Operator algebra workshop
The joint meeting of the British Mathematical Colloqium (BMC) and the British Applied Mathematics Colloquium (BAMC) in 2021 will be held at the University of Glasgow, Tuesday 6th – Friday 9th April 2021. As part of the colloquium, a workshop on operator algebras will take place, organised by Christian Voigt (Glasgow) and Stuart White (Oxford).
Speakers
- Christian Bönicke (Glasgow)
- Yemon Choi (Lancaster)
- Matthew Daws (UCLAN)
- Magnus Goffeng (Lund)
- Jacqui Ramagge (Durham), TBC
- Simon Schmidt (Glasgow)
- Alina Vdovina (Newcastle)
- Runlian Xia (Glasgow)
- Makoto Yamashita (Oslo)
Schedule
There will be two sessions on Tuesday 6th and Wednesday 7th April. The tentative schedule is as follows:
Tuesday 6 April
| 11:30 – 12:05 | Matt Daws |
| 12:10 – 12:30 | Simon Schmidt |
| Lunch break | |
| 14:00 – 14:35 | Alina Vdovina |
| 14:40 – 15:15 | Yemon Choi |
Wednesday 7 April
| 11:30 – 11:50 | Runlian Xia |
| 11:55 – 12:30 | Magnus Goffeng |
| Lunch break | |
| 14:00 – 14:20 | Christian Bönicke |
| 14:20 – 14:55 | Jacqui Ramagge (TBC) |
| 14:55 – 15:30 | Makoto Yamashita |
Abstracts
Christian Bönicke, University of Glasgow: Dynamic asymptotic dimension and groupoid homology
Abstract: Dynamic asymptotic dimension is a dimension theory for group actions and more generally for étale groupoids developed by Guentner, Willett, and Yu, which generalizes Gromov’s theory of asymptotic dimension. Having finite asymptotic dimension is known to have important implications for the structure of the involved C*-algebras. In this talk I will report on recent joint work with Dell’Aiera, Gabe, and Willett in which we prove a homology vanishing result for groupoids with finite dynamic asymptotic dimension. Our result allows us to present the first abstract class of groupoids satisfying Matui’s HK conjecture, which claims that the K-theory of a groupoid C*-algebra is completely encoded in the homology of the groupoid.
Matt Daws, UCLAN: An introduction to (quantum) symmetries of (quantum) graphs
Abstract: We will give a survey about quantum automorphism groups, concentrating upon automorphisms of graphs. We plan to quickly introduce the notion of a compact quantum group, describe how quantum groups can (co)act on spaces and algebras, and then describe a universal construction due to Wang which leads to the idea of quantum automorphisms of a finite set. Viewing a finite (simple) graph as a finite set of vertices with a relation describing the edges allowed Banica to define the compact quantum group of automorphisms of a graph. Surprisingly, this construction has recently appeared, repeatedly, in quantum information theory, and we will give a brief indication of how this is. Time allowing, we will also discuss “quantum graphs”, a non-commutative generalisation of a graph, and their quantum automorphisms. The talk will concentrate upon setting the scene and describing some of the technical machinery which occurs.
Yemon Choi, University of Lancaster: Completely almost periodic elements of group von Neumann algebras
Abstract: On a locally compact group $G$, one may consider those $h\in L^\infty(G)$ whose left (or right) translates form a relatively compact subset of $L^\infty(G)$ in the norm topology: such functions are said to be (Bochner-)almost periodic, and they form a unital $C^{\ast}$-subalgebra ${\rm AP}(G)\subseteq L^\infty(G)$, whose Gelfand spectrum coincides with the Bohr compactification of $G$. In particular, if $G$ is compact then ${\rm AP}(G)=C(G)$.
There is a natural analogue of this construction where $L^\infty(G)$ is replaced by ${\rm VN}(\Gamma)$ for a locally compact group $\Gamma$, and the action of the algebra $L^1(G)$ on $L^\infty(G)$ is replaced by the action of the Fourier algebra ${\rm A}(\Gamma)$ on ${\rm VN}(\Gamma)$. Operator space considerations suggest that we should replace the usual notion of compactness by one which takes into account matricial structure, and the resulting space ${\rm CAP}(\widehat{\Gamma})$ is the subject of this talk. We will sketch a proof that ${\rm CAP}(\widehat{\Gamma})$ is always a unital $C^{\ast}$-subalgebra of ${\rm VN}(\Gamma)$, significantly extending previous results of Runde who established this under amenability/injectivity assumptions, and we will indicate how the question “is ${\rm CAP}(\widehat{\Gamma})$ equal to $C^{\ast}_r(\Gamma)$ whenever $\Gamma$ is discrete?” is equivalent to an open problem concerning the uniform Roe algebra.
