Thu, 27 Jun 2024

15:15 - 16:15
C1

Cartan subalgebras of twisted groupoid $C^*$-algebras with a focus on $k$-graph $C^*$-algebras

Rachael Norton
(St Olaf College)
Abstract

The set $M_n(\mathbb{R})$ of all $n \times n$ matrices over the real numbers is an example of an algebraic structure called a $C^*$-algebra. The subalgebra $D$ of diagonal matrices has special properties and is called a \emph{Cartan subalgebra} of $M_n(\mathbb{R})$. Given an arbitrary $C^*$-algebra, it can be very hard (but also very rewarding) to find a Cartan subalgebra, if one exists at all. However, if the $C^*$-algebra is generated by a cocycle $c$ and a group (or groupoid) $G$, then it is natural to look within $G$ for a subgroup (or subgroupoid) $S$ that may give rise to a Cartan subalgebra. In this talk, we identify sufficient conditions on $S$ and $c$ so that the subalgebra generated by $(S,c)$ is indeed a Cartan subalgebra of the $C^*$-algebra generated by $(G,c)$. We then apply our theorem to $C^*$-algebras generated by $k$-graphs, which are directed graphs in higher dimensions. This is joint work with J. Briones Torres, A. Duwenig, L. Gallagher, E. Gillaspy, S. Reznikoff, H. Vu, and S. Wright.

On metric approximate subgroups
Hrushovski, E Fanlo, A Journal of Mathematical Logic (12 Jul 2024)
Geometric Martingale Benamou-Brenier transport and geometric Bass martingales
Backhoff-Veraguas, J Loeper, G Obloj, J (06 Jun 2024)
Local characterization of block-decomposability for multiparameter
persistence modules
Lebovici, V Lerch, J Oudot, S (26 Feb 2024) http://arxiv.org/abs/2402.16624v3
Twistor sigma models for quaternionic geometry and graviton scattering
Adamo, T Mason, L Sharma, A Advances in Theoretical and Mathematical Physics volume 27 issue 3 623-681 (06 Jun 2024)
Optimal maps and local-to-global property in negative dimensional spaces with Ricci curvature bounded from below
Magnabosco, M Rigoni, C Tohoku Mathematical Journal volume 75 issue 4 (01 Dec 2023)
The strong Brunn–Minkowski inequality and its equivalence with the CD condition
Magnabosco, M Portinale, L Rossi, T Communications in Contemporary Mathematics 2450025 (21 Aug 2024)
Sums of Linear Transformations in Higher Dimensions
Mudgal, A The Quarterly Journal of Mathematics volume 70 issue 3 965-984 (12 Sep 2019)
Subscribe to