### Abstracts of talks given at the SFB-workshop

# Groups, Dynamical Systems and C*-Algebras

### August 20 - August 24, 2013

### Zappa-Szép products of semigroups and their C*-algebras, Nathan Brownlowe

We examine a class of Zappa-Szép products of semigroups which generalise both the self-similar actions of Nekrashevych and the quasi-lattice ordered groups of Nica. We consider the C*-algebras of these products in the sense of Li, and we give an alternative presentation involving isometric representations of the semigroups. We discuss examples including self-similar actions of groups, the semigroup $\mathbb{N}\rtimes\mathbb{N}^\times$, and the Baumslag-Solitar groups. We also define a quotient C*-algebra we call the boundary quotient.

### Partial Crossed Product Description of the Cuntz-Li Algebras, Giuliano Boava

Abstract: A few years ago, Cuntz and Li introduced the C*-algebra A[R] associated to an integral
domain R with finite quotients. We show that A[R] is a partial group algebra with suitable relations.
We identify the spectrum of these relations as being homeomorphic to the profinite completion of R.
By using partial crossed product theory, we reconstruct some results proved by Cuntz and Li.
Among them, we prove that A[R] is simple by showing that the action is topologically free and minimal.
**Slides**

### Rudolph's theorem on the Furstenberg conjecture., Joachim Cuntz

Abstract: Rudolph's theorem states: Let p and q be two natural numbers which are relatively prime. Assume that $\mu$ is a measure on the unit interval which is invariant and ergodic for the two transfornations given by multiplication by p and q mod 1. If $\mu$ is not Lebesgue measure, then both transformations have entropy 0 with respect to $\mu$. We try to sketch a proof of this theorem and discuss consequences.

### The classification of C*-algebras with finitely many ideals, Soren Eilers

Although only in very few cases something one may establish by a direct appeal to classification results for simple C*-algebras, our insight into the classification theory of C*-algebras has taken dramatic steps forward over the last five years or so. Emphasizing applications of relevance to the focus of the conference, I will attempt to give an overview of the status quo of the area.

### Adjoining spectral projections to rotation algebras, George Elliott

In work with Zhuang Niu, it is shown that adjoining the logarithms of the canonical generators of the irrational rotation C*-algebra results in an AF algebra---and that the embedding is in some sense just the Pimsner-Voiculescu one. This leads to questions concerning more general crossed products.

### Graphs, groups and self-similarity, Ruy Exel

Abstract: We study a family of C*-algebras generalizing both Katsura algebras and certain algebras introduced by Nekrashevych in terms of self-similar groups.

### Lie groups actions, spectral triples and generalized crossed products, Oliver Gabriel

The aim of this talk is to generalise the constructions of spectral triples on noncommutative tori and Quantum Heisenberg Manifolds (QHM) to broader settings. After a few reminders about noncommutative tori and spectral triples, we prove that an ergodic action of a compact Lie group G on a unital C*-algebra A yields a natural spectral triple structure on A. In the second part, we investigate "permanence properties" for the previous sort of spectral triples. We first introduce the notion of Generalised Crossed Product (GCP) and illustrate it by the case of QHM. A GCP contains a sub-C*-algebra called its "basis". A spectral triple on the basis can induce a spectral triple on the GCP, under some assumptions which we make explicit.

### Factors arising as fixed point algebras under xerox type actions, Thierry Giordano

Thirty years ago, M. Baker and R. Powers began the study of von Neumann algebras
associated to product states on the gauge invariant and rotationnally invariant
CAR algebras. In this talk I will present new results concerning the type III factor case.

Joint work with Radu Munteanu

### Simplicity of partial skew groups rings, Daniel Gonçalves

Let R_0 be a commutative ring, G a group and \alpha a partial action by ideals that contain local units. In this talk we will describe a simplicity criterion for the partial skew group ring R_0*G in terms of maximal commutativity and G-simplicity of R_0. If time permits we will sketch how to use this result to give a new proof of the simplicity criterion for Leavitt path algebras.

