# THEMATIC PROGRAMS

July 23, 2016

## Thematic Program on Operator Algebras

### Ontario Non-Commutative Geometry and Operator Algebras Seminars July 2007 - June 2008

 Archive of talks 2002-2003 Archive of talks 2003-2004 Archive of talks 2004-2005 Archive of talks 2005-2006 Archive of talks 2006-2007 Back to Thematic Year home

 July 3 Nadish de Silva July 1 Canada Day- No seminar June 26 Leonel Robert June 24 Aaron Tikuisis June 19 Kevin Teh June 17 Aaron Tikuisis June 12 Alin Ciuperca June 10 Greg Maloney June 5 Greg Maloney June 3 Greg Maloney May 13 Alan Ciuperca May 8 --2:10pm, Rm 210 Greg Maloney May 6, --2:10pm, Rm 210 Greg Maloney April 17 Aaron Tikuisis April 15 Greg Maloney April 5 John Phillips, University of Victoria An Index Theory for Certain Gauge Invariant KMS Weights on C*-algebras. April 1 Aaron Tikuisis Mar. 20 Asger Tornquist and Roman Sasyk On the (non)classification of factors in vN algebras Mar. 18 Fernando Mortari Mar. 13 Trieu Le Mar. 11 Leonel Robert Mar. 6 Alin Ciuperca Mar. 4 Leonel Robert Feb. 28 Takeshi Katsura Nonseparable UHF algebras Feb. 26 Maria Grazia Viola Feb. 19 Sergio Doplicher, University of Rome 1 Quantum Spacetime and Noncommutative Geometry Abstract: We investigate the interplay between the universal differential calculus and other known algebraic structures, like Hochschild boundary on one side, and the C*-structure on the other.The latter provides natural norms one can evaluate on forms; we will discuss a relevant application in the case of the algebra of Quantum Spacetime, that will be discussed and physically motivated.One finds that, while the Algebra itself is fully translation and Lorentz invariant, the four dimensional Euclidean distance is a positive operator bounded below by a constant of order one (in Planck units); the area operator and the four volume operator are normal operators, the latter being a Lorentz invariant operator with pure point spectrum, whose moduli are also bounded below by a constant of order one. While the spectrum of the 3 volume operator includes zero. These findings are in perfect agreement with the physical intuition suggested by the Spacetime Uncertainty Relations which are implemented by the Algebra of Quantum Spacetime. The formulations of interactions between quantum fields on Quantum Spacetime will be discussed. The various approaches to interactions, equivalent to one another on the Minkowski background, yield to different schemes on Quantum Spacetime, with the common feature of a breakdown of Lorentz invariance due to interactions. In particular one of these schemes will be discussed and motivated, which leads to fully Ultraviolet-Finite theories. We will conclude with remarks on the fact that, in presence of Gravity, the commutators of the coordinates might in turn depend on the quantum fields, giving rise to a quantum texture where fields and spacetime coordinates cannot be separated. Feb. 14 Takeshi Katsura Cocycle crossed products Feb. 12 Leonel Robert Feb. 7 Sniggy Mahanta On a moduli space problem in Noncommutative Geometry Feb. 5 Greg Maloney Jan. 24 Luis Santiago Jan. 22 Kris Coward Jan. 17 Trieu Le Jan. 15 Leonel Robert Jan. 10 Alin Ciuperca Jan. 7 Luis Santiago Dec. 6 Yasuhiko Sato An application of the Evans-Kishimoto intertwining argument I will explain about an existence of an extension of automorphisms on unital Kirchberg algebras. This theorem is an application of the Evans-Kishimoto intertwining argument. So I will explain about the homotopy lemma for the stability and proof of this theorem. Huaxin Lin, University of Oregon Approximate unitary equivalence in simple C*-algebras of tracial rank one It will be an informal report. We will present a theorem which tells us when two monomorphisms from some AH-algebra into a unital simple C*-algebra of tracial rank one are approximate unitarily equivalent. We will revisit approximately multiplicative completely positive linear contractions from C(X) into a matrix algebra. We will discuss the problem when they are close to a homomorphism. Then we will discuss approximately multiplicative completely positive linear contractions from C(X) into an interval algebra. Perhaps we will also discuss when the general cases can be obtained from these two cases. Dec. 4 David Kyed (Copenhagen University) L^2-Betti numbers for compact quantum groups L^2-Betti numbers have proved themselves important within the theory of discrete groups and it is therefore reasonable to ask for an extension of this concept to the world of quantum groups. I will discuss one natural way to obtain such an extension for compact quantum groups with tracial Haar state, and thereafter discuss computational results and an extension of classical result (about L^2-Betti numbers for amenable groups) to the quantum setting. Nov. 29 Toshihiko Masuda Classification of minimal actions of a compact Kac algebras with amenable dual on the AFD factor of type II_1 I will explain the uniqueness of minimal actions of a compact Kac algebra with amenable dual on the AFD