Mini-workshop
"Operator Algebras and Applications"
Monday July 28 - Wednesday July 30, 2025 in Erlangen
Monday July 28 - Wednesday July 30, 2025 in Erlangen
Address: Exercise Room Ü2 at the Department of Mathematics at Friedrich-Alexander-Universität Erlangen-Nürnberg
Cauerstraße 11, 91058 Erlangen, Germany
Invited Speakers:
Maria Stella Adamo: An operator algebraic construction of 2D extensions of the Heisenberg conformal net
Abstract: We discuss the explicit construction of 2D conformal nets that extend the non-rational Heisenberg chiral conformal net, where irreducible representations (also known as DHR sectors) are all automorphisms.
To construct 2D local algebras, we introduce charged fields acting as (twisted) shifts between DHR sectors, which can be labelled by an additive subgroup $Q$ of the real linear space of charges. When $Q$ is the diagonal subgroup of the left and right charges, the 2D local conformal net can be obtained by including the charge shift operators. In the non-diagonal case, $Q$ is an even lattice, and we recover locality by introducing a 2-cocycle on $Q$ that encodes the interplay between the DHR categories of the left and right chiral components.
This talk is based on a joint project with L. Giorgetti, Y. Tanimoto, arXiv:2301.12310, arXiv:2506.01008.
Abstract: We will associate two particular objects with a countable group $\Gamma$. First, subgroup space-$\text{Sub}(\Gamma)$, the collection of all subgroups of $\Gamma$. Secondly, the group von Neumann algebra $L(\Gamma)$. Recently, Glasner and Lederle have introduced the notion of Boomerang subgroups. They generalize the notion of normal subgroups. They strengthen the well-known Margulis's normal subgroup Theorem among many other remarkable results.
More recently, in a joint work with Hartman and Oppelmayer, we introduced the notion of Invariant Random Algebra (IRA), an invariant probability measure on the collection of subalgebras of $L(\Gamma)$. Motivated by the works of Glasner and Lederle, in ongoing joint work with Yair Glasner, Yair Hartman, and Yongle Jiang, we introduce the notion of Boomerang subalgebras in the context of $L(\Gamma)$. In this talk, we shall make a connection between these two very seemingly distant notions. If time permits, we shall show that every Boomerang subalgebra of a torsion-free non-elementary hyperbolic group comes from a Boomerang subgroup. We'll also talk about its connection to understanding IRAs in such groups.
Abstract: I will present examples of non-amenable actions of free groups on coset spaces which extend to actions which are amenable on the corresponding Stone-{\v C}ech boundaries. This will give examples of $C^*$-algebras with a unique ideal, namely the ones associated to the given quasi-regular representations. The proposed approach relies on understanding the dynamics on certain extensions of these boundaries.
Abstract: I begin with the problem of the existence of an equivariant lift for a G-equivariant completely positive contractive map between G-C*-algebras, when G is a locally compact second countable group. In particular, I show that an equivariant lift exists when the codomain algebra is proper. Then I describe how the latter result ties in with a description of G-equivariant KK-theory in terms of extensions of C*-algebras. Finally, I explain the generalization of the above results to the context of groupoid actions.
This is joint work with S. Bhattacharjee.
Abstract: In recent work, we introduced a notion of geometric Banach property (T) for metric spaces, which parallels the concept of Banach property (T) for groups. While general metric spaces lack group structure—making the characterization of their geometric Banach property (T) challenging—we can derive more elegant descriptions for sequences of Cayley graphs, as the inherent group structure of each graph provides a coherent translation structure. In this talk, we employ nonstandard analysis to construct limit groups for Cayley graph sequences and prove that the geometric Banach property (T) of a Cayley graph sequence implies the Banach property (T) of its limit groups. Moreover, these properties become equivalent under certain conditions. Using the group structure of Cayley graphs, we also study the "coarse group actions" of Cayley graph sequences and investigate the relationships between Banach coarse fixed point property and geometric Banach property (T). This work is based on joint research with Jin Qian and Qin Wang.
Abstract: In this talk, I will present some results on almost elementariness, which was introduced as a new finite approximation property for locally compact Hausdorff \'{e}tale groupoids and implies
the (tracial) Z-stability for groupoid C*-algebras. I will demonstrate that for a large class C of Elliott invariants and for any unital C*-algebra A whose Elliott invariant Ell(A) in C, the C*-algebra A is classifiable if and only if A has an almost elementary groupoid model. As an application, every strongly self-absorbing C*-algebra satisfying the UCT has an almost elementary groupoid model. This is based on a joint work with Jianchao Wu.
Abstract: Roe algebras were introduced in the late 1990's in the study of indices of elliptic operators on (locally compact) Riemannian manifolds. Roe was particularly interested in coarse equivalences of metric spaces, which is a weaker notion than that of quasi-isometry. In fact, soon thereafter it was realized that the isomorphism class of these class of C*-algebras did not depend on the coarse equivalence class of the manifold. The converse, that is, whether this class is a complete invariant, became known as the 'Rigidity Problem for Roe algebras'. In this talk we will discuss an affirmative answer to this question, and how to approach its proof in a functorial manner. This will allow us to compute the outer automorphism groups of these algebras. This is based on joint work with Federico Vigolo.
