Texas A&M:

Abstracts

Amenable actions, C*-crossed products and applications to von Neumann algebrasNate Brown, Penn State University |

About five years ago Narutaka Ozawa used C*-results and techniques to prove his remarkable solidity theorem for certain group von Neumann algebras. Later work, some in collaboration with Sorin Popa, pushed these ideas further and gave nonisomorphism results for certain tensor products and free products. The goal of my lectures is to explain these fundamental contributions, highlighting the role of C*-crossed products and amenable actions. |

Invariants of the Cuntz-Pimsner algebra of a vector bundleMarius Dadarlat, Purdue University |

The Cuntz-Pimsner algebra of a complex vector bundle E of rank n over a compact space X is a locally trivial bundle O(E) with fiber the Cuntz algebra O(n). We will explore the following question suggested to us by Cuntz: What invariants of the vector bundle E are captured by the isomorphism class of the continuous field O(E)? |

Irreducible sofic shifts and non-simple Cuntz-Krieger algebrasSøren Eilers, University of Copenhagen |

Irreducible sofic shifts are given as the collection of all doubly infinite sequences which one may read off a labeled and strongly connected graph such as the one indicated. Classifying such objects up to so-called flow equivalence is an important but demanding challenge; perhaps surprisingly demanding in view of the fact that the case of a shift of finite type - where all edges are labelled uniquely - was solved in a very satisfactory way by Franks in the mid-eighties. |

The construction of Cuntz and Krieger relates flow equivalence of shifts of finite type to stable isomorphism of simple C*-algebras in a way which may be generalized to the sofic case by work of Matsumoto. In this case non-simple Cuntz-Krieger algebras arise, allowing us to bring recent progress in the understanding of these objects in play in parallel with tools more internal to sofic shifts. |

I will attempt a complete report on the status of this problem and present results by Bates, Boyle, Carlsen, Huang, Johansen, Pask, Restorff, Ruiz, Samuel and myself with direct or indirect implications for its solution. In particular, I can report on partial flow classification results by Boyle, Carlsen and myself. |

The commutant of B(H) in its ultrapowerIlijas Farah, York University |

For an infinite-dimensional separable Hilbert space H let B(H) be its algebra of bounded linear operators. Fix a free ultrafilter U on N and consider the ultrapower of B(H). Eberhard Kirchberg asked whether the commutant of B(H) in its ultrapower is equal to the scalar multiples of the identity. I will answer 3/4 of this question and give some information about the remaining 1/4. |

This is a joint work with N. Christopher Phillips and J. Steprans. |

Prime factors and amalgamated free productsCyril Houdayer, UCLA |

I will show that any non-amenable factor arising as an amalgamated free product of von Neumann algebras over an abelian von Neumann algebra is prime, i.e. cannot be written as a tensor product of diffuse factors. This gives new examples of prime factors of type II_1 and of type III. I will moreover discuss some applications in orbit equivalence. This is joint work with Ionut Chifan. |

Turbulence, representations, and trace-preserving actionsDavid Kerr, Texas A&M University |

The last twenty years have witnessed an explosion of activity in the study of the complexity of classification problems from a descriptive set theory perspective. I will give a survey of this field and discuss recent contributions of Hanfeng Li, Mikael Pichot, and myself in which we apply C*-algebra theory and Hjorth's notion of turbulence to analyze the complexity of various classes of dynamical systems, following a line of development that traces through work of Kechris and Sofronidis, Hjorth, and Foreman and Weiss. |

On a class of II_1 factors with at most one Cartan subalgebra [slides]Narutaka Ozawa, University of Tokyo |

In recent few years, the classification program of finite von Neumann algebras has seen a remarkable progress. I will review this subject and report some new results obtained by S. Popa and myself. |

Group cocycles and the ring of affiliated operatorsJesse Peterson, University of California, Berkeley |

