## Noncommutative geometry

### Organisers: Piotr M. Hajac and Tomasz Maszczyk

*Venue: Ul. Śniadeckich 8, room 321, Mondays*

*To see the video recording of a talk, click on its title.*

**Academic year 2017/2018:**

### 2 October 2017, 10:30-12:00

**RANK-TWO MILNOR IDEMPOTENTS FOR THE MULTIPULLBACK QUANTUM COMPLEX PROJECTIVE PLANE**

The *K _{0}*-group of the C*-algebra of the multipullback quantum complex projective plane is known to be

*Z*, with one generator given by the C*-algebra itself, one given by the section module of the noncommutative (dual) tautological line bundle, and one given by the Milnor module associated to a generator of the

^{3}*K*-group of the C*-algebra of the Calow-Matthes quantum 3-sphere. In this talk, we outline a proof that these Milnor modules are isomorphic either to the direct sum of section modules of the noncommutative tautological and dual tautological line bundles, or to a complement of this sum in the rank-four free module. We also argue that the direct sum of these noncommutative line bundles is a noncommutative vector bundle associated to the

_{1}*SU*(2)-prolongation of the Heegaard quantum 5-sphere

_{q}*S*viewed as a

^{5}_{H}*U(1)-*quantum principal bundle. Finally, we demonstrate that the above Milnor modules always split into the direct sum of the rank-one free module and a rank-one non-free projective module that is not associated with

*S*. Based on joint work with Carla Farsi, Tomasz Maszczyk and Bartosz Zieliński.

^{5}_{H}PIOTR M. HAJAC (IMPAN)

### 2 October 2017, 14:00-15:30

**ON THE QUANTUM ROSENBERG CONJECTURE**

The classical Rosenberg conjecture states that the reduced group C*-algebra of a discrete group is quasidiagonal if and only if the group itself is amenable. The conjecture was proved in the 2017 Annals paper of Tikuisis, White and Winter. Recently, Dadarlat and Sato improved this result by proving it without the UCT assumption. In this talk, we use the Dadarlat-Sato approach to obtain an analogous result for discrete unimodular quantum groups.

### 9 October 2017, 10:30-12:00

**AROUND PROPERTY (T) FOR QUANTUM GROUPS**

Kazhdan's Property (T) for quantum groups was introduced by Fima, and later studied by Kyed, Sołtan, Li, Ng, Arano and others. Here I will discuss recent related developments, especially concerning connections between property (T) and Bekka and Valette's property (T)^{{1,1}}. These turn out to be equivalent for discrete unimodular quantum groups. I will present the consequences of this result for 'typical' representations (à la Kerr-Pichot) and operator ergodic theory (à la Connes and Weiss). This is based on joint work with Daws and Viselter, and on recent results of Brannan and Kerr.

### 9 October 2017, 14:00-15:30

**THE INVARIANCE OF HOCHSCHILD AND CYCLIC HOMOLOGY UNDER ROW EXTENTIONS**

Goodwillie’s theorem states that the periodic cyclic homology is invariant under nilpotent extensions. We introduce a special type of nilpotent extensions of unital algebras (called row extensions) for which we prove a stronger result: the invariance of Hochschild and cyclic homology. The row extensions appear in abundance. They are always *H*-unital but generically non-unital and noncommutative. A very specific type of a row extension appears naturally in the construction of the Chern-Galois character. If *P* is an algebra with a principal coaction, and *B* is its coaction-invariant subalgebra, then the Chern-Galois character factors through the row extension of *B* by the nilpotent ideal consisting of the invariant universal differential one-forms on *P*. When *P* is a principal comodule algebra, one can identify this ideal with the kernel of the multiplication map restricted to the algebra of the associated Ehresmann-Schauenburg quantum groupoid. Based on joint work with Piotr M. Hajac.

TOMASZ MASZCZYK (Uniwersytet Warszawski)

### 16 October 2017, 10:30-12:00

**(CO)HOMOLOGY OF TOPOLOGICAL ALGEBRAS VIA COALGEBRAS**

Abrams and Weibel proved that continuous cohomology of profinite algebras with their natural topology can be calculated from the ordinary (co)algebraic cohomology of certain coalgebras, thus circumventing topological considerations. We are going to investigate how this result can be used to calculate continuous cohomology of a class of topological algebras that manifest themselves as the dual of incidence coalgebras. (This is a joint work with M. Kanuni and S. Sutlu.)

ATABEY KAYGUN (Istanbul Technical University)

### 16 October 2017, 14:00-15:30

**ON FILTRATION-PRESERVING QUANTUM SYMMETRY GROUPS OF NONCOMMUTATIVE TORI**

We will present necessary conditions for the existence of an action of a compact quantum group on the noncommutative *n*-torus *T ^{n}_{θ}* that preserves (in the sense of T. Banica and A. Skalski) a filtration orthogonal with respect to a given state. Then we will construct a family of compact quantum groups

*G*such that, for all

^{n}_{θ}*θ*, the quantum group

*G*is the final object in the category of all compact quantum groups acting on

^{n}_{θ}*T*in a filtration preserving way. We shall describe the structure of the C*-algebra of

^{n}_{θ}*G*, and discuss the representation theory of

^{n}_{θ}*G*. Joint work with M. Marciniak.

