Periodic Reporting for period 3 - AMAREC (Amenability, Approximation and Reconstruction)
Période du rapport: 2022-10-01 au 2024-03-31
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.
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.
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(s’ouvre dans une nouvelle fenêtre) 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(s’ouvre dans une nouvelle fenêtre) 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(s’ouvre dans une nouvelle fenêtre) 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(s’ouvre dans une nouvelle fenêtre) Martínez starts the study of whether the reduced C*-algebra of an inverse semigroup is quasi-diagonal.
In https://arxiv.org/abs/2108.04670(s’ouvre dans une nouvelle fenêtre) 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(s’ouvre dans une nouvelle fenêtre) 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(s’ouvre dans une nouvelle fenêtre) 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(s’ouvre dans une nouvelle fenêtre) 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(s’ouvre dans une nouvelle fenêtre) 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.
In https://arxiv.org/abs/2209.06507(s’ouvre dans une nouvelle fenêtre) 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(s’ouvre dans une nouvelle fenêtre) 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(s’ouvre dans une nouvelle fenêtre) 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(s’ouvre dans une nouvelle fenêtre) Martínez starts the study of whether the reduced C*-algebra of an inverse semigroup is quasi-diagonal.
In https://arxiv.org/abs/2108.04670(s’ouvre dans une nouvelle fenêtre) 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(s’ouvre dans une nouvelle fenêtre) 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(s’ouvre dans une nouvelle fenêtre) 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(s’ouvre dans une nouvelle fenêtre) 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(s’ouvre dans une nouvelle fenêtre) 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.
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.