Interactions between C*-algebraic KK-theory and homotopy theory

Aim

Throughout the mid to late 20th century, K-theory has been an important tool in its various algebraic, topological, and analytic manifestations. Homotopy theory has long been a central language for studying algebraic and topological K-theory, in particular. Recently, there has been more work connecting the methods of homotopy theory and infinity categories to C*-algebraic K-theory as well, and the potential applications of this new perspective are still to be seen.

The aim of this 2-week online workshop will be to bring together researchers and students with backgrounds in either C*-algebraic KK-theory or homotopy theory to learn about the growing interactions between the two fields. The first week will consist of separate introductions to classical C*-algebraic KK-theory and infinity categories, which will be followed by a modern infinity categorical approach to C*-algebraic KK-theory. The second week will consist of research talks related to the interactions of these two fields.

There will be time during the second week for shorter contributed talks by early career mathematicians (PhD students and postdocs). We encourage those who are interested in speaking to submit a title and abstract in the registration form below by December 11, 2024.

Time

Dates: 7 - 10 January, and 13 - 17 January, 2025
Time: 16:00 - 18:30 CET / 09:00 - 11:30 CST
Zoom access data: Has been sent out to registered participants via E-Mail.
The specific time is chosen so that the workshop is hopefully accessible to people in various different time zones.

Speakers

Talks:

Pierre Albin
Ulrich Bunke
Bastiaan Cnossen
Marius Dadarlat
Ivo Dell'Ambrogio
Eugenia Ellis
Alexander Engel
Markus Land
Ralf Meyer
Devarshi Mukherjee
George Nadareishvili
Shintaro Nishikawa
Ulrich Pennig
Christian Voigt

Short presentations:

Anupam Datta
Arturo Jaime
Georg Lehner
Georgii Makeev
Vikram Nadig
Chrisil Ouseph
Valerio Proietti
Luuk Stehouwer

Schedule

Week 1:

	Tuesday 07 Jan	Wednesday 08 Jan	Thursday 09 Jan	Friday 10 Jan
16:00 - 17:00 (CET)	Pierre Albin	Shintaro Nishikawa	Ulrich Bunke	Ulrich Pennig
17:30 - 18:30	Bastiaan Cnossen	Bastiaan Cnossen	Christian Voigt	Short Presentations

Week 2:

	Monday 13 Jan	Tuesday 14 Jan	Wednesday 15 Jan	Thursday 16 Jan	Friday 17 Jan
16:00 - 17:00 (CET)	Eugenia Ellis	Devarshi Mukherjee	Ivo Dell'Ambrogio	Markus Land	Alexander Engel
17:30 - 18:30	Ulrich Bunke	George Nadareishvili	Ralf Meyer	Short Presentations	Marius Dadarlat

Short presentations, session 1: Anupam Datta, Georg Lehner, Georgii Makeev, Valerio Proietti
Short presentations, session 2: Arturo Jaime, Vikram Nadig, Chrisil Ouseph, Luuk Stehouwer

Abstracts

Week 1:

Pierre Albin: An introduction to KK theory; Video, Notes

I will give a brief introduction to KK theory from the point of view of the Atiyah-Singer index theorem.

Ulrich Bunke: KK-theory from the point of view of homotopy theory; Video, Notes

The classical group-valued KK-theory can be organized into a triangulated category. In this talk I will explain that classical KK-theory is the homotopy category of a stable infinity category. I will discuss various aspects of constructions of such a stable infinity category.

Bastiaan Cnossen : Introduction to Stable ∞-Categories; Video 1, Video 2

These two talks provide an introduction to (stable/symmetric monoidal) ∞-categories for mathematicians familiar with ordinary categories but new to higher categorical methods. The aim is to provide foundations for some of the later talks in the workshop.
Talk 1: General introduction to ∞-categories
Talk 2: Stability and symmetric monoidality

Shintaro Nishikawa: The Baum-Connes Conjecture: An Introduction and Survey; Video

This talk introduces the Baum-Connes Conjecture (BC) and explores a few subtopics. After a standard/traditional introduction to the BC, we discuss the distinctions and parallels between the BC and the Farrell-Jones Conjecture, examine the Higson-Kasparov theorem, and highlight the contributions of Julg and Lafforgue to the BC in the context of groups with Kazhdan's Property (T).

Ulrich Pennig: Twisted K-theory via homotopical algebra; Video, Notes

There are (at least) two ways of defining twisted K-theory: From an operator-algebraic perspective it is given by K-theory of a bundle of C*-algebras, from a homotopy theoretic viewpoint it is defined using homotopy classes of sections of a bundle of spectra. In this talk I will outline how these two pictures nicely fit together when using the language of infinity-categories.

Christian Voigt: The Baum-Connes conjecture and quantum groups; Video, Notes

The Baum-Connes conjecture provides a powerful method to calculate the K-theory of group C*-algebras and crossed products. Some of this can be applied beyond the original setting considered by Baum-Connes, via the approach to the Baum-Connes conjecture based on triangulated categories developed by Meyer-Nest. In this talk I will survey how this can be done in the setting of discrete quantum groups.

