Abstract: Finite nuclear dimension is a regularity property of C*-algebras that have played a pivotal role in the Elliott classification program of C*-algebras. It has been a key problem in the field to verify this property for crossed product C*-algebras associated to topological (as well as C*-) dynamical systems. Previous results have largely focused on the case of free actions, since existing techniques are based on Kakutani-Rokhlin-type towers in one way or another.

This mini lecture series will be based on our recent joint paper, in which we show that any action of a finitely generated virtually nilpotent group on a finite-dimensional space gives rise to a crossed product C*-algebra with finite nuclear dimension. This is achieved by introducing a new topological-dynamical concept called the long thin covering dimension of an action, which involves a suitable version of Kakutani-Rokhlin-type towers for possibly non-free actions. The result covers additional examples beyond the case of virtually nilpotent groups, including boundary actions of hyperbolic groups as well as allosteric groups of certain wreath products, such as the lamplighter group; the latter examples are of particular recent interest, as they cannot be handled using almost finiteness techniques. Further applications involve graph C*-algebras and (twisted) group C*-algebras of virtually polycyclic groups.

The series will be divided into four lectures. The first lecture will be a general overview, and the last three lectures will delve into details, but are designed to be largely independent of one another.

Lecture #1: General overview. We review the background and past results, state our main results, and showcase some applications (Jianchao).

Lecture #2: Arbitrary actions of groups of polynomial growth have finite long thin covering dimension. We outline the proof of this theorem and indicate how the long thin covering dimension can be bounded in terms of the covering dimension of the space and the growth of the group using a new related concept we introduce, namely the large-scale packing constant. Time permitting, we will also indicate how to bound the long thin covering dimension for boundary actions of hyperbolic groups with the help of the so-called equivariant asymptotic dimension (Jianchao).

Lecture #3: Survey of the proof of our main theorem. We present an equivalent form of the long thin covering dimension and indicate how it is used to bound the nuclear dimension of a crossed product (Ilan).

Lecture #4: Allosteric actions. We survey the construction of allosteric profinite actions by wreath products and show how the main theorem is used to bound the nuclear dimension of the crossed product (Ilan).

Abstract: Dynamic asymptotic dimension is a dimension theory for topological dynamical systems introduced by Guentner, Willett, and Yu in 2017. Since its inception it has found numerous applications to the structure theory of C*-algebras as well as homology and K-theory. In this talk I will discuss some aspects of this dimension theory and its applications to C*-Algebras.

Abstract: The concept of non-commutative topological dimension has been a central theme in the study of C*-algebras. Among several notions, the nuclear dimension, introduced by Winter and Zacharias, serves as a non-commutative analogue of the Lebesgue covering dimension and has been pivotal in the classification program of simple unital C*-algebras.

A natural progression in the field is the classification of maps between C*-algebras. Similar to the classification of algebras, this endeavor necessitates strong regularity conditions, such as nuclear dimension, but applied at the level of the maps themselves rather than the algebras. In this talk, I will present a description of *-homomorphisms with nuclear dimension equal to zero.

This is joint work with Robert Neagu.

Abstract: Building on the notion of diagonal dimension introduced by Li, Liao, and Winter in 2023, we introduce the Cartan dimension for an inclusion pair of C*-algebras. We present a computation of an upper bound for the Cartan dimension of two cases: the inclusion of the unitization of c_0 (\mathbb N) in the Toeplitz Algebra, and the inclusion of C(X) in the crossed product by topologically free actions of the integers on the Cantor space X. This is joint work with W. Winter.

Abstract: Uniform property Gamma is a divisibility property of C*-algebras that has played a major role in the advances of classification theory.  Recently, a version of uniform property Gamma for Cartan sub-C*-algebras has been developed which is tightly related to the small boundary property, when the Cartan pair arises from a topological dynamical system. We will discuss these relations and further potential applications of the regularity theory of C*-pairs.

The talk reports on joint work with H. Liao, A. Tikuisis and A. Vaccaro, and on joint work with D. Kerr and S. Petrakos.

Abstract: 

Associated to every shift space, the Cuntz-Krieger algebra (C-K algebra for abbreviation) is an invariant of conjugacy defined and developed by K. Matsumoto, S. Eilers, T. Carlsen and many of their collaborators in the last decade. In particular, Carlsen defined the C-K algebra to be the full groupoid C*-algebra of the “cover”, which is a topological system consisting of a surjective local homeomorphism on a zero-dimensional space induced by the shift space. 

In 2022, K. Brix proved that the C-K algebra of the Sturmian shift has finite nuclear dimension, where the Sturmian shift is the (unique) minimal shift space with the smallest complexity function: p_X(n)=n+1. In  recent results (joint with Z. He), we show that for any minimal shift space with finitely many left special elements, its C-K algebra always have finite nuclear dimension. In fact, this can be further applied to the class of aperiodic shift spaces with non-superlinear growth complexity. 

Abstract: We show that the nuclear dimension of a (twisted) group C*-algebra of a virtually polycyclic group is finite. This prompts us to make a conjecture relating finite nuclear dimension of group C*-algebras and finite Hirsch length, which we then verify for a class of elementary amenable groups beyond the virtually polycyclic case. In particular, we give the first examples of finitely generated, non-residually finite groups with finite nuclear dimension. A parallel conjecture on finite decomposition rank is also formulated and an analogous result is obtained. Our method relies heavily on recent work of Hirshberg and the second named author on actions of virtually nilpotent groups on C0(X)-algebras.