \subseteq \mathbb{C}^d$$ and $$T$$ is a $$d$$-tuple of commuting operators. But how does one define a functional calculus for non-commuting operators? In this talk we will discuss non-commutative functions, and how they can be used to construct a functional calculus for non-commuting $$d$$-tuples.
As an application, consider an equation like $X^3 + 2 X^2 Y + 3 XYX + 4 YX^2 + 5 XY^3 + 6 Y^2XY + 7 XY + 8 YX + 9 X^2 = 10$ If $$(X,Y)$$ is a pair of matrices that satisfy this equation, then, generically in $$X$$, we will show that $$Y$$ must commute with $$X$$. The talk is based on joint work with Jim Agler.
• Paul Muhly, University of Iowa
Applications of Geometric Invariant Theory to Free Analysis
Abstract: In this talk, which is based upon joint work with Erin Griesenauer and Baruch Solel, I will discuss advances in the problem of identifying Arveson's boundary representations and $$C^*$$-envelopes of subalgebras of homogeneous $$C^*$$-algebras built from algebras of generic matrices. Specifically, let $$X$$ be a compact subset of the $$d$$-tuples of $$n\times n$$ matrices, $$M_n^d(\mathbb{C})$$, and let $$\mathbb{G}(d,n, X)$$ be the closed subalgebra of $$C(X,M_n(\mathbb{C}))$$ generated by the "coordinate functions", $$\mathfrak{z} \to Z_k$$, where $$\mathfrak{z} = (Z_1,Z_2,\ldots, Z_d)\in M_n^d(\mathbb{C})$$. The problems that we address include: Describe the $$C^*$$-subalgebra of $$C(X,M_n(\mathbb{C}))$$ generated by $$\mathbb{G}(d,n, X)$$, $$C^*(\mathbb{G}(d,n, X))$$. Calculate sufficiently many boundary representations of $$C^*(\mathbb{G}(d,n, X))$$ for $$\mathbb{G}(d,n, X)$$ to determine the Shilov boundary ideal of $$C^*(\mathbb{G}(d,n, X))$$ for $$\mathbb{G}(d,n, X)$$.
• Gelu Popescu, UT San Antonio
Operator theory on noncommutative polyballs
Abstract: The talk is a survey of several aspects of operator theory on noncommutative polyballs including an analogue of Sz.-Nagy-Foias theory of contractions, a theory of free holomorphic functions on polyballs and their automorphisms, as well as results concerning the curvature invariant and Euler characteristic associated with the elements of the polyball and an extension of Arveson's version of Gauss-Bonnet-Chern theorem from Riemannian geometry. Several open problems are pointed out.
• Sarah Reznikoff, Kansas State University

• David Sherman, University of Virginia
Setting boundaries
Abstract: I will start by surveying some of the main mathematical concepts around the Choquet boundary for unital function spaces, due in largest part to Bishop and de Leeuw in 1959. Then I will discuss how the entire framework generalizes, or should generalize, to a noncommutative version. Many of the principal ideas originated in Arveson's 1969 work, but the actual results have been arriving gradually, and some of them are quite recent. I will point out places where there is more left to do.
• Roger Smith, Texas A&M A Galois correspondence for crossed products
Abstract: If a discrete group $$G$$ acts on an operator algebra $$A$$ (C$$^*$$ or von Neumann) the question arises of whether the algebras between $$A$$ and its crossed product by $$G$$ can be characterized by subgroups of $$G$$. When $$A$$ is a simple C$$^*$$-algebra and $$G$$ acts by outer automorphisms, a positive answer has been given by Landstad-Olesen-Pedersen when $$G$$ is abelian, by Choda (with some rather restrictive extra hypotheses) and by Izumi when $$G$$ is finite. In this talk I will give a positive solution for all discrete groups $$G$$ and discuss some consequences.
This is joint work with Jan Cameron.
• Mark Tomforde, University of Houston Classification of Graph Algebras
Abstract: Over the past two decades, graph $$C^*$$-algebras have emerged as a class of $$C^*$$-algebras that is simultaneously large and tractable. In addition to being used to define the construction, the graphs provide useful tools for analyzing and codifying the structure of their associated $$C^*$$-algebras. Based on the success of this approach, researchers have also introduced algebraic counterparts of the graph $$C^*$$-algebras, known as Leavitt path algebras, for which many similar results have been obtained. In the past few years great strides have been made in the classification of graph algebras, including results for both the graph $$C^*$$-algebras and Leavitt path algebras. These classification results have illuminated the relationships among not only the graph, the algebra, and the $$C^*$$-algebra, but also among related objects such as the graph groupoid, the shift space of the graph, and the diagonal subalgebra of the $$C^*$$-algebra. This talk will survey recent results for the classification of graph $$C^*$$-algebras and Leavitt path algebras. We will discuss the significance of these results and also describe some open problems and questions remaining to be answered.