Set theory and C*-algebras

January 23 to January 27, 2012

at the

American Institute of Mathematics, San Jose, California

organized by

Ilijas Farah and David Kerr

Original Announcement

This workshop will be devoted to applications of set theory to C*-algebras. It would be hard to imagine a workshop on set theory and C*-algebras as recently as ten years ago, and this meeting will bring together researchers in these two fields. Dramatic progress has brought set theory and operator algebras closer together over the last decade, and this meeting will invigorate the collaboration between the two subjects.

The workshop will concentrate on the following topics:

- Set-theoretic analysis of Elliott's classification program for nuclear C*-algebras. This would be a continuation of the broad endeavor to study the comparative complexity of classification problems prominent in descriptive set theory in the last twenty years. Progress in C*-algebra theory over the last decade has brought into focus the role of regularity properties like Z-stability as a means for understanding the effectiveness and limitations of K-theoretic invariants for the purpose of classification. One outstanding problem is to determine the jump in Borel complexity at the boundary of this effectiveness. While descriptive techniques have been successfully applied to classification problems in ergodic theory, most notably via Hjorth's notion of turbulence, and many nuclear C*-algebras arise from dynamical systems as crossed products, the fact that some dynamical information gets lost in the crossed product construction in an often mysteriously complicated way suggests that the analysis of Borel complexity in context of C*-algebras might require the development of new tools and lead to new insights in the application of descriptive set theory to classification problems.

-Potential applications of combinatorial set theory to C*-algebras, such as the question whether Naimark's problem can have a positive answer in some model of set theory or whether the Calkin algebra admits a K1-reversing automorphism. Negative answers to both of these questions are known to be relatively consistent with ZFC.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

The generator rank for C*-algebras
by  Hannes Thiel
The isomorphism relation for separable C*-algebras
by  George A. Elliott, Ilijas Farah, Vern Paulsen, Christian Rosendal, Andrew S. Toms, Asger Tornquist,  Math. Res. Lett. 20 (2013), no. 6, 1071-1080  MR3228621