factor of type II$_1$. This particularly implies the uniqueness of minimal actions of a compact group on the AFD factor of type II_1. This is a joint work with R. Tomatsu. Takeshi Katsura The Evans-Kishimoto intertwining argument I present the classification result of Z^2-actions on UHF algebras which was obtained by joint work with Hiroki Matui. I talk about the Evans-Kishimoto intertwining argument, which is one of the ingredient of the proof, with emphasis on possibility (or impossibility) of generalizing our argument to more general groups and algebras. Nov. 27 Francesc Perera (Universitat Autonoma de Barcelona) Purely infinite corona algebras. Nov. 22 Aidan Sims, University of Wollongong, Australia Compactly aligned product systems and Cuntz-Nica-Pimsner algebras In his seminal paper, Pimsner introduced a class of C*-algebras O_X associated to C*-correspondences X over A. Now known as Cuntz-Pimsner algebras, these C*-algebras simultaneously generalise Cuntz-Krieger algebras and crossed products by Z. In Pimsner's theory, if a in A satisfies a \cdot x = 0 for all x in X, then the canonical image of a in O_X is itself equal to zero; a situation which is at odds with the point of view suggested by graph algebras as generalised Cuntz-Krieger algebras. Recently, Katsura has modified Pimsner's construction to remedy this issue, and used the resulting C*-algebras to define and analyse his topological graph C^*-algebras. Discrete product systems X of C*-correspondences and associated C*-algebras O_X were introduced by Fowler with a view to simultaneously generalising Pimsner's construction and work of Nica and others on Toeplitz algebras associated to quasi-lattice ordered groups (G,P). Fowler's construction has the same drawbacks as Pimsner's when nonzero elements of A act trivially on the left of some fibres of X, but the example of higher-rank graph C*-algebras show that more subtle issues may arise even when each nonzero a acts nontrivially on each fibre. In this talk I shall show how to modify Fowler's construction to circumvent these issues. I will indicate how the resulting Cuntz-Nica-Pimsner algebras simultaneously generalise Katsura's O_X, higher-rank graph C^*-algebras, and Crisp and Laca's boundary-quotient algebras. This talk is based on joint work with Yeend, and with Carlsen, Larsen and Vittadello. Gunnar Restorff, University of Copenhagen Computing Kirchberg's Idealrelated KK-theory KK-theory is an important tool in the theory of classification of C*-algebra. Kirchberg's development of an idealrelated version of KK-theory has been shown to be important for the classification of (non-simple) purely infinite C*-algebras. To obtain a univariant (classifying) functor instead of using the bivariant KK-functor, one needs some results along the lines of Rosenberg and Schochet's UCT and Dadarlat and Loring's UMCT. Even though some progress has been made, this has not at all been solved yet. I will start out by introducing the usual UCT and UMCT. Then I will focus on the case where the specified idealstructure is just a single ideal, and talk about a UCT by Bonkat and some related problems that arise in this case. This is joint work with Sï¿½ren Eilers and Efren Ruiz. Nov. 20 Robin Deeley (University of Victoria) The Orbit Operator Given a bounded linear operator and a vector in a Hilbert space on which it acts, we associate a linear map which we call the orbit operator. The orbit operator has trivial kernel if and only if the vector is a cyclic vector of the original operator. We discuss the connections between the orbit operator and invariant subspaces and the differing behaviour of the orbit operators associated with contractions and strict contractions. Nov. 8 Luis Santiago Extended traces on a C*-algebra and the Cuntz semigroup Oct. 25 Ivan Dynov (Max Planck Institute for Mathematics) The type III_1 factor generated by the regular representations of the infinite-dimensional group $(B_0)^Z$. Martin Mathieu (Queen's University Belfast) Spectral characterizations of Jordan homomorphisms I will try to explain and motivate a problem on operators on II_1 factors that I am currently thinking about. Tuesday, 23 October 2007, Ilan Hirshberg (Ben Gurion University) C(X)-algebras and strongly self-absorbing C*-algebra Let X be a finite dimensional space, and A a separable C(X)-algebra. I'll discuss the following result: if D is a K_1-injective strongly self-absorbing C*-algebra, and each fiber A_x of the C(X)-algebra is D-absorbing, then A is D-absorbing. This is joint work with Mikael Rordam and Wilhelm Winter. Oct. 18 Ilijas Farah (York University) All automorphisms of the Calkin algebra are inner Phillips and Weaver have proved that the Continuum Hypothesis implies the existence of an outer automorphism of the Calkin algebra. On the other hand, Todorcevic's Open Colouring Axiom, OCA, implies that all the automorphisms of the Calkin algebra are inner. I will discuss OCA and present key parts of the proof of the latter theorem. No knowledge of the set theory is