K-theory and C*-algebras

Becky Armstrong (University of Sydney)
Sara Azzali (University of Potsdam)
Sarah Browne (University of Kansas)
José Carrión (Texas Christian University)
Kristin Courtney (University of Münster)
Marius Dadarlat (Purdue University)
Robin Deeley (University of Colorado)
Clément Dell’Aiera (University of Hawaiʻi)
Jintao Deng (Texas A&M University)
Jamie Gabe (University of Woolongong)
Guihua Gong (University of Puerto Rico)
Sherry Gong (University of California Los Angeles)
Hao Guo (Texas A&M University)
Xin Li (Queen Mary University of London)
Huaxin Lin (University of Oregon and East China Normal University)
Shintaro Nishikawa (Penn State University)
Mikael Rørdam (University of Copenhagen)
Yi Wang (University of Buffalo)
Jianchao Wu (Texas A&M University)

Becky Armstrong, Twisted C∗-algebras of Deaconu–Renault groupoids
In 2014, Brown, Clark, Farthing, and Sims proved that the C ∗ -algebra of an amenable Hausdorff ´ etale groupoid is simple if and only if the groupoid is minimal and effective. This result does not hold for the more general class of twisted groupoid C ∗ -algebras, because, for instance, the irrational rotation algebras are simple twisted C ∗ -algebras of non-effective groupoids. In 2015, Kumjian, Pask, and Sims used groupoid techniques to give a characterisation of simplicity of twisted C ∗ -algebras of cofinal, row-finite, source-free higher-rank graphs. The groupoids involved belong to the strictly larger class of minimal Deaconu–Renault groupoids. In this talk, I will give a characterisation of simplicity of all twisted C ∗ -algebras of minimal Deaconu–Renault groupoids. (This is joint work with Nathan Brownlowe and Aidan Sims.)

Sara Azzali, KK-theory with real coefficients, traces, and discrete group actions
The groups of KK-theory were introduced by Kasparov in the 1980’s and have important applications to many geometric and topological problems which are tackled by C∗-algebraic techniques. In this talk, we investigate KK-theory groups with coefficients in R. By construction, the adding of real coefficients provides natural receptacles for classes coming from traces on C∗-algebras. We focus on applications to the study of discrete groups actions on C ∗ -algebras. We show that in equivariant KK-theory with coefficients one can “localise” at the unit element“ of the discrete group, and this procedure has interesting consequences on the Baum–Connes isomorphism conjecture. Based on joint works with Paolo Antonini and Georges Skandalis.

Sarah Browne, The Baum-Connes correspondence for the pure braid groups on 4 strands
The Baum-Connes conjecture is known to be true for the pure braid groups. We compute the left and right hand side of the Baum-Connes correspondence for the pure braid groups on four strands, by providing explicit calculations for both sides. This is joint with Sara Azzali, Maria Paula Gomez Aparicio, Lauren Ruth and Hang Wang.

José Carrión, K-theoretic ingredients in the classification of morphisms
We will focus on the K-theoretic aspects involved in the classification of *-homomorphisms between classifiable C*-algebras obtained in joint work with J. Gabe, C. Schafhauser, A. Tikuisis, and S. White.

Kristin Courtney, Defining C*-structure on images of completely positive order zero maps
A completely positive map is called order zero when it preserves orthogonality. Such maps enjoy a rich structure, which has made them extremely important in the study of nuclear C*-algebras. In this talk, we consider the structure of the image of a cp order zero map. When the domain is separable and unital, we can define multiplication and a corresponding pre-C*-norm on the image. Generalizing this construction, we are able to characterize when a self-adjoint subspace of a C*-algebra is the image of a unital C*-algebra under a cp order zero map. In both directions, the argument is constructive. This is joint work with Wilhelm Winter.

Marius Dadarlat, Connective groups
A connective countable amenable group G has the property that the reduced K-homology of its C*-algebra can be realized as homotopy classes of asymptotic representations {π t :G → U(∞)} t∈[0,∞) . We plan to review properties of connectivity and survey examples of connective groups. This is based on joint work with Ulrich Pennig, Andrew Schneider and Ellen Weld.

Robin Deeley, University of ColoradoThe K-theory of the stable and stable Ruelle algebras of a Wieler solenoid
Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Through examples I will discuss how this allows one to compute the K-theory of the stable algebra, S, and the stable Ruelle algebra, S o Z. These computations involve writing S as a stationary inductive limit and S o Z as a Cuntz-Pimsner algebra. These constructions reemphasize the view point that Smale space C*-algebras are higher dimensional generalizations of Cuntz-Krieger algebras.

Clément Dell’Aiera, Laplacians for geometric C∗-algebras and K-exactness
When the Cayley graph of a group contains isometrically a sequence of expanders, Ozawa showed that its reduced C ∗ -algebra is not K-exact. We will show how his proof can be extended to new settings, thanks to a Laplacian type operator. This is work in progress.

Jintao Deng, The Novikov conjecture and group extensions
The Novikov conjecture is an important problem in higher dimensional topology. It claims that the higher signatures of a compact smooth manifold are invariant under orientation preserving homotopy equivalences. The Novikov conjecture is a consequence of the strong Novikov conjecture in the computation of the K-theory of group C*-algebras. In this talk, I will talk about the Novikov conjecture for groups which are extensions of coarsely embeddable groups.

Jamie Gabe, Classification of nuclear *-homomorphisms
I will talk about some recent developments which provide a completely new approach to the classification of nuclear simple C*-algebras through classifying nuclear ∗-homomorphisms. This is joint work with Jos´ e Carrión, Chris Schafhauser, Aaron Tikuisis, and Stuart White.

