Schedule for: 17w5127 - Future Targets in the Classification Program for Amenable C*-Algebras

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In Tikuisis-White-Winter's substantial breakthrough, it was shown that any faithful tracial state of a separable nuclear C*-algebra is quasidiagonal under the assumption of UCT. In my talk, we produce an alternative approach to their theorem, which has the feature that it avoids the UCT assumption. This is a joint work with M. Dadarlat.

Let A be a simple, nuclear, Jiang-Su stable C*-algebra, and let G be an amenable group. We conjecture that the following properties for an action of G on A are equivalent: strong outerness, the weak tracial Rokhlin property, finite Rokhlin dimension, and absorption of a model action on the Jiang-Su algebra. This equivalence has been confirmed for algebras whose tracial state space is a Bauer simplex with finite dimensional extreme boundary, and for a large class of groups. We will sketch the proof in the case of finite groups, and give an idea of how to extend the argument further. As an application, it follows that strongly outer actions of (residually) finite groups have Rokhlin dimension at most one. Parts of this talk are joint work with Ilan Hirshberg, and parts are joint work with Chris Phillips and Qingyun Wang.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Let A be a unital, separable, simple C*-algebra with stable rank one. Every positive element in the stabilization of A induces a rank function on the space of tracial states of A. We show that all possible rank functions are realized this way. This implies that the Cuntz semigroup of A is almost divisible as soon as it is almost unperforated. Assuming additionally that A has locally finite nuclear dimension (for example, that A is an ASH-algebra), we obtain that strict comparison of positive elements in A implies that A tensorially absorbs the Jiang-Su algebra. In particular, this verifies the Toms-Winter conjecture for ASH-algebras with stable rank one.

I’ll discuss the role of a ’tracial gluing’ condition in the implications of the Toms-Winter conjecture currently relying on trace space hypotheses. Based on joint work with Jorge Castillejos, Sam Evington, Aaron Tikuisis and Wilhelm Winter.

A tracial state on a C*-algebra $A$ is called \emph{matricial field} (MF) if there is a net of self-adjoint, linear, finite rank maps $\varphi_n$ on $A$ which approximately preserve the multiplication and approximately preserve the trace. To date, there is no known example of a trace which is not MF. We will discuss some recent progress on verifying the MF property for traces (notably, the Tikuisis-White-Winter Theorem) and some new examples of MF traces. For traces on crossed products, the best known results involve actions of free groups on nuclear C*-algebras. For instance, in joint work with Tim Rainone, we showed if $A$ is AH with real rank zero and $F$ is a free group, all traces on $A \rtimes_r F$ are MF. More, recently, this result has been extended beyond the real rank zero setting assuming $A$ has the ideal property and $\mathrm{K}_1(A)$ is a torsion group. The new step involves a finite-dimensional approximation result for states on Cuntz semigroups.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We discuss recent results relating the mean dimension of a dynamical system to the radius of comparison of its crossed product.

The Rokhlin property for flows on C*-algebras is a concept introduced and studied by Kishimoto, which was shown to occur in various natural examples. In this talk, I will report on progress in the classification of Rokhlin flows on C*-algebras, which in part is inspired by a similar recent theory in von Neumann algebras due to Masuda and Tomatsu. I will explain a positive solution to a conjecture of Kishimoto, which asserts that every Kirchberg algebra carries a unique Rokhlin flow up to cocycle conjugacy. We will also discuss a non-simple generalization of this statement, as well as a classification of Rokhlin flows on classifiable KK-contractible C*-algebras.

I will talk about my recent joint work with Ilan Hirshberg on bounding the nuclear dimension for (twisted) crossed products associated to actions of finitely generated virtually nilpotent groups on spaces with finite covering dimension.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Semigroup C*-algebras have been studied for several classes of semigroups. In this talk, we focus on monoids given by particular presentations (i.e., generators and relations), with an emphasis on monoids defined by a single relation. We discuss structural properties of the corresponding semigroup C*-algebras, and also K-theory.

A C*-dynamical system is said to have the ideal separation property if the ideals of the corresponding reduced crossed product can be described in terms of the invariant ideals of the underlying C*-algebra. For commutative exact C*-dynamical systems, a characterization of this property was recently obtained by Kawabe. I will discuss a characterization of the ideal separation property for arbitrary exact C*-dynamical systems in terms of noncommutative boundaries. This is joint work with Christopher Schafhauser.

