# NCG Learning Seminar

## Welcome to the NCG Learning Seminar.

The seminar is running on a weekly basis via Zoom.

For the **Zoom link (permanent)** or the **mailing list (weekly) **send an email to

Ryszard Nest or Alexander Frei.

Similarly feel encouraged to also send an email, if you are interested in **providing a talk** yourself.

Moreover, on behalf, we would also like to share with you the

Göttingen Oberseminar Noncommutative Geometry,

if interested, feel free to contact Ralf Meyer,

as well as the

Functional Analysis and Operator Algebras seminar,

organized by Aristides Katavolos.

Looking forward to seeing you online.

## Upcoming talks:

Currently on hold.

## Past talks and slides:

**September 16, 2020:**Jens Kaad, SDU Odense**,**Exterior products of compact quantum metric spaces.

** **The theory of compact quantum metric spaces was initiated by Rieffel in the late nineties. Important inspiration came from the fundamental observation of Connes saying that the metric on a compact spin manifold can be recovered from the Dirac operator. A compact quantum metric space is an operator system (e.g. a unital C*-algebra) equipped with a seminorm which metrizes the weak-*-topology on the state space via the associated Monge-Kantorovich metric. In this talk we study tensor products of compact quantum metric spaces with specific focus on seminorms arising from the exterior product of spectral triples. On our way we obtain a novel characterization of compact quantum metric spaces using finite dimensional approximations and we apply this characterization to propose a completely bounded version of the theory.

September 9, 2020:

Elmar Schrohe, The local index formula of Connes and Moscovici and equivariant zeta functions for the affine metaplectic group, slides here.

We consider the algebra $A$ of bounded operators on $L^2(\mathbb{R}^n)$ generated by quantizations of isometric affine canonical transformations.

This algebra includes as subalgebras the noncommutative tori and toric orbifolds.

We introduce the spectral triple $(A, H, D)$ with $H=L^2(\mathbb R^n, \Lambda(\mathbb R^n))$ and the Euler operator $D$, a first order differential operator of index $1$.

We show that this spectral triple has simple dimension spectrum: For every operator $B$ in the algebra $\Psi(A,H,D)$ generated by the Shubin type pseudodifferential operators and the elements of $A$, the zeta function $\zeta_B(z) = Tr (B|D|^{-2z})$ has a meromorphic extension to $\mathbb C$ with at most simple poles and decays rapidly along vertical lines.

Our main result then is an explicit algebraic expression for the Connes-Moscovici cyclic cocycle.

As a corollary we obtain local index formulae for noncommutative tori and toric orbifolds.

(Joint work with Anton Savin, RUDN, Moscow)

September 2, 2020:

Juan Orendain, Double categories of factors, slides here.

The Haagerup $L^2$-space construction, introduced by Haagerup in the 70's, associates a standard form to every von Neumann algebra, without any reference to weights, and is thus regarded as a coordinate free version of the GNS construction. The Haagerup standard form and the Connes fusion tensor product organize von Neumann algebras and their representations into a bicategory. This bicategory encodes weak Morita equivalence as a formal homotopy relation and Jones index as a categorical dimension.

Bicategories are a specific type of categorical structure of second order, corresponding to globular sets. The second order categorical structures corresponding to cubical sets are double categories. Results studying relations between cubical and globular categories have been obtained continually since the 60's, mainly in nonabelian homotopy theory, but more recently in areas ranging from algebraic geometry to dynamical systems. I will explain results of this type regarding the existence of two non-equivalent double categories of representations of factors constructed as 'lifts' of the bicategory of von Neumann algebras, and how these structures relate to questions of functoriality of the Haagerup standard form and the Connes fusion tensor product.

July 8, 2020:

Mikkel Munkholm, KL-obstruction to quasidiagonality, slides here and video here.

The talk will cover an analysis of the KK-theory of the trace-kernel extension, leaning on the fairly recently developed techniques by Christopher Schafhauser and James Gabe. It will appeal to an audience with a keen eye on classification (of certain simple nuclear C*-algebras) alongside the interplay between amenable traces and quasidiagonal traces. It will feature a biproduct of Christopher Schafhauser’s methods in his AF-embedding paper, whereby the KK - and KL-theory of the key extension, the trace-kernel extension, are shown to coincide. I will make an effort to emphasize on how uniqueness results are connected to said KK (and KL) biproduct and flag up the various key ingredients.

