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qmul school of mathematical sciences
Abstracts
Cho-Ho Chu, Queen Mary University of London
Jordan structures in C*-algebras and dynamics
We discuss the non-associative Jordan structures in C*-algebras and explain how one can use them to study holomorphic dynamics on unit balls of C*-algebras.
Joachim Cuntz, University of Münster
Left regular C*-algebras for semigroups from number theory
We describe the left regular C*-algebras associated with semigroups arising from the ring of algebraic integers in a number field. This includes a determination of their K-theory as well as of their KMS states.
Søren Eilers, University of Copenhagen   [slides]
Orbit and flow equivalence versus diagonal-preserving *-isomorphism of Cuntz-Krieger algebras
In their beautiful recent paper, Matsumoto and Matui proved that a simple Cuntz-Krieger algebra remembers the orbit and flow equivalence class of the irreducible shift of finite type defining it, provided that the canonical diagonal subalgebra is considered as a part of the data. Having such rigidity results for general shift spaces would provide a better understanding of the classification problem for general Cuntz-Krieger algebras, recently solved, but the proof by Matsumoto and Matui goes through a result by Boyle and Handelman which does not readily extend to the reducible case. I will present a different approach which establishes rigidity in general. The work presented is joint work with Arklint, Carlsen, Ortega, Restorff, Ruiz and Sørensen in various constellations.
George Elliott, University of Toronto
Recent progress in the classification of amenable C*-algebras
The class of unital simple separable UCT C*-algebras with finite nuclear dimension is now quite well understood. The non-unital ones may not be far behind (work in progress with Z. Niu).
Adrian Ioana, UC San Diego
Cocycle superrigidity for translation actions of product groups
Given a compact group G together with two countable dense subgroups Γ and Λ, one can consider the left-right translation of Γ×Λ on G. I will present a recent result which asserts that, under fairly general conditions, any cocycle for such an action with values into a countable group is cohomologous to a homomorphism. I will also discuss applications of this result to orbit equivalence and W*-superrigidity. This is based on joint work with Damien Gaboriau and Robin Tucker-Drob.
Evgenios Kakariadis, Newcastle University   [slides]
Operator algebras associated with subshifts
A subshift is characterized by a set of allowable words on d symbols and encodes the allowable operations an automaton performs. In the late 1990's Matsumoto constructed a C*-algebra associated to a subshift, deriving initially his motivation from the work of Cuntz and Krieger. These C*-algebras were then studied in depth in a series of papers.
In a recent work with Shalit we take another look at this context and study subshifts in terms of classification programmes on nonselfadjoint operator algebras and Arveson's Programme on the C*-envelope. We investigate two types of nonselfadjoint operator algebras and we show that they completely classify subshifts: (a) up to the same allowable words, and (b) up to local conjugacy of the quantized dynamics. In the case of sofic subshifts the quantized dynamics are encoded through the follower set graph and the two types of nonslefadjoint operator algebras offer classification: (a) up to label isomorphisms, and (b) unlabeled isomorphism. In addition we discover that the C*-algebra fitting Arveson's Programme is the quotient by the generalized compacts, rather than taking unconditionally all compacts as Matsumoto does. Actually there is a nice dichotomy that depends on the structure of the monomial ideal.
Nevertheless in the process we accomplish more in different directions. This happens as our case study is carried in the intersection of C*-correspondences, subproduct systems, dynamical systems and subshifts. In this talk we will give the basic steps of our results with some comments on their proofs. This is joint work with Shalit and with Barrett.
Elias Katsoulis, East Carolina University
Crossed products of operator algebras
We give the definition of the crossed product of an operator algebra by a locally compact group of completely isometric automorphisms using the notion of a relative crossed product. It seems that in general there are two competing candidates for defining the (full) crossed product. We will explain the merits of both candidates and show that in the case where G is amenable, both choices coincide. We will demonstrate that Takai duality holds in the non-selfadjoint setting as well and we will explain how the Hao-Ng isomorphism problem from C*-algebra theory relates to our concept of a crossed product of non-selfadjoint operator algebras. We will solve one important case of that problem using our theory.
Matthew Kennedy, University of Waterloo
An intrinsic algebraic characterization of C*-simplicity for discrete groups
In this talk I will present recent results that provide an intrinsic algebraic characterization of discrete groups that are C*-simple, meaning that they have simple reduced C*-algebras.
Volodymyr Nekrashevych, Texas A&M University
Groups and rings of dynamical origin
We will talk about recent results on groups and rings associated with dynamical systems (mostly the topological full group and its analogs, and cross-product rings). In particular, we will talk about their growth, amenability, finite generation, simplicity, and classification. We will show how easily defined dynamical systems produce interesting groups and rings that answer some previously open problems.
Piotr Nowak, Institute of Mathematics of the Polish Academy of Sciences
Spectral gaps, warped cones and the coarse Baum-Connes conjecture
The coarse Baum-Connes conjecture is a large-scale geometric variant of the Baum-Connes conjecture. In 1999 Higson showed that certain expanders are counterexamples to the conjecture. I will discuss the construction of a warped cone over a group action and explain why such warped cones over actions with spectral gaps should also be counterexamples to the coarse Baum-Connes conjecture. This is joint work with Cornelia Drutu.