Guihua Gong, On the classification of unital simple separable C∗-algebras of finite nuclear dimension
In this talk, I will give a survey for the classification of simple C*algebras. In particular, I will present the my joint work with Huaxin Lin, Zhuang Niu, and with George Elliott, Huaxin Lin and Zhuang Niu.

Sherry Gong, The Novikov conjecture, the group of volume preserving diffeomorphisms, and Hilbert-Hadamard spaces
We discuss the Novikov conjecture and explain that a version of it holds for manifolds whose fundamental groups admit proper and isometric actions on certain non-positively curved infinite dimensional spaces called Hilbert-Hadamard spaces. This talk is based on joint work with Jianchao Wu and Guoliang Yu (arxiv1811.02086).

Hao Guo, Callias quantization, a coarse approach
I will report on some recent joint work with Peter Hochs and Mathai Varghese on index theory of a class of elliptic operators on non-compact manifolds that are invertible outside of a small set, with respect to a locally compact group of symmetries on the manifold. The index we study takes values in the maximal group C ∗ -algebra. As an application, I will mention a version of Guillemin and Sternberg’s quantization commutes with reduction principle for equivariant indices of Spin-c Callias-type operators.

Xin Li, Constructing Cartan subalgebras in all classifiable C*-algebras
I will explain how to construct Cartan subalgebras in all classifiable stably finite C*- algebras, and I will discuss the Jiang-Su algebra as a particular example.

Huaxin Lin, Simple C*-algebras with tracial nuclear dimension finite
We will present some version of tracial finite nuclear dimension. We will also discuss the connection of tracial finite nuclear dimension with tracially Z-stable C*-algebras. Some properties of simple C*-algebras with finite nuclear dimension will be presented.

Shintaro Nishikawa, The Baum-Connes Conjecture and Proper Kasparov Cycles
Around 1988, Gennadi Kasparov introduced equivariant KK-theory and used it to prove the Novikov conjecture for all groups which act properly and isometrically on a complete, simply connected Riemannian manifold of non-positive sectional curvature or on a homo- geneous space G/K for an almost connected group G and its maximal compact subgroup K. His method, which we currently call the gamma element method (or the Dirac and dual-Dirac method), became a powerful and versatile approach for attacking the Novikov conjecture and the Baum-Connes conjecture. I will review his work from the viewpoint of the Baum-Connes conjecture. I will also describe my recent work which in a way simplifies and offers a new perspective on his work.

Mikael Rørdam, Traces on residually finite-dimensional C*-algebras
I will talk about different classes of traces on RFD C*-algebras, in general, and on the universal unital free product of two copies of a matrix algebra, in particular. The latter relates to factorizable maps (quantum channels) on matrix algebras. This is joint work with Magdalena Musat.

Yi Wang, Arveson-Douglas Conjecture on the Segal-Bargmann space
The Arveson-Douglas Conjecture is about essential normality of quotient modules asso- ciated to complex varieties. The original conjecture is formulated on analytic functions spaces on the unit ball. A positive result on the Arveson-Douglas Conjecture will lead to an element in the K-homology on the variety. In this talk, I will talk about an on-going research about an analogue of the Arveson-Douglas Conjecture on the Segal-Bargman space.

Jianchao Wu, C*-algebras associated to nonpositively curved infinite dimensional spaces and their K-theory
I will outline the construction of C*-algebras associated to Hilbert-Hadamard spaces, understood as a kind of (typically infinite dimensional) nonpositively curved manifolds. Under mild assumptions, these C*-algebras retain a remnant of Bott periodicity, which we exploit to prove the Novikov conjecture of geometrically discrete groups of diffeomorph- isms. Our construction extends and simplifies that of Higson-Kasparov-Trout, which has played important roles in the some of the best results in the Novikov conjecture, the Baum-Connes conjecture and the UCT so far. This is joint work with Sherry Gong and Guoliang Yu.

The talks will take place in Kuykendall Hall, room 101. See the abstracts here
The mathematics department is located in Keller Hall.

We will have an informal meeting at Kaimana beach on Monday evening, starting at 4:30pm. Ask around for rides!

Accommodation
Almost all hotels in Honolulu are in Waikiki, the main tourist area. This is around 3km away from campus, and a common option for visitors to the university (see below for transport options): it is by a beach, and there are plenty of food and entertainment options aimed at tourists. There are often good accommodation options on AirBnB and similar sites. A third possibility is Lincoln Hall on campus https://www.eastwestcenter.org/about-ewc/housing/housing-facilities/lincoln-hall, which often has rooms available (try early). This is reasonably priced, and very conveniently located for the campus itself. Transport
Getting between Waikiki and campus (and often getting around other places), your best option is probably the local bus service: see google maps for directions. The bus costs $2.75 each way (pay on the bus, no change given). Walking is also very possible, but it’s a relatively long, and possibly also quite hot, walk. Other options for getting around include a bike hire scheme

Gobiki

as well as Uber and Lyft, which are very active here and usually cheaper and more convenient than traditional taxi services. Car rental is a good option if you want to explore the island more generally, but be warned that parking both in Waikiki and on campus can be difficult and / or expensive. Lunch options
There are two main indoor food options on campus, at the Campus Center and Paradise Palms. There are also quite a few food trucks and outdoor seating options available. Ask one of the locals for recommendations!

See some places to eat here and here on campus.

Nate Brown (Pennstate University)
Clément Dell'Aiera (UH Manoa)
Rufus Willett (UH Manoa)
Guoliang Yu (Texas A&M)