Keywords: KK/KL-theory, (tracial) ultrapowers, UCT, quasidiagonal and amenable traces, the trace-kernel extension, nuclear absorption.

July 1, 2020: No seminar.

June 24, 2020:

Mario Klisse, TU Delft, Graph product Khintchine inequalities and applications to Hecke C*-algebras, slides here.

A graph product of groups is a group theoretic construction that generalizes both free products and Cartesian products. It admits an operator algebraic counterpart, which interpolates between free products and tensor products. Both constructions preserve many properties of the underlying groups/algebras. Important examples of operator algebras that can be realized in terms of graph products are right-angled Hecke algebras, special cases of mixed q-Gaussian algebras as well as several group C*-algebras. In this talk we will discuss so called Khintchine inequalities for general C*-algebraic graph products. These are inequalities which estimate the operator norm of a reduced operator of a given length with the norm of certain Haagerup tensor products of column and row Hilbert spaces. Inequalities of this kind turn out to be very useful. We will demonstrate this in the case of (right-angled) Hecke C*-algebras, by investigating their ideal structures.

June 17, 2020:

Walter van Suijlekom, Spectral truncations in non-commutative geometry and operator systems, slides here.

We extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces (i.e. imposing an ultraviolet cutoff in momentum space). In our new approach the traditional role played by C*-algebras is taken over by so-called operator systems. Essentially, this is the minimal structure required on a space of operators to be able to speak of positive elements, states, pure states, etc. We consider C*-envelopes and introduce a propagation number for operator systems, which we show to be an invariant under stable equivalence and use to compare approximations of the same space.

We illustrate our methods for concrete examples obtained by spectral truncations of the circle. These are operator systems offinite-dimensional Toeplitz matrices and their dual operator systems which are given by functions on the circle whose Fourier series have only a finite number of modes. It turns out that the cones of positive elements and the pure state spaces for these operator systems possess a very rich structure which we analyze including the metric aspect, i.e., the distance on the state space associated to the Dirac operator. We also show that both truncations converge to the circle (in the Gromov-Hausdorff sense).

(based on joint work with Alain Connes)

June 10, 2020:

Adam Dor-On, Introduction to non-commutative convex geometry, slides here.

Abstract: We explain the Webster-Winkler categorical duality between operator systems and matrix convex sets in the finite dimensional situation. Beyond recent applications of operator systems to non-commutative geometry (see the next scheduled talk), this categorical duality provides us with ample geometric intuition for studying operator systems, and conversely, we can use the categorical approach of operator systems to better understand matrix convex sets.

We show some applications of operator systems to convexity, and discuss two candidates for non-commutative extreme points of matrix convex sets, one due to Webster--Winkler, and the other due to Arveson. We show that these non-commutative notions of extreme points behave very differently when one asks for analogues of Krein-Milman and Minkowski theorems for them."

June 3, 2020:

Makoto Yamashita, Groupoid homology, slides here.

Homology of étale groupoids, as defined by Crainic and Moerdijk, is an interesting generalization of cohomology with compact support for spaces on the one hand, and homology of discrete groups on the other, which will live on opposite the opposite (co)homological degrees. On the “integral” side of theory, I will review the categorical approach based on derived category of equivariant sheaves, and connection to K-theory recently popularized by Matui. On the “rational” side, I will review the connection to cyclic homology due to Crainic.

May 27, 2020:

May 20, 2020:

Christian Voigt, Introduction to compact quantum groups, slides here.

May 13, 2020:

Valerio Proietti, Index theory and solenoidal Tori, slides here.

I will discuss principal solenoidal Tori and their foliated manifold structure. By applying the index theorem for measure foliations on such spaces, we are able to obtain information on some K-theoretic "characteristics numbers" for associated topological dynamical systems. Most of the talk will be based on earlier work by Benameur, Oyono-Oyono, Kaminker, and Putnam.

May 6, 2020:

Thomas Wasserman, From 3-dimensional TQFT to rational CFT, video here.

In this talk, I will give an interpretation of the celebrated Fuchs-Runkel-Schweigert description of rational conformal field theories (CFTs) in terms of special symmetric Frobenius algebras in representation categories of vertex operator algebras. After sketching a model for CFTs, I will proceed with briefly introducing topological quantum field theories (TQFTs) and their defects. I will then sketch how to obtain rational CFTs as defects for three dimensional TQFTs, and how this leads to the FRS description.