^{n}_{θ}MICHAŁ BANACKI (Universytet Gdański)

### 23 October 2017, 10:30-12:00

**INVARIANT MARKOV SEMIGROUPS ON QUANTUM HOMOGENEOUS SPACES**

First, we show how to classify the convolution semigroups of states and the semigroups of *G*-invariant Markov operators on quantum spaces with an action of a compact quantum group* G*. Then, we discuss three types of spheres that are homogeneous spaces of quantum orthogonal groups *O _{Nx}* as examples. Furthermore, given

*O*-invariant Markov operator semigroups on these spheres, (

_{Nx}*T*) with

_{t}*t*greater or equal to zero, we calculate eigenvalues of their generator operators. The eigenvalues are described by some orthogonal polynomials.

XUMIN WANG (Université de Franche-Compté)

### 23 October 2017, 14:00-15:30

**KAZHDHAN PROPERTY (T) FOR QUANTUM GROUPS: SUMMARY AND NEW RESULTS**

The discovery of Property (T) was a cornerstone in group theory, and the last decade saw its importance in many different subjects like ergodic theory, abstract harmonic analysis, operator algebras, and some of the very recent topics like C*-tensor categories. It is quite natural to explore the possibility of extending the notion of Property (T) to the realm of quantum groups. This was done in the following sequence: ﬁrst within the framework of Kac algebras, then for algebraic quantum groups and discrete quantum groups, and ﬁnally for locally compact quantum groups by Joita, Petrescu, Fima, Soltan, Kyed, Skalski, Viselter, Daws, Brannan and Kerr. Thus far Property (T) has been mainly studied for quantum groups with the trivial scaling automorphism group. A recent result of Y. Arano states that the Drinfeld quantum double of the Woronowicz compact quantum group *SU _{q}(2n+1)* has Property (T), thus producing a concrete example of a quantum group with non-trivial scaling automorphism group which has Property (T). This necessitates the need to generalize the results obtained before for quantum groups with the trivial scaling action to quantum groups non-trivial scaling actions. In this talk, our main objective is to push forward the methodologies and techniques developed in all the above-mentioned previous works about Property (T) to accommodate some quantum groups with non-trivial scaling action, thereby complementing the previously obtained results. Based on an ongoing joint work with P. Salmi.

BISWARUP DAS (Uniwersytet Wrocławski)

### 18 December 2017, 10:30-12:00

**MATRICIAL STABILITY FOR COMMUTATIVE C*-ALGEBRAS**

All known results about matrices almost commuting with respect to the operator norm are partial answers to the following open question: which commutative C*-algebras *C*(*X*) are matricialy stable? We answer this question under the assumption that *X* has finite covering dimension. This result is obtained by working out new permanence results for matricial stability, and by proving a purely topological result about finite-dimensional spaces, which can be of independent interest. To end with, we will discuss applications of the main result. Based on joint work with Dominic Enders.

TATIANA SHULMAN (IMPAN)

### 18 December 2017, 14:00-15:30

STRICT INCREASING APPROXIMATE UNITS OF C*-ALGEBRAS

It is well known that every C*-algebra has an increasing approximate unit with respect to the usual partial order on the positive unit ball. We consider the strict order << instead, where *a* << *b* means *a* = *ab*. Here again it is well known that every separable or σ-unital C*-algebra has a <<-increasing approximate unit, but the general case remained unresolved. In this talk, we outline our recent work showing that this extends to Ω_{1}-unital C*-algebras but not, in general, to Ω_{2}-unital C*-algebras. In particular, we consider C*-algebras defined from Kurepa trees which are scattered, and hence *LF* but not *AF* in the sense of Farah and Katsura. It follows that whether all separably representable *LF*-algebras are *AF* is independent of *ZFC*. (Based on joint work with Piotr Koszmider.)

TRISTAN BICE (IMPAN)

### 8 January 2018, 10:30-12:00

**LOCAL TRIVIALITY FOR ACTIONS OF COMPACT QUANTUM GROUPS**

In noncommutative geometry, there is no straightforward generalization of the concept of local triviality of a principal bundle because it is not clear how to define an open cover. In the compact case, one can get away by using finite closed covers to define piecewise triviality. The latter is only slightly more general than local triviality, and works well in the noncommutative setting through multi-pullbacks of unital C*-algebras. However, the multi-pullback construction does not work for simple C*-algebras. As a remedy, we introduce a definition of local triviality for actions of compact quantum groups on unital C*-algebras. It is inspired by the Rokhlin dimension used in and around the classification of unital simple separable nuclear C*-algebras. We show that, for commutative C*-algebras, our definition recovers the standard definition of local triviality of compact principal bundles. We also prove that locally trivial actions of compact quantum groups are automatically free. Finally, we apply this new notion to make progress with the noncommutative Borsuk-Ulam-type conjecture: we prove that it holds for actions of compact quantum groups with a classical subgroup whose induced action is locally trivial. Based on joint work with Eusebio Gardella, Piotr M. Hajac and Jianchao Wu.

MARIUSZ TOBOLSKI (IMPAN)

### 8 January 2018, 14:00-15:30

SAFOURA ZADEH (IMPAN)

### 15 January 2018, 10:30-12:00

PAUL F. BAUM (Penn State / IMPAN)