Short Course

Simplicity of the reduced crossed product C(X) ⋊r Γ and its relation with dynamics

Tattwamasi Amrutam

                        Institute of Mathematics of the Polish Academy of Sciences

About the speaker

Dr. Tattwamasi Amrutam is an (adjunct) assitant professor at the Institute of Mathematics of the Polish Academy of Sciences. Previously, he obtained his PhD in 2021 from the University of Houston under the supervision of Mehrdad Kalantar and conducted postdoctoral research at Ben-Gurion University of the Negev. His research focuses on the intersection of group dynamics and operator algebras.

Abstract

Let Γ be a discrete group and X a minimal Γ space (by this, we mean that X is compact, Hausdorff and the action Γ ↷ X is by homeomorphisms). For the unital Γ-C∗-algebra C(X), the reduced crossed product C(X) ⋊rΓ, encodes the information of C(X) and the group Γ (much similar to the construction of G ⋉ H, a semi-direct product of two groups G and H). In 1972, Furstenberg introduced “boundary” to study certain properties of lattices of semisimple Lie groups (e.g., think of SLn(Z) inside SLn(R)). Glasner generalized this notion to make sense of a boundary over a compact Hausdroff space X (which is called a generalized boundary these days). Building on the works of Kalantar-Kennedy [4], Breuillard-Kalantar-Kennedy-Ozawa [2], Haagerup [3], Naghavi [6], Kawabe [5], and our joint work with Ursu [1], we establish the equivalence of the following.

1. C(X) ⋊rΓ is simple (assuming that Γ ↷ X is minimal).

2. Γ ↷ ∂F(Γ, X) is (topologically) free.

3. C(X) ⋊rΓ has generalized Powers’ averaging.

When X is one point space, we recover the result of Kalantar-Kennedy [4], which gave a dynamical characterization of C∗-simplicity. In particular, they showed that a group Γ is C∗- simple if and only if the action Γ ↷ ∂FΓ on the Furstenberg boundary ∂FΓ is (topologically) free.

Time and venue:

Thursday 04/17 and Friday 04/18, 14:30-17:30, Zhizhi Building, Room 32 

Monday 04/21, 9:00-12:00, Zhizhi Building, Room 22

Thursday 04/24 and Friday 04/25, 14:30-17:30, Zhizhi Building, Room 32

References

1. Tattwamasi Amrutam and Dan Ursu.

   A generalized powers averaging property for commutative crossed products.

   Transactions of the American Mathematical Society, 375(3):2237–2254, 2022.

2. Emmanuel Breuillard, Mehrdad Kalantar, Matthew Kennedy, and Narutaka Ozawa.

   C∗-simplicity and the unique trace property for discrete groups.

   Publications mathématiques de l’IHÉS, (1):35–71, 2017.

3. Uffe Haagerup.

   A new look at C*-simplicity and the unique trace property of a group.

   In Toke M. Carlsen, Nadia S. Larsen, Sergey Neshveyev, and Christian Skau, editors, Operator Algebras and Applications, pages 167–176, Cham, 2016.

   Springer International Publishing.

4. M. Kalantar and M. Kennedy.

   Boundaries of reduced C∗-algebras of discrete groups.

   Journal für die Reine und Angewandte Mathematik, 727:247–267.

5. Takuya Kawabe.

   Uniformly recurrent subgroups and the ideal structure of reduced crossed products.

   arXiv e-prints, page arXiv:1701.03413, 2017.

6. Zahra Naghavi.

Furstenberg Boundary of Minimal Actions.

Integral Equations and Operator Theory, 92(2), 2020.