I will present some results (joint work with Andreas Thom)
on cocycles from a group into its left regular representation and
also into the ring of affiliated operators of the group von Neumann
algebra. Specifically I will be interested in when a group Γ
has positive first l^{2}-Betti number. I will include a strong
generalization of a result of Lück and Gaboriau which states that if
Δ is a finitely generated normal subgroup of a group Γ
with 0<β_{1}^{(2)}(Γ)<∞ then either |Δ|<∞ or [Γ:Δ]<∞. |

Connected masas in UHF algebrasChris Phillips, University of Oregon |

It has been an open question for some time whether there
is a maximal abelian subalgebra in a UHF algebra which is isomorphic
to C(X) for a connected space X. In this talk, we describe a method
for producing uncountably many mutually nonconjugate maximal abelian
subalgebras in the CAR algebra (the 2^{∞} UHF algebra), each
isomorphic to C([0,1]). |

This is joint work with Simon Wassermann. |

Automorphisms of operator algebras: from local properties to ultrapowersDavid Sherman, University of Virginia |

There are many ways in which one might say that an automorphism is close to being inner. In the first part of this talk I will give background and applications for the new notion of "local innerness." Then I will explain the relations between local innerness, approximate innerness, and their variations, especially for automorphisms of ultrapowers. Some of these results, and others which involve approximate equivalence, can be reinterpreted in terms of continuous (not classical) logic. |

Graph algebras and the Classification ProgramMark Tomforde, University of Houston |

In the past decade, the technique of associating a C*-algebra to a directed graph has provided a useful tool for modeling many well-known classes of C*-algebras --- particularly, classes of C*-algebras related to dynamical systems. More recently, algebraists have been interested in associating algebras, known as Leavitt path algebras, to directed graphs. It turns out that each Leavitt path algebra sits as a dense *-subalgebra in a graph C*-algebra, and the structures of the two objects are intimately related. We will discuss interactions of the graph algebra theories with the Classification Program. In particular, we'll see that graph algebras can be used to model and provide insights for certain classifiable C*-algebras; and conversely, we'll show how classification results have applications to the graph algebra theories. |

Classification of smooth transformation group C*-algebrasAndrew Toms, York University |

I: Overview of classification and its connections to dynamics II: Recursive subhomogeneous C*-algebras in crossed products III: Continuation of II, and some applications |

Perturbations of C*-algebraic invariantsStuart White, University of Glasgow |

Consider two C*-algebras concretely represented on the same Hilbert space. Kadison and Kastler asked 'Under what conditions are they close?' i.e. there exists a small ε>0 such that every element in the unit ball of one algebra is within ε of the other algebra. In the late 1970's, Erik Christensen showed that any algebra sufficiently close to an injective von Neumann algebra M must be of the form uMu*, where u is a unitary close to 1. Examples, due to Barry Johnson, rule out a unitary conjugacy result for close separable nuclear C*-algebras, but it is still open as to whether sufficiently close separable nuclear C*-algebras must be isomorphic. In this talk, we'll examine invariants of close C*-algebras, with the aim of using the classification program to obtain an isomorphism. In particular, we'll see that a C*-algebra (satisfying the UCT) which is sufficiently close to a Kirchberg algebra A (also satisfying the UCT) must be isomorphic to A. This is joint work with Erik Christensen, Allan Sinclair and Roger Smith. |

Decomposition rank and Z-stabilityWilhelm Winter, University of Nottingham |

For a simple, unital C*-algebra, finite decomposition rank implies Z-stability, where decomposition rank is a notion of topological dimension for nuclear C*-algebras and Z denotes the Jiang-Su algebra. We outline the proof of this result and explain its consequences for Elliott's program to classify nuclear C*-algebras by their K-theory data. As an application, we can now complete the classification of C*-algebras associated to smooth, minimal, uniquely ergodic dynamical systems (with compact and finite dimensional base spaces) by their ordered K-groups. |