Operator Algebras and Dynamics Seminar  

Fall 2014

All talks are from 3:45-4:45 p.m. in the Seminar room, unless otherwise specified.

  • Nov 24
    Convex Sets Associated to C$^*$-algebras Scott Atkinson University of Virginia Time: 03:45 PM

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    Given a separable C$^*$-algebra $A$, we can associate to $A$ an invariant given by a family of convex separable metric spaces. Each convex separable metric space is given by equivalence classes of $*$-homomorphisms of $A$ into a McDuff factor $M$. This family is closely related to the trace space of $A$, and in some cases this invariant appears to be finer than the trace space invariant. This is an ongoing project based off of a 2011 paper by Nate Brown.
  • Nov 21
    Representations of the Cuntz-Krieger algebras and Bratteli diagrams Prof. Sergey Bezuglyi The University of Iowa Time: 03:45 PM

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    The talk is devoted to the study of a new class of representations of the Cuntz-Krieger algebras constructed by semibranching function systems which are naturally related to stationary Bratteli diagrams. We show that isomorphic semibranching function systems generate unitarily equivalent representations of the Cuntz-Krieger algebras. We work with Markov measures defined on the path space of stationary Bratteli diagrams to construct isomorphic representations of these algebras. To do this, we associate a strongly directed graph to a stationary simple Bratteli diagram, and show that isomorphic graphs generate isomorphic semibranching function systems. We also consider a class of monic representations of the Cuntz-Krieger algebras, and classify them up to unitary equivalence. The talk is based on a joint paper with Palle E.T. Jorgensen.
  • Nov 17
    Cartan Pairs and Extensions of Inverse Semigroups Prof. David Pitts University of Nebraska Time: 03:45 PM

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    A pair (M,D) consisting of a von Neumann algebra M containing a MASA D is a Cartan pair if the span of the unitaries in M which normalize D has weak-* dense span in M and if there is a normal conditional expectation of M onto D. In the 1970s, Feldman and Moore used measured equivalence relations and a cohomology theory for equivalence relations to classify the family of all separably acting Cartan pairs. In this talk, I will describe joint work with Allan Donsig and Adam Fuller. We use extensions of inverse semigroups to classify all Cartan pairs. Our approach is more algebraic than the approach of Feldman-Moore. Also, our approach does not explicitly use much measure theory, instead the measure theory is encoded into the meet semi-lattice structure of a fundamental inverse semigroup constructed from the Cartan pair. In a famous paper, Muhly, Saito and Solel asserted a spectral theorem for bimodules, and used it to describe maximal subdiagonal algebras of von Neumann algebras which contain a Cartan MASA. Unfortunately, their proof of the spectral theorem contains a serious gap, and hence the validity of their description of maximal subdiagonal algebras is in question. As applications of our description of Cartan pairs, I will show how to rephrase the Spectral Theorem for Bures Closed Bimodules of Cameron-Pitts-Zarikian in terms of data associated with the pair (M,D) closer in the spirit of the original Muhly-Saito-Solel assertion. This leads to a description of all maximal subdiagonal algebras in the same spirit as the original Muhly-Saito-Solel description.
  • Nov 14
    Extendable endomorphisms of factors (part 2) Prof. Alexis Alevras USNA Time: 03:45 PM

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    An endomorphism of a factor M acting standardly on a Hilbert space H gives rise, via the conjugation operator, to an endomorphism of the commutant M'. The endomorphism is called extendable if there is a simultaneous extension of the actions on M and M' to an endomorphism of B(H). We will review recent results of Bikram, Izumi, Srinivasan and Sunder on extendability, with applications to the classification of endomorphism semigroups of factors of type II and III.
  • Nov 10
    Symbolic dynamics for three dimensional flows with positive topological entropy Yuri Lima University of Maryland Time: 03:45 PM

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    Since Hadamard, the construction of symbolic models for dynamical systems has been successfully implemented in many scenarios. In a joint work with Sarig, we deal with flows with positive speed on three dimensional manifolds. These include geodesic flows on surfaces, and Reeb vector fields. Provided the flow has positive entropy, we code it by a suspension over a countable Markov shift. Here is an application: for almost every metric on the two-sphere, there are positive real numbers C and h such that there are at least Ce^{Th}/T closed geodesics of size at most T.
  • Oct 20
    Extendable endomorphisms of factors (part 1) Prof. Alexis Alevras USNA Time: 03:45 PM

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    An endomorphism of a factor M acting standardly on a Hilbert space H gives rise, via the conjugation operator, to an endomorphism of the commutant M'. The endomorphism is called extendable if there is a simultaneous extension of the actions on M and M' to an endomorphism of B(H). We will review recent results of Bikram, Izumi, Srinivasan and Sunder on extendability, with applications to the classification of endomorphism semigroups of factors of type II and III.
  • Oct 09
    Diagonality and Idempotents, Jasper's frame theory problem, and Schur-Horn theorems Prof. Gary Weiss University of Cincinnati Time: 03:45 PM

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    (Joint work with Jireh Loreaux.) We explore various relations of operators to their diagonals which study we coin diagonality and explain. This will include how to use 0-diagonality (operators with zero diagonal in some basis) to answer a frame theory question equivalent to an idempotent question of Jasper's, for which I will give some background. Then we will explore recent and past work on Schur-Horn theorems on possible diagonals a positive compact operator can have. I hope to give an overview of some core steps in the development of both. This work was inspired by work of Gohberg, Markus, Arveson, Kadison, Jasper, Fan, Fong, Herrero, and others, which will be explained.
  • Oct 03
    The Kadison-Singer Problem (part 4) Vrej Zarikian USNA Time: 03:45 PM

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    In the first 3 lectures we developed the tools of the Marcus, Spielman, and Srivastava proof of KS, culminating with a probabilistic statement about the maximum root of the characteristic polynomial of a certain class of random positive-semidefinite matrices. In this lecture we outline how, by a series of clever reductions, the aforementioned probabilistic statement implies a positive solution to Anderson's Matrix Paving Problem (known to be equivalent to KS).
  • Sep 22
    The Kadison-Singer Problem (part 3) Vrej Zarikian USNA Time: 03:45 PM

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    Continuation of the previous talks, emphasizing multilinearity properties of the mixed characteristic polynomial, and statements about "comparison with the mean".
  • Sep 15
    The Kadison-Singer Problem (part 2) Vrej Zarikian USNA Time: 03:45 PM

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    We continue to develop the tools required for Marcus, Spielman, and Srivastava's proof of the Kadison-Singer Problem. In particular, real stability and the mixed characteristic polynomial.
  • Sep 05
    The Kadison-Singer Problem (part 1) Vrej Zarikian USNA Time: 03:45 PM

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    The Kadison-Singer Problem was a famous open problem in the theory of operator algebras, posed in 1959. It was recently (and completely unexpectedly) solved (in the affirmative) by three computer scientists: Adam Marcus, Daniel Spielman, and Nikhil Srivastava. Their proof is both novel and (in a sense) elementary. In this series of lectures I will give a detailed account of their proof. The first lecture will explain the problem as well as some of its many reformulations. I will also introduce and explore the properties of "real stability" for complex polynomials, a key ingredient of the proof.
  • Aug 21
    On diagonal actions of branch groups and corresponding characters Artem Dudko Stony Brook Time: 03:45 PM Abstract

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