Smale spaces are a class of dynamical system defined by Ruelle to axiomatize properties of basic sets of an Axiom A diffeomorphism. From a Smale space one can construct a number of C*-algebras; each is obtained from an equivalence relation. When the original system is mixing each of these C*-algebras is simple, separable, nuclear, and stably finite. I will discuss recent results on the structure of these algebras along with group actions on them. This talk is based on joint work with Karen Strung and Allan Yashinski.

Nicolas Monod has in a recent paper introduced a new class of groups, groups with fixed-point property for cones, characterized by always admitting a non-trivial fixed-point whenever they act on cones (under some additional hypothesis). He showed that this class contains all groups of sub-exponential growth and is contained in the class of supramenable groups. (It is not known if these three classes are distinct!) He proved a number of equivalent conditions to be a group with the fixed-point property for cones, and he established a list of permanence properties for this class of groups. Monod’s results have relevance for the existence of invariant traces on a (non-unital) C*-algebra with an action of a group. The purpose of my talk will be to explain some of Monod’s results and some of their applications to C*-algebras.

I will report on recent developments in noncommutative dimension theories of uniform Roe algebras associated to metric spaces with bounded geometry.\\ In a joint work with Rufus Willett, we show that for uniform Roe algebras, being AF, having stable rank one, having cancellation, and having finite decomposition rank, are all equivalent to the underlying metric space having asymptotic dimension zero. A countable group has asymptotic dimension zero if and only if it is locally finite. In a joint work with Hung-Chang Liao, we show that uniform Roe algebras of locally finite countable groups can be completely classified by $K_0$ groups together with units. To our best knowledge, this is the first classification result for non-separable and non-simple $C^*$-algebras. As a contrast, if the metric space $X$ is non-amenable and has asymptotic dimension one, then the $K_0$ group of the uniform Roe algebra over $X$ is always zero. Finally, we answer negatively to a question of Elliott and Sierakowski about the vanishing of $K_0$ of the uniform Roe algebras of non-amenable groups with high asymptotic dimension.\\ If time permits, we will discuss the relation between nuclear dimension of uniform Roe algebra and asymptotic dimension of its underlying metric space.

A C*-algebra is symmetric if it is isomorphic to its opposite algebra, or equivalently if it has a conjugate-linear automorphism. Not all nuclear C*-algebras, not even all homogeneous C*-algebras, are symmetric, but all currently classifiable ones are. A sufficiently nonsymmetric C*-algebra (one not KK-equivalent to its opposite algebra) would even be a counterexample to the UCT. A better understanding of symmetry will be needed to extend classification beyond its current limits, especially to nonsimple C*-algebras. I will discuss some aspects of symmetry and th rigidity of complex structure in C*-algebras, with some examples. The talk will be rather speculative, with more questions than answers.

We revisit the topology of the KK-groups and the role of the UCT in classification theory. We show (without assuming the UCT) that if $A$ is a separable exact residually finite dimensional C*-algebra with finitely generated K-homology, then $A$ embeds in the CAR algebra.

This talk will report on joint work in progress with Joan Bosa, Jianchao Wu, and Joachim Zacharias. A notion of action by a group on a Cuntz semigroup will be defined, and the various categories involved described. We also define a semigroup that encodes the action and indicate its role in the description of purely positive elements in certain crossed products.

In this talk, I will present a classification theorem of unital simple separable Z-stable C*-algebras of rationally generalized tracial rank at most one, due to Gong-Lin-Niu. Also I will present the reduction theorem due to Elliott-Gong-Lin-Niu.

I will outline a new proof of Kirchberg's classification of all separable, nuclear, $O_\infty$-stable C*-algebras using ideal related KK-theory. In particular, this gives a new, short and more elementary proof of the Kirchberg-Phillips theorem. The new key ingredient is an elementary trick which combines infiniteness and quasidiagonality of representations.

Quasi-lattice ordered groups and their Toeplitz algebras were introduced by Nica in 1992. A quasi-lattice ordered group is amenable if the concrete Toeplitz subalgebra acting on $l^2(P)$ is isomorphic to the universal one. Laca-Raeburn used "controlled maps" to find sufficient conditions for amenability. Here I will discuss a more general notion of controlled map. This is joint work with Ilija Tolich and Iain Raeburn.

I will outline a construction of a simple approximately homogeneous C*-algebra such that its Elliott invariant admits an automorphism which is not induced by an automorphism of the algebra. This is joint work with N. Christopher Phillips.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.