Magnus Goffeng, University of Lund: Exotic examples in Fell algebras
Abstract: As a partial result towards classifying the C*-algebras that arises from Smale spaces, Robin Deeley and Allan Yashinski studied a Fell algebra arising from Smale spaces with totally disconnected stable sets. The Fell algebra in question has compact spectrum and trivial Dixmier-Duoady class. Using work of Robin Deeley and Karen Strung, the existence of a projection in this Fell algebra was the missing piece needed to finish their classification and a projection was later on (tautologically enough) constructed using classification results. Fell algebras with compact spectrum and trivial Dixmier-Duoady invariant is just a Hausdorff assumption away from being a unital commutative C*-algebra so one might naively think that general approaches from noncommutative geometry will let us put Fell algebras with smooth spectrum on equal footing with manifolds. For instance, one might suspect that all Fell algebras with compact spectrum and trivial Dixmier-Duoady invariant are stably unital. In this talk I will discuss some examples where this fails. Based on joint work with Robin Deeley and Allan Yashinski.
Jacqui Ramagge, University of Durham: An algebraist in operator algebras – a self-similar perspective
Simon Schmidt, University of Glasgow: On the quantum symmetry of distance-transitive graphs
Abstract: To capture the symmetry of a graph one studies its automorphism group. We will talk about a generalization of automorphism groups of finite graphs in the framework of Woronowicz’s compact matrix quantum groups. An important task is to see whether or not a graph has quantum symmetry, i.e. whether or not its quantum automorphism group is commutative. We will see that a graph has quantum symmetry if its automorphism group contains a certain pair of automorphisms. Then, focussing on distance-transitive graphs, we will discuss tools for proving that the generators of the quantum automorphism group commute and deduce that several families of distance-transitive graphs have no quantum symmetry.
Alina Vdovina, University of Newcastle: Buildings, $C^*$-algebras and new higher-dimensional analogues of the Thompson groups
Abstract: We present explicit constructions of infinite families of CW-complexes of arbitrary dimension with buildings as the universal covers. These complexes give rise to new families of $C^*$-algebras, classifiable by their $K$-theory. The underlying building structure allows explicit computation of the $K$-theory. We will also present new higher-dimensional generalizations of the Thompson groups, which are usually difficult to distinguish, but the $K$-theory of $C^*$-algebras gives new invariants to recognize non-isomorphic groups.
Runlian Xia, University of Glasgow: Non-commutative Hilbert transforms and Cotlar-type identities
Abstract: The Hilbert transform $H$ is a basic example of a Fourier multiplier. Its behaviour on Fourier series is the following:
$$
\sum_{n\in \mathbb{Z}}a_n e^{inx} \longmapsto \sum_{n\in \mathbb{Z}}m(n)a_n e^{inx},
$$
with $m(n)=-i\,{\rm sgn} (n)$. Riesz proved that $H$ is a bounded operator on $L_p(\mathbb{T})$ for all $1<p<\infty$. We study Hilbert transform type Fourier multipliers on group algebras and their boundedness on corresponding non-commutative $L_p$-spaces. The pioneering work in this direction is due to Mei and Ricard, in which they prove $L_p$-boundedness of Hilbert transforms on free group von Neumann algebras using a Cotlar identity on von Neumann algebras. In this talk, we introduce a new form of Cotlar identities for groups that are not necessarily free products and study their validity for lattices of ${\rm SL}_2(\mathbb{C})$ and some other groups acting on trees. Joint work with Adri\’an Gonz\’alez and Javier Parcet.
Makoto Yamashita, University of Oslo: Homology and K-theory of torsion-free ample groupoids
Abstract: The problem of connecting the integral homology to the K-groups of C*-algebra for ample groupoids was recently popularized by Matui. As an answer for this, we construct a spectral sequence starting from the groupoid homology which ends at the K-groups when the groupoid satisfies the Baum-Connes conjecture and has torsion free stabilizers. The construction crucially relies on the Meyer-Nest theory of triangulated structure on equivariant KK-categories. The same technique allows us to incorporate Putnam’s homology theory for Smale spaces in place of groupoid homology.
Contact
Any registered participant in the BMC/BAMC is welcome to attend the workshop. For instructions on how to register see here.
If you have any questions regarding the workshop please use the contact form, or contact one of the organisers directly.
BMC/BAMC Glasgow 2021 