Amenability, Approximation and Reconstruction

Projektinformationen

AMAREC

ID Finanzhilfevereinbarung: 834267

DOI

10.3030/834267

Projekt abgeschlossen

EK-Unterschriftsdatum 10 Mai 2019

Startdatum 1 Oktober 2019

Enddatum 30 September 2025

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 1 596 017,00

EU-Beitrag

€ 1 596 017,00

Koordiniert durch

UNIVERSITAET MUENSTER
Germany

Periodic Reporting for period 3 - AMAREC (Amenability, Approximation and Reconstruction)

Berichtszeitraum: 2022-10-01 bis 2024-03-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

The project aims at analysing finite dimensional approximations of operator algebraic structures in an amenable context. We study how such approximations arise, how they encode pertinent information, and to what extent the systems are rigid in the sense that they determine the underlying structure and allow to recover it.

Amenability phenomena exist in abundance in theoretical mathematics. They provide a common viewpoint on structures and problems arising in geometry, dynamics, algebra, and functional analysis. In this project we focus on amenable (aka nuclear) C*-algebras, with a particular interest in those coming from amenable dynamical systems.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

The project has so far resulted in a number of preprints and peer-reviewed publications by various authors.

In https://arxiv.org/abs/2209.06507 Gardella, Geffen, Naryshkin, and Vaccaro establish Z-stability for crossed products of outer actions of amenable groups on Z-stable C∗-algebras under a mild technical assumption.

In https://arxiv.org/abs/2201.03409 Gardella, Geffen, Kranz, and Naryshkin show that all amenable, minimal actions of a large class of nonamenable countable groups on compact metric spaces have dynamical comparison.

In https://arxiv.org/abs/2111.15221 Lledó and Martínez show that the main C*-algebras describing the canonical commutation relations of quantum physics, i.e. the Weyl and resolvent algebras, are in the class of Følner C*-algebras.

In https://arxiv.org/abs/2107.14725 Martínez starts the study of whether the reduced C*-algebra of an inverse semigroup is quasi-diagonal.

In https://arxiv.org/abs/2108.04670 Naryshkin shows that a minimal action of a finitely generated group of polynomial growth on a compact metrizable space has comparison.

In https://arxiv.org/abs/2002.03287 Geffen bounds the nuclear dimension of crossed products associated to some partial actions of finite groups or ℤ on finite dimensional locally compact Hausdorff second countable spaces.

In https://arxiv.org/abs/2012.03650 Evington obtains an improved upper bound for the nuclear dimension of extensions of O_\infty-stable C^*-algebras.

In https://arxiv.org/abs/2105.05587 Evington and Girón Pacheco study the H^3 invariant of a group homomorphism ϕ:G→Out(A), where A is a classifiable C^*-algebra.

In https://arxiv.org/abs/2101.08556 Armstrong, de Castro, Orloff Clark, Courtney, Lin, McCormick, Ramagge, Sims, and Steinberg show how to recover a discrete twist over an ample Hausdorff groupoid from a pair consisting of an algebra and what they call a quasi-Cartan subalgebra.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

We seek to make progress on the UCT problem and on the Künneth formula for nuclear C*-algebras. There is work in progress on the non-existence of irrational projections in strongly self-absorbing C*-algebras which points in this direction.