## 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 (University of Warsaw)

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

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

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

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

BISWARUP DAS (Uniwersytet Wrocławski)

### 18 December 2017, 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)

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

TATIANA SHULMAN (IMPAN)