NCG&T Prague

usually takes place each Tuesday at 16:00 Institute of Mathematics CAS, Blue lecture room, Zitna 25, Praha 1
Chair: Tristan Bice, Karen Stung

What is noncommutative geometry and topology? The idea stems from the Gelfand theorem which states that the category of compact Hausdorff spaces and commutative C*-algebras are dual. If we drop the condition of commutativity from our C*-algebras, we arrive at the notion of a noncommutative topological space. This can be carried further into the realm of noncommutative of geometry by equipping *-algebras with geometric structures.
Our research focusses on both quantum algebraic and operator algebraic aspects of noncommutative geometry and topology. This includes research in Hopf algebras, quantum groups, and noncommutative complex geometry, while on the operator algebra side, we study C*-algebras, with particular focus on C*-algebras arising from dynamical constructions such as minimal actions, groupoids, and semigroups.
Partially supported by GAČR project 20-17488Y Applications of C*-algebra classification: dynamics, geometry, and their quantum analogues and PRIMUS grant Spectral Noncommutative Geometry of Quantum Flag Manifolds.

Nowhere scatteredness and the Global Glimm Problem

Eduard Vilalta

Universitat Autònoma de Barcelona

Tuesday, 29. March 2022 - 16:00 to 17:00

This talk will take place in the blue seminar room, back building, Žitná 25.

It will also be broadcast on Zoom:
https://cesnet.zoom.us/j/91975183920?pwd=SGdlRUhLS21HUnVFTXBKcE1vYXlrQT09
Meeting ID: 919 7518 3920
Passcode: 102707

In this talk, I will define nowhere scattered C*-algebras and explain their relation with the Global Glimm Problem. These algebras, which can be thought of as C*-algebras that are very far from being scattered, can be characterized in a number of ways, such as the absence of nonzero elementary ideal-quotients, or topological properties of the spectrum.

Every C*-algebra with the Global Glimm Property (in the sense of Kirchberg and Rørdam) is nowhere scattered. The Global Glimm Problem asks if the converse holds. That is to say, it asks if every nowhere scattered C*-algebra has the Global Glimm Property.

I will explain how both nowhere scatteredness and the Global Glimm Property can be characterized in terms of divisibility properties of the Cuntz semigroup. I will also mention how these characterizations can provide a new approach to the problem.

The talk is based on ongoing joint work with Hannes Thiel.

Braided quantum symmetries of graph C*-algebras

Suvrajit Bhattacharjee

Charles University

Tuesday, 5. April 2022 - 16:00 to 17:00

A braided compact quantum group (over \mathbb{T}) is, roughly speaking, a “compact quantum group” object in the category of \mathbb{T}-C*-algebras equipped with a twisted monoidal structure. In this talk, we shall explain the existence of a universal braided compact quantum group acting on a graph C*-algebra in the category mentioned above. Time permitting, we shall sketch the proof, constructing along the way a braided analogue of the free unitary quantum group. Finally, as an example, we shall compute this universal braided compact quantum group for the Cuntz algebra.

Split extensions and KK-equivalences for quantum projective spaces

Sophie Emma Mikkelsen

Syddansk Universitet, Odense

Tuesday, 12. April 2022 - 16:00 to 17:00

In this talk, I will present an explicit KK-equivalence between the noncommutative C^*-algebra of continuous functions on the Vaksman–Soibelman quantum complex projective space C(\mathbb{C}P_q^n) and the commutative algebra \mathbb{C}^{n+1}. The KK-equivalence is constructed by finding an explicit splitting for the short exact sequence of C^*-algebras \mathcal{K}\to C(\mathbb{C}P_q^n)\to C(\mathbb{C}P_q^{n-1}). In the construction of the splitting it is crucial that C(\mathbb{C}P_q^n) can be described as a graph C^*-algebra due to Hong and Szymański.

This talk is based on joint work with Francesca Arici.

Uniform Roe algebras of uniformly locally finite metric spaces are rigid

Alessandro Vignati

University of Paris

Tuesday, 26. April 2022 - 16:00 to 17:00

Coarse geometry is the study of metric spaces when one forgets about the small scale structure and focuses only on large scales. Objects of interests are coarse spaces, while relevant maps between them are coarse equivalences, required, in short, only to send close points to close points, and far points to far points. Typical examples are finitely generated groups with word metrics, and discretizations of non-discrete spaces such as Riemannian manifolds. To a coarse space one associates, after Roe, several C*-algebras aiming to detect algebraically the geometrical behaviour of the spaces. Chief among the is the uniform Roe algebra of X, C_u^*(X), key for its applications to index theory, mathematical physics, and recently topological phases of matter.

It is natural to ask how much of the underlying coarse geometry can be detected by these C*-algebras. We answer this question, by showing that if X and Y are uniformly locally finite metric spaces whose uniform Roe algebras... more

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