Ryszard Nest: A secondary index for chirality operators associated with quantum walks on the binary tree
Abstract: The Witten index is a K-theoretic invariant associated to a couple of almost anti-commuting symmetries on a Hilbert space. Its appearance in physics is connected for example quantum random walk on integers and related chirality operators. We will recall the construction of random quantum walk on graphs, the associated chiral operators and the computation of the associated K-theoretic invariant in the case of quantum random walk on a binary tree. This is a joint work with Toshikazu Natsume.
Abstract: The talk gives an overview on index theory applied to topological insulators, with particular focus on the spectral localizer.
Abstract: The landmark completion of the Elliott classification program for unital separable simple nuclear C*-algebras saw three regularity properties rise to prominence: Z-stability, a C*-algebraic analogue of von Neumann algebras' McDuffness; finite nuclear dimension, an operator algebraic version of having finite Lebesgue dimension; and strict comparison, a generalization of tracial comparison in II_1 factors. Given their relevance to classification, most of the investigations into their interplay have focused on the simple nuclear case.
The purpose of this talk is to advertise the general study of these properties and discuss their applications both within and outside operator algebras. Concretely, I will explain how characterizing when certain twisted group C*-algebras are Z-stable can provide new partial solutions to a well-known problem in generalized time-frequency analysis; this is joint work with U. Enstad. If time allows, I will also briefly discuss how a different incarnation of tracial comparison (finite radius of comparison) for non-commutative tori relates to the existence of smooth Gabor frames; this last part is joint work with U. Enstad and also H. Thiel.
Abstract: The classical result of Gelfand duality says that one can recover a compact Hausdorff space from the C*-algebra of continuous functions on it. In the same spirit, one might ask when can one recover a geometric structure completely from the corresponding topological version of algebras. For example, given the algebra of smooth functions on a smooth manifold, equipped with the topology of compact convergence on all derivatives, can one recover the smooth manifold merely from this data? We provide a generalized version of Gelfand duality that answers this latter question in the affirmative. We will also describe how our result applies to a very large class of topological spaces that includes not only all locally compact Hausdorff spaces, but also all metrizable spaces as well as all CW complexes. Time permitting, in parallel to the theory of Kac algebras and the theory of locally compact quantum groups, with the help of some deep structural results relating to Hilbert’s fifth problem, we will describe how one can study locally compact groups satisfying a mild condition as the spectrum of a suitable topological Hopf algebra that admits a satisfactory duality theory.
Abstract: We introduce the notion of geometric Banach property (T) for metric spaces, which simultaneously generalizes both Banach property (T) for groups and geometric property (T) for metric spaces. This generalization is achieved through representations of the uniform Roe algebra of a metric space on various Banach spaces, including Lp-spaces and uniformly convex Banach spaces, among others. We demonstrate that geometric Banach property (T) is a coarse invariant and establish several equivalent characterizations, including one based on the existence of Kazhdan projections in Banach-Roe algebras. Our investigation is motivated by extending beyond Hilbert space geometry as the model space framework, aiming to deepen the understanding of coarse embeddings and the coarse Baum-Connes conjecture for metric spaces in the context of Banach space geometry. This is joint work with Liang Guo.
Abstract: In this talk, we study Fourier multiplier commutators on a twisted crossed product . We will characterise their Schatten p-class membership by that of their symbols in the associated Besov space. In addition, we show a formula on the Dixmier trace, which also gives us a characterization of the weak Schatten p-class membership of these commutators by a Sobolev space. As an example, our results applied to noncommutative Euclidean Space.
Abstract: In this talk, we study inner and outer supports for an ideal in the reduced groupoid C*-algebra $C^*_r(G)$ for a locally compact Hausdorff and \’{e}tale groupoid $G$. By introducing the notion of ghostly ideals, we establish a sandwiching result for the ideal structure of $C^*_r(G)$. Moreover, we use the dynamics of the groupoid $G$ to characterise when the inner and outer supports of a given ideal in $C^*_r(G)$ coincide. Finally, we will show several applications including regular ideals, tracial ideals and the Thompson group. This is a joint work with Kang Li.
Participants: Michal Doucha, Kang Li, Yoh Tanimoto
Workshop Dinner: Tuesday July 29 at 18:00
Accommodation: we will book hotel rooms at Altmann's Stube (10 minutes walk from Erlangen Central Station) for invited speakers and hotel costs will be directly covered by the workshop.
Airports near Erlangen
Nuremberg (NUE): very few flight connections but is the nearest airport to Erlangen only 12 km distance and 35 min by bus.
Munich (MUC): good flight connections and 2h 30 min by bus or by train.
Frankfurt (FRA): very good flight connections and 3h by train.
Train tickets to Erlangen Central Station: I recommend you to purchase Flexpreis tickets at Deutsche Bahn.
Last modified: 22-07-2025