Week 2:

Ulrich Bunke: Homotopy theory for K-theory of crossed products

If a group G acts on a C*-algebra A, then one can form the crossed product A ⋊ G. In this talk I discuss the problem of calculating the K-theory K(A ⋊ G) of the crossed product in terms of the spectrum K(A) with the induced G-action. This talk should motivate to consider the K-theory of a C*-algebra as a spectrum rather than a graded abelian group.

Marius Dadarlat: Asymptotic Homotopy, Asymptotic Commutants, and K-Theoretic Duality

The talk aims to highlight connections between certain asymptotic properties of C*-algebras and KK-theory. It draws on joint work with Loring, Pennig, Willett, and Wu.

Ivo Dell'Ambrogio: Tensor triangular geometry for equivariant KK-theory

In 2006, Ralf Meyer and Ryszard Nest proved that the equivariant KK-theory of a locally compact group G carries the structure of a tensor triangulated category. As with any tensor triangulated category, one can then try to apply the methods of Paul Balmer's tensor triangular geometry to it. In this talk I will introduce these ideas and explain their latest applications to KK-theory, in the form of classifications of thick and localizing subcategories inside the G-equivariant bootstrap class for certain finite groups G (joint work with Rubén Martos).

Eugenia Ellis: Homotopy structures realizing algebraic kk-theory

Algebraic kk-theory, introduced by Cortiñas - Thom is a bivariant K-theory defined on the category of algebras over a commutative unital ring l. It consists of a triangulated category kk endowed with a functor j : Alg → kk that is the universal excisive, homotopy invariant and matrix-stable homology theory. Moreover, we can recover Weibel's homotopy K-theory from kk since we have kk(l,A) = KH(A) for any algebra A. In this work we see that the category of algebras with fibrations the split surjections and weak equivalences the kk-equivalences is a stable category of fibrant objects, whose homotopy category is kk. Using this we prove that kk_∞ the Dwyer-Kan localization of the ∞-category of algebras at kk-equivalences is a stable infinity category whose homotopy category is kk. (joint with Emanuel Rodriguez-Cirone)

Alexander Engel: C*-Categories

After giving a brief introduction to C*-categories I will explain how to extend KK-theory from C*-algebras to C*-categories and why this is useful. In the second half of my talk I will focus on the (2,1)-categorical nature of C*-categories arising from considering unitary equivalences.

Markus Land: K- and L-theory of C*-algebras

I will explain how to use the ∞-categorical enhancement of Kasparov’s KK-cateogry to study the relation between (topological) K- and L-theory spectra of (complex) C*-algebras. This is largely motivated by the quest for a comparison of the Baum-Connes conjecture and the (L-theoretic) Farrell-Jones conjecture, and I will explain how to obtain such a comparison. Furthermore, I will explain that the comparison between K- and L-spectra is slightly more subtle than was initially expected. Finally, if time permits, I will indicate how to describe L-theory of arbitrary real C*-algebras in terms of topological K-theory; in contrast to the complex case, the L-groups of real C*-algebras have not previously been described in terms of topological K-groups.

Ralf Meyer: Classifying C*-algebras up to KK-equivalence

Several classification results for C*-algebras say that an equivalence between two C*-algebras in a suitable bivariant KK-theory that encodes extra structure such as ideals or a group action, lifts to an isomorphism of those C*-algebras, suitably compatible with the extra structure. This leads to the question how to classify objects in a bivariant KK-theory up to isomorphism using some simpler K-theoretic invariant. One method to produce prove that an invariant works for this is to show that any object has a projective resolution of length 1 in a certain sense. There are several examples where this exists. A rather sophisticated theorem about this due to my doctoral student Köhler is for C*-algebras with an action of a group of prime order. More generally, if an object has a projective resolution of length 2, then one may use a more general classification result proven by Bentmann and me. Besides the invariant used to set up the projective resolutions, this invariant contains an extra piece of information, which we call the obstruction class. This classification method applies, for instance, to C*-algebras with an action of the circle group or to graph C*-algebras with finite ideal lattice.

George Nadareishvili: Universal Coefficient Theorems in KK-theory

During the talk, we briefly describe the general machinery for deriving Universal Coefficient Theorems in different categories of Kasparov's KK-theory. We go on to consider the KK-category of finite group actions that are equivalent to one on a Type I C*-algebra; we explain how after localising at the group order, we can specialize to the appropriate Universal Coefficient Theorem. The talk follows a joint work with Ralf Meyer.

Short presentations:

Anupam Datta: Model categories and equivariant KK-theory

KK-theory can be seen as a homotopy theory for C*-algebras. It is plausible to ask whether it can be cast in the framework of abstract homotopy theory. In the non-equivariant case, Joachim-Johnson showed in 2007 that KK-groups are in fact the homotopy groups of a stable model structure on pro-C*-algebras. I will discuss similar results in the equivariant case, with a locally compact group action on the algebras involved. Some new techniques involving universal equivariant algebras were needed in the construction which would also be discussed. Lastly, a comparison could be made to the more recent infinity categorical equivariant KK-theory á la Bunke. This is joint work with Michael Joachim.

