# Dynamics and C*-Algebras: Amenability and Soficity (14w5161)

## Organizers

George Elliott (University of Toronto)

Thierry Giordano (University of Ottawa)

David Kerr (Texas A & M University)

Andrew Toms (Purdue University)

## Objectives

Progress over the last several years at the interface of C*-algebra classification theory and dynamics has generated a host of tantalizing conceptual and technical questions involving regularity properties of C*-algebras, dimensional versions of entropy, noncommutative dimension theory, Rokhlin tower decompositions, perforation, orbit equivalence, and amenability. The workshop will pursue these lines of investigation in a way that integrates the expertise of specialists in dynamical systems who may not be versed in C*-classification theory. This is especially timely as recent developments in C*-classification have honed in on issues that have long been central in dynamics, such as the tension between additive and multiplicative structure that is characteristic of amenability and the distinction between internal and external approximation that figures prominently in the currently very active study of sofic groups.

Since its inception in the 1920s, the theory of operator algebras has developed in large part through ideas of a dynamical nature and has enjoyed a rich variety of connections to the study of group actions. Topological dynamics has in particular played a significant role in C*-algebra theory by providing a tool for the coordinatization of algebraic structure. The crossed product construction permits the exploitation of symmetry through the acting group and is generous enough to produce a variety of C*-algebraic phenomena. A natural problem is to gauge the precise extent of this variety, and in particular to determine when crossed products by group actions on compact metrizable spaces fall within the scope of the Elliott classification program for separable nuclear C*-algebras in its original $K$-theoretic formulation. There now exist classification results for large classes of crossed products, the origins of which can be traced to the circle algebra decompositions of Putnam for crossed products of minimal homeomorphisms of the Cantor set and of Elliott and Evans for irrational rotation algebras.

In one of the more recent breakthroughs, Toms and Winter proved, by applying Winter's innovative approach to C*-classification and building on work of Lin and Phillips, that crossed products of uniquely ergodic minimal homeomorphisms of infinite compact metrizable spaces with finite covering dimension are classified by ordered $K$-theory. On the other hand, Giol and Kerr showed that the Euler class obstructions that were first used by Villadsen in the analysis of simple C*-algebras can be recast in a dynamical form so as to place some crossed products of minimal homeomorphisms beyond the reach of $K$-theoretic classifiability in the manner of Toms's counterexamples to the Elliott conjecture. Complementing this line of investigation in the stably finite setting, R{\o}rdam and Sierakowski have recently explored the connection between paradoxicality for actions of nonamenable groups on the Cantor set and pure infiniteness for crossed products, in particular as it relates to the Kirchberg-Phillips classification of separable simple purely infinite nuclear C*-algebras satisfying the universal coefficient theorem.

Remarkable as all of this progress has been, many of the pieces of the puzzle have yet to be put together. One vexing question concerns the general relationship between $K$-theoretic classifiability and the topological-dynamical notion of mean dimension. Examples suggest that the latter is directly related to the degree to which comparability of positive elements in the crossed product can be detected by evaluation on traces, which is a fundamental issue in C*-classification theory. An outstanding problem is to determine the degree to which the multiplicative structure reflected in mean dimension can be reconciled with the additive structure in the Hirshberg-Winter-Zacharias notion of Rokhlin dimension based on tower decompositions. This ties at the C*-algebra level to a conjecture of Toms and Winter relating regularity properties relevant to the Elliott classification program. A primary aim of the workshop is to promote the interaction between C*-algebraists and dynamical experts specializing in recurrence and mixing-type phenomena so as to make headway on understanding how all of these ideas might fit together at a technical level.

In another largely unexplored direction one can ask about the classifiability and dimensional behaviour of crossed products of minimal actions of more general amenable groups, such as the Grigorchuk group. This raises the question of whether the measure-theoretic Rokhlin lemma can be used to model topological actions, in particular under the kind of regularity assumptions that underpin C*-classifiability. A wide-ranging problem in a related vein asks what a topological orbit equivalence theory might look like for groups beyond $\mathbb{Z}^d$ and/or spaces that are not zero-dimensional. To embark on these new projects that seek to widen the scope of C*-classification, we plan to enlist the expertise of researchers in combinatorial and geometric group theory.

Like Connes's classification in the von Neumann algebra case, the Elliott program is predicated on the notion of amenability, which is rooted in the idea of approximation by internal finite or finite-dimensional structures. This kind of finite approximation is the basis of the classical Kolmogorov-Sinai theory of dynamical entropy. The notion of external finite approximation for groups was formalized by Gromov and Weiss in a property called soficity, which can be viewed as a combinatorial analogue of the existence of tracial microstates for finite von Neumann algebras. Soficity is the basis of a remarkable and surprising expansion of the classical theory of dynamical entropy beyond amenability that was pioneered by Bowen a few years ago and further developed by Kerr and Li. Although entropy is known to be unrelated to the structure of the crossed product for minimal homeomorphisms of the Cantor set, various connections to operator algebras have nevertheless been revealed over the last decade through the work of Bowen on Bernoulli actions, of Li and Thom on $L^2$-torsion, and of Kerr and Li on combinatorial independence. Another fundamental goal of the workshop is to provide a forum for discussions on how these ideas might impact the analysis of C*-algebra structure through the concepts of order, dimension, and perforation germane to C*-classification theory.