required. Marius Dadarlat (Purdue University) Trivialization of continuous fields of C*-algebras with strongly self-absorbing fibers Let A be a separable unital continuous field over a finite dimensional compact metric space X. Suppose that each fibre of A is isomorphic to the same strongly self-absorbing and K1-injective C*-algebra D. Then A is trivial field, i.e., A is isomorphic to C(X, D). The class of strongly self-absorbing C*-algebras was introduced by Winter and Toms. The only known examples are the UHF algebras of infinite type, the Cuntz algebras O(2) and O(infinity), the Jiang-Su algebra and tensor products of O(infinity) with UHF algebras of infinite type. All these examples are K1-injective. Tuesday, 16 October 2007, Bruce Blackadar (University of Nevada, Reno) Nonstable K-theory for properly infinite C*-algebras and unital free products. Oct. 11 Snigdhayan Mahanta (Max Planck Institute for Mathematics) Holomorphic vector bundles over noncommutative tori The category of the holomorphic vector bundles over noncommutative 2-tori will be discussed. Given any irrational $\theta$, a faithful exact functor from the category of finite dimensional representations of $\mathbb{Z} + \theta\mathbb{Z}$ to the category of holomorphic vector bundles over the noncommutative torus $\mathbb{T}_\theta$ will be constructed. Some homotopy theoretic consequences will be discussed. The flavour of the talk will be mostly algebraic (and geometric), rather than operator algebraic. This is a joint work with W. D. van Suijlekom. Mike Whittaker (University of Victoria) C*-algebras from Tilings and Infinite Rotational Symmetry Tilings with infinite rotational symmetry and long range order, such as the Pinwheel Tiling, are still not well understood. In this talk we will construct C*-algebras from the dynamical system associated with such a tiling. When the tiling has a substitution rule we produce a particularly tractable C*-subalgebra which is classifiable. This work extends the construction of Kellendonk and Putnam. Tuesday, 9 October 2007, Walter van Suijlekom, Radboud University Renormalization of gauge fields using Hopf algebras We discuss the Connes-Kreimer Hopf algebra of renormalization in the case of gauge theories. We show that the Slavnov-Taylor identities - which are the quantum analogues of the classical gauge symmetry - are compatible with the Hopf algebra structure, in that they generate a Hopf ideal. Consequently, the quotient Hopf algebra is well-defined and has those identities built in. This provides a rigorous proof of compatibility of the Slavnov-Taylor identities with renormalization. Oct. 4 Roman Sasyk Oct. 2 Lon Mitchell (Virginia Commonwealth University) Oct. 1 Soren Eilers (University of Copenhagen and the Fields Institute) From substitutions to tilings and C*-algebras A (rather complicated) computation of the K-groups of the C*-algebras associated by work of Matsumoto to substitutionaldynamical systems leads to a new and computable invariant for flow equivalence of such systems. The description of this invariant, obtained in joint work with Carlsen, as a stationary inductive system involves certain so-called augmented matrices, and the fact that these lead to flow invariants is far from intuitive. However, an alternative approach by Barge and Smith gives substantial input to the understanding of this phenomenon and promises a better understanding of the equivalence relation induced by stable isomorphism of Matsumoto algebras. Sept. 27 Yasuyuki Kawahigashi (University of Tokyo) Conformal field theory and representation theory of von Neumann algebras. Sept. 25 Stefanos Orfanos (Purdue University) Sept. 13 Ken Davidson Operator algebras of rank 2 graphs Sun Wei (University of Oregon) Dynamical systems on products of Cantor set and the circle Sept. 11 Nikolay Ivanov The K-Theory of Toeplitz C*-Algebras of Right-Angled Artin Groups Sept. 6 Benjamin Itza-Ortiz Minimal automorphisms of C*-algebras Sept. 4 Eberhard Kirchberg (Humboldt-Universität zu Berlin) Aug. 23 Adam Sierakowski Transformation group and C*-algebras Aug. 23 Leonel Robert Aug. 23 Todd Kemp (MIT) Haagerup inequalities and semigroup contractions properties in free probability Aug. 16 Daniel Markiewicz (Technion - Israel Institute of Technology) Aug. 14 Alin Ciuperca (University of Toronto) Aug. 9 Dan-Virgil Voiculescu (University of California, Berkeley) Free analysis Aug. 7 Santanu Dey (Ernst-Moritz-Arndt-Universitat) Aug. 2 Ivan Dynov (Max-Planck-Institute for Mathematics) Type III factors generated by regular representations of infinite-dimensional nilpotent groups. Abstract: We discuss analogues of regular representations of two groups of upper-triangular matrices of arbitrary order as defined and studied by Alexander Kosyak. The representations are defined in terms of gaussian measures on the space of infinite upper-triangular matrices. Further we consider von Neumann algebras generated by these representations and discuss their type (in the factorial case). July 31 Trieu Le (University of Toronto)