Arturo Jaime: The topology on KK(A,B) via controlled KK-theory

An interesting feature of KK-theory is that for any pair of separable C*-algebras, (A,B), KK(A,B) admits a canonical (possibly non-Hausdorff) topology which makes it into a topological group. The Rørdam group, KL(A, B), the quotient of KK(A, B) by the closure of {0}, has played an important role in the classification of C*-algebras. Recently, Willett and Yu have used a controlled picture of KK-theory to describe KL(A,B) as an inverse limit of controlled KK-groups. Taking advantage of this structure on KL allows for a more explicit description of its topological structure. We show that this allows to classify the possible topologies realized on KK(A,B) up to homeomorphism. This is work in progress.

Georg Lehner: An approach to the Rosenberg Conjecture

The Rosenberg conjecture states that for any (unital) real or complex Banach algebra, the algebraic and topological K-theory agree in mod n coefficients in non-negative degrees and can be thought off as a variation of the (solved) Karoubi conjecture. Recent progress using condensed mathematics allows one to reduce the Rosenberg conjecture to a statement about rigidity in algebraic K-theory. We will discuss a new proof strategy that further reduces the Rosenberg conjecture to the question of whether a concrete functor of (1)-categories induces a homotopy equivalence.

Georgii Makeev A higher category approach to the Connes-Higson E-theory

We are going to introduce the 2-category of sufficiently nice endofunctors of C*-algebras and homotopy classes of natural transformations between them and obtain an unsuspended description of the Connes-Higson E-theory in terms some generalized homotopies of *-homomorphisms associated with such endofunctors. Along the way we expose some coarse Baum-Connes conjecture type result in the endofunctorial context; formulate an E-theoretic analog of the isomorphism between the Kasparov KK₁-theory and the theory of invertible extensions; and discuss the relative K-homology in our setting.

Vikram Nadig: Takai Duality and Treumann Duality

I will examine two duality theorems - the classical Takai duality, which considers C*-algebras equipped with an action of a locally compact abelian group; and a modern homotopy theoretic duality result of Treumann concerning modules over KU equipped with an action of a finite abelian group. I will then state a comparison result between them, in the framework of the equivariant infinity-categorical KK-theory developed by Bunke, Engel, and Land.

Chrisil Ouseph: Contractible Cuntz Classes

The Cuntz semigroup of a C*-algebra A is a sensitive invariant which can be thought of as a refinement of the Murray-von Neumann semigroup. It consists of Cuntz equivalence classes of positive elements in the stabilization of A, and these classes come in two flavors: Compact (classes of projections) and non-compact (the rest). When A has real rank zero, any class in its Cuntz semigroup is the supremum of an increasing sequence in the positive cone of K₀ (A).
We show that if A is unital, simple and Z-stable, then the set of positive elements in A belonging to a fixed non-compact Cuntz class is contractible. Combined with results of Zhang, Jiang, and Hua for compact classes, this completes the calculation of the homotopy groups of Cuntz classes for these algebras. This work is joint with Andrew S. Toms.

Valerio Proietti: Chern character and rational K-theory of ample groupoids

Building on previous work by Davis and Lück, and recent constructions of KK-theory as a stable ∞-category, I will sketch the construction of a Chern character running from the left-hand side of the Baum–Connes conjecture for ample groupoids with torsion-free isotropy to the periodicized homology groups of the given groupoid. This map is a rational isomorphism, thereby establishing a modified form of Matui’s HK conjecture (after in-tegral counterexamples have been found). This construction also computes the rational homotopy type of the algebraic K-theory spectrum of ample groupoids as defined in a recent work by X. Li. This is joint work-in-progress with M. Yamashita.

Luuk Stehouwer: The failure of Bott periodicity for interacting topological phases

K-theory of C*-algebras has proven to be an excellent tool in understanding non-interacting topological phases of matter. An organizing principle known as the tenfold way is closely related to Bott periodicity of K-theory. The approach to interacting topological phases using topological field theories has been very successful but also difficult to relate to Hamiltonian lattice models. Freed and Hopkins have proposed a purely homotopy-theoretic way to compare these two approaches. In joint work with Debray, Krulewski and Pacheco-Tallaj we show how the tenfold way breaks down in this approach to interacting phases, corresponding to the fact that the spin bordism spectrum MSpin is not 8-periodic.

Registration

The normal registration period has passed. To receive a Zoom-link, please reach out to one of the organizers.

Organizers

Benjamin Dünzinger (Regensburg)
Yigal Kamel (Illinois)
Fredrick Mooers (Illinois)

HomeAboutPeopleEventsResearchRTGGuest ProgrammeImpressum