### Automorphisms and conjugacy of MASAs in the Cuntz algebras, Jeong Hee Hong

Abstract :
In this talk, we discuss recent progress in investigations of the structure of the automorphism group of the Cuntz algebra
${\mathcal O}_n$. In particular, we describe the full and the restricted Weyl group, respectively. Then we give examples of
outer but not inner conjugate MASAs in ${\mathcal O}_n$.

This talk relies on the recent joint work with Roberto Conti (Rome) and Wojciech Szyma\'{n}ski (Odence)

### A quasi-category approach to E- and KK-theory, Nikolay Ivankov

Quasi-categories in the sense of Joyal enrich categories by allowing higher levels of (n+1)-arrows between n-arrows for all n in an elegant way. We enrich equivariant E- and KK-theory categories to quasi-categories. The most difficult aspects are the composition products in E- and KK-theory, which are also encoded in our quasi-categories, together with their associativity.

### K-theory for crossed products of totally disconnected dynamical systems, Xin Li

I explain a method to compute K-theory for crossed products attached to
totally disconnected dynamical systems. As illustrations, we discuss concrete
examples like semigroup C*-algebras or graph C*-algebras.

One part of this is joint work with J. Cuntz and S. Echterhoff, and another
part is joint work with M. Norling.

### Crossed products for crossed module actions on C*-algebras, Ralf Meyer

Crossed modules describe non-Hausdorff quotient groups in the same way that groupoids describe non-Hausdorff quotient spaces. We define actions of crossed modules on C*-algebras and crossed products for them. We explain how to decompose this crossed product construction into simpler operations. We show that completely non-Hausdorff Lie crossed modules are equivalent to Abelian crossed modules and extend Takesaki-Takai duality to Abelian crossed modules. (This is joint work with Alcides Buss.)

### Hamiltonian actions of quantum groups, Ryszard Nest

abstract to be added

### Hilbert Modules associated with C*-dynamical systems, Justin Peters

Following recent work of Courtney, Muhly and Schmidt, we look at the question of when an endomorphism of a C*- or W*-dynamical system can be implemented by a family of Cuntz isometries. This can be viewed as an extension of the original endomorphism, and one can ask which properties of the original endomorphism carry over to the extension. Our approach is by means of Hilbert modules. The is a preliminary report based on joint work with Evgenios Kakariadis.

### A homology theory for smale spaces, Ian Putnam

Abstract of lecture: Smale spaces were defined by David Ruelle as abstract topological versions
of the basic sets for Smales Axiom A systems. Later, he also showed how (several) C*-algebras
could be constructed from such systems. Smale spaces include shifts of finite type and here
the C*-algebras are the well-known Cuntz-Krieger algebras, as well as their AF-cores.
Rufus Bowen conjectured the existence of a homology theory for Smale spaces which would
provide a Lefschetz-type formula to count the number of periodic points. This was first done
for shifts of finite type by Bowen-Franks and Krieger (independently). The Krieger invariant
was the K-theory of the AF-algebra. Here, we provide a solution to Bowen's conjecture which
is heavily based on the dimension group invariant. This should also provide a method of
computation of the K-theory of the C*-algebras for general Smale spaces.
** Slides**

### Exotic coactions, John Quigg

If a group acts on a C*-algebra, we have both full and reduced crossed products, and each has a coaction. We investigate coactions in between, and our strategy is inspired by recent work of Brown and Guentner on new C*-group algebra completions. We have more questions than answers. Joint work with Steve Kaliszewski and Magnus Landstad.

