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Mini-course Week of June 9-13: (video of the talks)

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June 16-20, 2014 Workshop (abstracts)
Video of the talks
Organizers: Zhuang Niu and Luis Santiago

C*-Algebras and Dynamical Systems Abstracts

In this talk we focus on the fact that the map induced by a cpc order zero $\varphi : A \to B$ in the category Cu does not preserve the compactly containment relation. In particular, these kinds of maps are not in the category Cu, so that, in general, they may not be used in the classification of C*-algebras via the Cuntz Semigroup. Nevertheless, there is a subclass of these maps which preserves the relation, and so they can be used in the above mentioned classification. Our main result characterizes these maps via the positive element induced by the description of cpc order zero maps shown in [1]}.

References
[1] Winter, W. and Zacharias, J.,Completely positive maps of order zero,
Munster J. Math., 2, 2009, 311--324.

We survey work in progress by Dawn Archey, N. Christopher Phillips, and myself on various formulations of large subalgebras of C*-algebras. Such definitions provide abstract formulations of the properties observed in the approximating subalgebras used to study transformation group C*-algebras. Applications to the structure of crossed products will be presented.

We define a continuous analog of the Rokhlin property for circle actions, asking for a continuous path of unitaries instead of a sequence. Besides being classifiable, these actions enjoy a number of nice properties that do not hold in general for Rokhlin actions. This talk will focus on the connections between the continuous Rokhlin property and E-theory, with the goal of showing that if $\alpha\colon \mathbb{T}\to \mbox{Aut}(A)$ is an action with the continuous Rokhlin property on a nuclear C*-algebra $A$, then $A$ satisfies the UCT if and only if the fixed point algebra satisfies the UCT, if and only if the crossed product satisfies the UCT.

In a recent work by Thomas Schmidt and Klaus Thomsen on $C^\ast$-algebras arising from circle maps, they introduced orientation preserving groupoids as an intermediate step. It was shown, under some assumptions on the circle maps, that the oriented transformation groupoid algebras introduced there are classifiable by K-theory due to the Kirchberg--Phillips classification theorem. In the same spirit, the cores of the oriented transformation groupoid algebras, i.e., the fixed point algebras of a gauge action on the oriented transformation groupoid algebras, are classifiable by ordered K-theory using a classification result by Andrew Toms.

We give first a short overview on the -- possibly different -- notions of pure infiniteness and describe then a method to prove with help of the study of a ``filling family'' ``strong'' pure infiniteness, or of its permanence under the operations like e.g. tensor products or endomorphism crossed products. Always with some additional assumptions like e.g.\ exactness and generalized Rokhlin type conditions.

Related are Sections 3 and 6 of the joint work with A.Sierakowski: ``Strong pure infiniteness of crossed products''

Let : S^{2n+1} --> S^{2n+1} be a minimal homeomorphism (n a natural number ). We show that the crossed product C(S^{2n+1})x Z has rational tracial rank at most one.
More generally, let M be a connected compact metric space with finite covering dimension and with H^1(M, Z)=0. Suppose that K_i(C(M))=Z\oplus G_i for some finite abelian group $G_i,$ $i=0,1.$ Let M --> M be a minimal homeomorphism. We also show that A=C(M) x Z has rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces.

We consider the problem of three matrices, two unitaries $U$ and $V$ that commute with each other and the third matrix being Hermitian, with bounds on $\|H\|\leq L,\|H^{-1}\|\leq L$, so that $H$ almost commutes with $U$ and $V$. When can we deform such a system continuously, keeping all the exact and almost keeping the approximate relations, so that at the end of the path we have three commuting matrices? The answer, not surprisingly, has to do with $K$-theory. The problem is inspired by finite models of topological insulators and superconductors. To address all the flavors of topological insulators and superconductorswe need to consider one or two real structures on $\mathbf{M}_{n}(\mathbb{C})$and consider invariants in both $KU$ and $KO$.

This talk includes many joint results with Adam S\o rensen.