Ian Putnam, University of Victoria
Minimal Zd-actions on the Cantor set
The first part of the talk will discuss the development of a model for minimal actions of Z2 on the Cantor set. In the case of Z-actions, such a model was constructed by Richard Herman, Christian Skau and myself, building heavily on the ideas on Anatoly Vershik. This has become widely used in dynamical systems and introduced new invariants. Generalizations to higher rank actions led to a classification of minimal actions of finitely generated groups on the Cantor set. However, the original model remains available only for the case of Z. The emphasis will be on the role of the first cohomology group of the action. In the second part of the talk we will discuss the cohomological invariant for the particular case of Zd-odometers. This is joint work, in progress, with Thierry Giordano and Christian Skau.
Jean Renault, University of Orléans   [slides]
Random walks on Bratteli diagrams
Random walks on Bratteli diagrams appear naturally in the theory of hyperfinite von Neumann algebras. Indeed they provide models for normal states on these algebras. Connes and Woods have also introduced the notion of matrix-valued random walks which provides models for amenable G-spaces, and in fact for all amenable G-spaces as shown by Adams, Elliott and Giordano. I will show how this construction can be extended to the case when G is a groupoid and relate it to random walks on groupoids. This is a report on work in progress with T. Giordano.
Mikael Rørdam, University of Copenhagen   [slides]
Just infinite C*-algebras
There is a well-established notion of just infinite groups, i.e., infinite groups for which all proper quotients are finite. The residually finite just infinite groups are particularly interesting. They are either branch groups (e.g., Grigorchuk's group of intermediate growth) or hereditarily just infinite groups (eg. Z, the infinite dihedral group, and SLn(Z)). It is natural to consider the analogous notion for C*-algebras, whereby a C*-algebra is just infinite if it is infinite dimensional and all its proper quotients are finite dimensional. The study of these C*-algebras was motivated by a question of Grigorchuk if the group C*-algebra associated with his group might have this property. We give a classification of just infinite C*-algebras in terms of their primitive ideal space. We will discuss examples and properties of the residually finite dimensional just infinite C*-algebras; and we will also discuss the question of Grigorchuk. This is joint work with R. Grigorchuk and M. Musat.
Roman Sauer, Karlsruhe Institute of Technology
L2-Betti numbers of totally disconnected groups and approximation
We discuss a generalization of the approximation theorem to lattices in totally disconnected groups that converge in the sense of invariant random subgroups. This is joint work with Henrik Petersen and Andreas Thom.
Klaus Schmidt, University of Vienna
Homoclinic points of algebraic group actions
Quite a lot is known about various notions of entropy of algebraic actions of very general groups, but establishing more detailed dynamical properties of such actions seems quite difficult, even in simple examples. In this talk I am planning to discuss properties like expansiveness, specification and entropy for principal algebraic actions of the discrete Heisenberg group. This talk is based on joint work with Doug Lind.
Jan Spakula, University of Southampton
Coarse medians and Property A
Coarse medians were invented recently by Brian Bowditch (2011), with the aim of providing a common framework for both hyperbolic groups and mapping class groups. Loosely speaking, coarse median spaces are metric spaces which are locally approximable by CAT(0) cube complexes. The "rank one" situation corresponds exactly to the case of hyperbolic groups, which are (in the appropriate sense) locally approximable by trees (= rank one CAT(0) cube complexes).
Our main result is that coarse median spaces of finite rank have property A. This provides an alternative proof of the result of Y. Kida (2005) that mapping class groups have property A. This is a joint work with Nick Wright.
Aaron Tikuisis, University of Aberdeen
The Roe algebra as a relative commutant
Roe algebras are certain large C*-algebras which encode coarse information about a metric space, and which play a role in index theory. I will present a new perspective on these algebras, by realizing them as a certain relative commutant.
Stuart White, University of Glasgow
Quasidiagonality and amenability
Quasidiagonality is a concept originating in work of Halmos in operator theory; it asks for block diagonal approximations of an operator algebra. It's a pretty mysterious property of a somewhat topological nature, and as noted by Rosenberg and Voiculescu quasidiagonality always ntails some level of amenability. In this talk (based on joint work with Aaron Tikuisis and Wilhelm Winter) I'll discuss a kind of converse: when can we get quasidiagonality from amenability and what consequences does this have for group algebras and for simple amenable C*-algebras.
Rufus Willett, University of Hawaii at Manoa
Two exotic examples
I'll discuss two examples where relationships between dynamical objects and associated C*-algebras "go wrong". The first is a non-amenable etale groupoid whose maximal and reduced C*-algebras are the same. The second is a free action of a discrete group on a compact space where the correspondence between ideals in the crossed product and invariant open sets in the space is not bijective. In both cases, failures of exactness are an essential feature: I'll try to explain why, and also ask a lot of questions.
Wilhelm Winter, University of Münster
Finite approximations in C*-dynamics
Approximations by finite structures appear in C*-algebra theory time and time again, and in quite different senses. I will isolate some of these and look for common patterns, with the hope of gaining better insights into the structure of amenable C*-algebras and their symmetries.