### Hypergroupoids and C*-algebras, Jean Renault

Let (G; ) be a locally compact groupoid with Haar system and let (X; )
be a proper G-space with an equivariant system of measures. If the action of G is free,
then H = (X X)=G is a groupoid equivalent to G and C(H) is a C*-algebra Morita
equivalent to C(G). We shall study the case when the action of
G is no longer free and present some examples, among which hypergroups and Hecke pairs. This is joint work with Rohit Holkar.
**Slides**

### Purely infinite partial crossed products, Adam Sierakowsky

Several known stable and unital Kirchberg algebras can realised as C*-algebras associated to dynamical systems. In this framework many properties, such as ideal structure or pure infiniteness of these C*-algebras, can be described using the underlying properties of the dynamical systems. In this lecture we consider such results for partial dynamical systems. Let (A,G) be a partial dynamical system, where a discrete group G acts (by a partial action) on a C*-algebra A. We show that there is a bijective correspondence between G-invariant ideals of A and ideals in the partial crossed product A xr G provided the action is exact and residually topologically free. Assuming, in addition, a technical condition---automatic when A is abelian---we show that A xr G is purely infinite if and only if the positive nonzero elements in A are properly infinite in A xr G. As an application we verify pure infiniteness of various partial crossed products, including realisations of the Cuntz algebras O_n, O_A, O_N, and O_Z as partial crossed products.

### Permutative automorphisms of graph C*-algebras, Wojciech Szymanski

Abstract: I discuss recent progress in investigations of permutative automorphisms of graph C*-algebras, emphasizing the interplay of analytic and computer supported discrete methods.

### Central sequences, dimension, and Z-stability of C*-algebras, Aaron Tikuisis

Abstract: A current goal in C*-algebras is to firmly establish the correct notion of low-dimension, or regularity, as a refinement of nuclearity. In concrete terms, this goal amounts to showing that finite nuclear dimension, Z-stability, and regularity in the Cuntz semigroup are equivalent properties (outside of well-known, obvious obstructions). In this talk, I will focus on arguments to prove Z-stability. A necessary concept in such arguments seems to be the notion of approximately centrality, which is neatly captured by the central sequence C*-algebra. We will see how hypotheses of low dimension allows one to manipulate this object. This is largely joint work with Leonel Robert.

### Exactness and Continuity properties of C*-dynamical systems, Otgonbayar Uuye

Abstract: C*-exactness plays an important role in the study of C*-algebras and locally compact groups. It is well-known that C*-exactness implies C*-continuity, which is defined as the minimal tensor product or the reduced crossed product commutes with arbitrary sequential inductive limits. In this talk we show the converse. In fact, we extend the notion of C*-exactness and C*-continuity to C*-dynamical systems and show that they are equivalent. If time allows, we will explain how much of the same technique can be used prove the equivalence between K-exactness and K-continuity.

### Inducing Irreducible Representations, Dana Williams

Given a C*-dynamical system, the question of whether an irreducible representation "induced from a stability group"
is itself irreducible turns out to be a very subtle question. Even if the group acting is amenable -- so that the
famous Gootman-Rosenberg-Sauvagoet solution to the Effros-Hahn conjecture implies that every primitive ideal of the
crossed product is induced from a stability group -- it is still not know if inducing irreducible representations
results in irreducible representations (or even primitive ideals). Siegfried Echterhoff and I have conjectured that,
under appropriate circumstances, inducing irreducible representations from stability groups should always produce
irreducible representations. I plan to discuss this problem as well as results of Echterhoff and myself
supporting the conjecture. If time permits, I will outline some generalizations of these results, proved in
collaboration with Marius Ionescu, to much more general dynamical systems including the C*-algebras associated
to Fell bundles over locally compact groupoids. **Slides**

### Regularity in C*-algebras and topological dynamics, Wilhelm Winter

Abstract: I revisit certain C*-algebraic regularity properties which are important for the classification and structure theory of nuclear C*-algebras and interpret them from a dynamical point of view.

### Equivariant comparison of quantum homogeneous spaces, Makoto Yamashita

The q-deformation of simply connected simple compact Lie groups, and their homogeneous spaces
with respect to Poisson-Lie quantum subgroups can be obtained by the quantization of symplectic
leaves on the manifold of the classical case. We prove that these deformations are equivariantly
KK-equivalent to the classical one with respect to the action of maximal torus, extending the nonequivariant case of Neshveyev-Tuset.
As an application we obtain an analogue of the Borsuk-Ulam theorem for the quantum spheres conjectured by Baum-Hajac.
**Notes.**