For an action of a finite group on a C*-algebra, we present some conditions under which properties of the C*-algebra pass to the crossed product or the fixed point algebra. We mostly consider the ideal property, the projection property, topological dimension zero, and pure infiniteness. In many of our results, additional conditions are necessary on the group, the algebra, or the action. Sometimes the action must be strongly pointwise outer, and in a few results it must have the Rokhlin property. When the group is finite abelian, we prove that crossed products and fixed point algebras preserve topological dimension zero with no condition on the action. We give an example to show that the ideal property and the projection property do not pass to fixed point algebras (even for the two element group). The construction also gives an example of a C*-algebra which does not have the ideal property but such that the algebra of 2 by 2 matrices over it does have the ideal property; in fact, this matrix algebra has the projection property. This is joint work with N. Christopher Phillips, and it will be published in the Transactions of the A.M.S.

I will report on joint work with Marius Dadarlat. We showed that the Dixmier-Douady theory of continuous fields of C*-algebras with compact operators as fibers extends to a more general theory of fields with fibers stabilized strongly self-absorbing C*-algebras. The classification of the corresponding locally trivial fields involves a generalized cohomology theory obtained from the unit spectrum of topological K-Theory, which is computable via the Atiyah-Hirzebruch spectral sequence. An important feature is the appearance of characteristic classes in higher dimensions. We found a necessary and sufficient K-theoretical condition for local triviality of these continuous fields over spaces of finite covering dimension. If time permits I will also explain how the torsion elements in the classifying generalized cohomology group arise from locally trivial fields with fibers isomorphic to matrix algebras over the strongly self-absorbing algebra.

The Cuntz semigroup $W(A)$ of a C$^*$-algebra $A$ is an important ingredient, both in the structure theory of C$^*$-algebras, and also in the current format of the Classification Programme. It is defined analogously to the Murray-von Neumann semigroup $V(A)$ by using equivalence classes of positive elements. The lack of continuity of $W(A)$, considered as a functor from the category of C$^*$-algebras to the category of abelian semigroups, led to the introduction (by Coward, Elliott and Ivanescu) of a new category Cu of (completed) Cuntz semigroups. They showed that the Cuntz semigroup of the stabilized C$^*$-algebra is an object in Cu and that this assignment extends to a sequentially continuous functor.

We introduce a category W of (pre-completed) Cuntz semigroups such that the original definition of Cuntz semigroups defines a continuous functor from local C$^*$-algebras to W. There is a completion functor from W to Cu such that the functor Cu is naturally isomorphic to the completion of the functor W. Using this, we show that the functor Cu is continuous.
We also indicate how the category Cu should be recasted, by adding additional axioms. If time allows, we will discuss the construction of tensor products in the category Cu.

(This is joint work with Ramon Antoine and Hannes Thiel.)

Consider a continuous surjection of the circle which is piecewise monotone, but not locally injective. To this, we associate a locally compact étale groupoid -- the so called \emph{amended transformation groupoid} -- and study the relationship between the dynamical system, the groupoid, and the associated groupoid $C^*$-algebra. Under modest assumptions on the dynamics, we apply the work of Katsura on $C^*$-correspondences to develop an algorithm that reduces calculating the $K$-theory of the $C^'$-algebra to elementary linear algebra. This is joint work with Klaus Thomsen.

Xin Li's construction of C*-algebras for arbitrary left-cancellative semigroups S has raised interest in semigroup C*-algebras over the last years. Right LCM semigroups constitute a large class of left-cancellative semigroups. For instance, it encompasses semigroups associated to self-similar actions and suitable semidirect products of groups by semigroups. In this talk I will indicate how the right LCM property simplifies the study of the full semigroup C*-algebra C*(S). This leads to a uniqueness theorem for C*(S) based on its diagonal subalgebra (in the spirit of a result by Laca-Raeburn for quasi-lattice ordered groups from 1996). As a byproduct, we obtain a criterion to ensure that C*(S) is purely infinite simple. I will discuss several examples arising as semidirect products of groups by semigroups. This is joint work with Nathan Brownlowe and Nadia S. Larsen.

