

The 2017 Great Plains Operator Theory Symposium
Texas Christian University
Fort Worth, Texas
May 2226, 2017


Schedule and Abstracts
All talks will take place in Lecture Halls 14 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
Titles and abstracts received.
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 TomsWinter 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 BaumConnes type results). With Rufus
Willett and Guoliang Yu, we have formalized this into the
"Kcomputability 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 cooperation 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 Ktheory 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.
 Jens Kaad, Syddansk Universitet
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 measurepreserving setting and of
\(\mathcal{Z}\)stability in the C*algebraic setting. This
generalizes Matui's concept of the same name from the
zerodimensional 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 TomsWinter
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
CuntzPimsner algebras from subgroup
\(C^*\)correspondences
Abstract: Rieffelinduction 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 CuntzPimsner 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 RieszDunford 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 noncommuting operators? In this talk
we will discuss noncommutative functions, and how they can be
used to construct a functional calculus for noncommuting
\(d\)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.NagyFoias 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 GaussBonnetChern 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 LandstadOlesenPedersen
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.