Seminar in operator theory and operator algebras (MATH 9310)
Spring 2016

The seminar is organized by David Sherman. We meet Tuesdays 4-5 in Kerchof 326.

February 23

Joe Ball, Virginia Tech

The Schur-Agler class: the free noncommutative setting

The Schur-Agler class consists of functions over a domain satisfying an appropriate von Neumann inequality. Originally defined over the polydisk, the idea has been extended to general domains in multivariable complex Euclidean space with matrix polynomial defining function as well as to certain multivariable noncommutative-operator domains with a noncommutative linear-pencil defining function. Still more recently there has emerged a free noncommutative function theory (functions of noncommuting matrix variables respecting direct sums and similarity transformations -- the recent book of Kaliuzhnyi-Verbovetskyi--Vinnikov listed below provides a systematic self-contained treatment). The talk will discuss extensions of the Schur-Agler-class theory to the free noncommutative function setting. Time permitting, this will include the positive-kernel-decomposition characterization of the class, transfer-function realization and Pick interpolation theory. A special class of defining functions is identified for which the associated Schur-Agler class coincides with the contractive-multiplier class on an associated noncommutative reproducing kernel Hilbert space; in this case, solution of the Pick interpolation problem is in terms of the complete positivity of an associated Pick matrix which is explicitly determined from the interpolation data. The talk is based on joint work with Gregory Marx (Virginia Tech) and Victor Vinnikov (Ben Gurion University).

-- J.A. Ball, G. Marx, and V. Vinnikov, Noncommutative reproducing kernel Hilbert spaces, preprint.
-- J.A. Ball, G. Marx, and V Vinnikov, Interpolation and transfer-function realization for the noncommutatjive Schur-Agler class, preprint.
-- D.S. Kaliuzhnyi-Verbovetskyi and V. Vinnikov, Foundations of Free Noncommutative Function Theory, Math. Surveys and Monographs Vol. 199, AMS, 2014.

March 15

Julian Buck, Francis Marion University

Classification of C*-algebras arising from minimal discrete dynamical systems

The class of transformation group C*-algebras, that is, crossed product C*-algebras arising from the action of a homeomorphism h on a compact metric space X, has provided one of the best examples of the success of the classification program for nuclear C*-algebras. Beginning with the work of Putnam on minimal actions on the Cantor set, and continuing through the expansive work of Q. Lin and Phillips on compact manifolds, Toms and Winter on finite-dimensional spaces, and Elliott, Niu, and H. Lin on the general case of dynamical systems with mean dimension zero, we will survey the known results and successes that have led historically to the current understanding. We will also discuss what is known about crossed products by actions on algebras of the form C(X,A) (that is, continuous functions from X into an algebra A) in certain situations where A has nice properties. Throughout, an underlying theme will be the use of various subalgebras to approximate (or tracially approximate) properties of the crossed product.

March 29

Kristin Courtney, UVa

Residual finite-dimensionality of universal C*-algebras

Residually finite-dimensional (RFD) C*-algebras are C*-algebras that can be faithfully represented as a direct sum of finite dimensional C*-algebras, and as such, they resemble finite-dimensional C*-algebras in useful and meaningful ways. However, determining whether or not a given C*-algebra is RFD is often difficult and occasionally equivalent to the infamous Connes Embedding Problem. We will survey some existing results and techniques that can determine whether or not a given universal C*-algebra is RFD.

April 5

Kristin Courtney, UVa

Residual finite dimensionality of universal C*-algebras, part II

We will continue our survey of RFD C*-algebras by formalizing the notion of universal C*-algebras and discussing the role of lifting properties of relations on these algebras. We will conclude with results and questions about when combinations of RFD C*-algebras are again RFD.

April 12

Michael Hartz, University of Waterloo

von Neumann's inequality for commuting weighted shifts

von Neumann's inequality asserts that if T is a contraction on a Hilbert space and p is a polynomial, then

||p(T)|| ≤ sup { |p(z)| : |z| ≤ 1 }.

While Ando's dilation theorem implies an analogous inequality for pairs of commuting contractions, the corresponding statement for triples of commuting contractions is false. The first counterexamples were found by Kaijser-Varopoulos and Crabb-Davie in the early seventies, but this phenomenon is still not well understood. I will talk about a result which shows that von Neumann's inequality holds for a particularly tractable class of commuting contractions, namely multivariable weighted shifts. This provides a positive answer to a question of Lubin and Shields.

April 19

Corey Jones, Vanderbilt University

Rigid C*-tensor categories and their analytic properties

In this talk, we will discuss rigid C*-tensor categories. We describe the notions of amenability, the Haagerup property, and property (T) introduced for tensor categories by Popa and Vaes. We will explain how they can be understood through the representation theory of the tube algebra, and present some examples.

April 26

Benjamin Hayes, Vanderbilt University

1-bounded entropy and regularity problems in von Neumann algebras

We introduce and investigate the singular subspace of an inclusion of a tracial von Neumann algebra N into another tracial von Neumann algebra M. The singular subspace is a canonical N-N subbimodule of L2(M) containing the normalizer, the quasi-normalizer (introduced by Izumi-Longo-Popa), the one-sided quasi-normalizer (introudced by Fang-Gao-Smith), and the wq-normalizer (introduced by Galatan-Popa). By abstracting Voiculescu's original proof of absence of Cartan subalgebras, we show that the von Neumann algebra generated by the singular subspace of a diffuse, hyperfinite subalgebra is not L(F2). We rely on the notion of being strongly 1-bounded, due to Jung, and the 1-bounded entropy, a quantity which measures "how" strongly 1-bounded an algebra is. Our methods are robust enough to repeat this process by transfinite induction and we use this to prove some conjectures made by Galatan-Popa in their study of smooth cohomology of II1-factors. We also present applications to nonisomoprhism problems for Free-Araki woods factors, as well as crossed products by Free Bogoliubov automorphisms in the spirit of Houdayer-Shlyakhtenko.

May 3

Vrej Zarikian, US Naval Academy

Unique expectations and pseudo-expectations for C*-inclusions

A conditional expectation for a C*-inclusion of D in C is a contractive projection E from C to D. It can easily happen that a C*-inclusion admits no conditional expectations (for example, if C is injective, but D is not). Pitts introduced pseudo-expectations as a substitute for possibly non-existent conditional expectations. A pseudo-expectation is a completely positive map from C to I(D) that restricts to the identity on D. (Here I(D) is Hamana's injective envelope of D.) Pseudo-expectations generalize conditional expectations, but they must exist for any C*-inclusion, by injectivity. Typically there are many pseudo-expectations, but uniqueness is possible, depending on structural properties of the inclusion. For example, Pitts proves uniqueness for regular MASA inclusions. In this talk, based mostly on joint work with Pitts, we characterize when various natural classes of C*-inclusions admit a unique conditional expectation and/or pseudo-expectation. Our exposition will oscillate between specific examples and the general theory required to analyze them. Particular attention will be paid to abelian inclusions and W*-inclusions. Finally, as applications, we show how the existence of a faithful unique pseudo-expectation can simplify C*-envelope calculations, and how it relates to norming (in the sense of Pop, Sinclair, and Smith).

You can reminisce about previous semesters at the links below:
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