I will present some recent work on product systems of Hilbert bimodules and their corresponding C*-algebras. The focus will be on dynamical properties, including topological aperiodicity and (time permitting) KMS states. The talk will contain some results obtained in collaboration with Jeong Hee Hong, Bartosz K. Kwasniewski and Nadia S. Larsen.

In 2012, Ilan Hirshberg, Wilhelm Winter and Joachim Zacharias introduced the concept of Rokhlin dimension for actions of finite groups and the integers. Shortly thereafter, this was adapted to actions of Z^m. The main motivation for introducing this concept was that actions with finite Rokhlin dimension preserve the property of having finite nuclear dimension, when passing to the crossed product C*-algebra. Since then, this has been successfully used to verify finite nuclear dimension for a variety of non-trivial examples of C*-algebras, in particular transformation group C*-algebras. We extend the notion of Rokhlin dimension to cocycle actions of countable, residually finite groups. If the group in question has a box space of finite asymptotic dimension, then one gets an analogous permance property concerning finite nuclear dimension. We examine the case of topological actions and indicate that Rokhlin dimension is closely related to amenability dimension in the sense of Erik Guentner, Rufus Willett, and Guoliang Yu. Moreover, it turns out that the recent result concerning the Rokhlin dimension of free Z^m-actions on finite dimensional spaces generalizes to actions of finitely generated nilpotent groups. (joint work with Jianchao Wu and Joachim Zacharias)

We introduce the concepts of Cuntz-semirings and their modules. Natural examples are given by Cuntz semigroups of C*-algebras that are strongly self-absorbing and of C*-algebras that tensorially absorb such a C*-algebra.

We characterize the modules over the Cuntz-semiring of the Jiang-Su algebra as those Cuntz-semigroups that are almost unperforated and almost divisible.

Then, we study simple Cuntz-semirings. Under mild assumptions, they are automatically almost unperforated and almost divisible. We also classify all solid Cuntz-semirings. A semiring is called solid if the multiplication map induces an isomorphism of the tensor-square of the semiring with the semiring. One can think of solidity as an algebraic analog of being strongly self-absorbing.

(joint work with Ramon Antoine and Francesc Perera)

After a brief introduction, we will see how saturation in a continuous model theory setting for an abelian real rank zero C*-algebra without minimal projections corresponds to saturation of the associated Boolean algebra of projections, in the classical model theoretical sense. Moreover we show that the theory of this class of C*-algebras is complete, that is, that every two such C*-algebras are elementary equivalent.

This is a joint work with Christopher Eagle.

Tracial Rokhlin property for actions on unital simple C*-algebras has been proved to be very useful in determine the structure of the crossed product. But most of the results dealt with actions of finite groups or group of integers only. In this talk, we will give a definition of tracial Rokhlin property for actions of countable discrete amenable groups. We shall see that most of the previous results could be generalized to our case. Among other things, we show that the crossed products of actions with tracial Rokhlin property preserve the class of C* algebras with real rank zero, stable rank one and strict comparison for projections, and the crossed products of actions with weak tracial Rokhlin property preserve the class of tracially $\mathcal{Z}$-stable C*-algebra. We shall also give some interesting examples if time permits.

Lifting and perturbation results have played an important, but largely undervalued, role in the classification program thus far. Motivated by these applications, and also the difference between alternative characterisations of nuclear dimension and decomposition rank for C*-algebras with real rank zero, we introduce various local lifting properties. We show that these properties are fairly general (being satisfied by many classes of 'nice' algebras) and are useful. Our main application will be to show that simple, separable, unital, finite, nonelementary C*-algebras with finite nuclear dimension, real rank zero and finitely many extremal traces that are locally liftable (in a suitable sense) are TAF, and so those that also satisfy the Universal Coefficient Theorem are classifiable.