Abstract :

Let Γ be a discrete group and A, an unital Γ-C ∗ -algebra. We can associate with it  a C ∗ -algebraic object A ⋊r Γ called the reduced crossed product. Similarly, for a Γvon Neumann algebra M, we can associate a von Neumann algebraic crossed product  denoted by M ⋊ Γ. When $A =C,$ this construction gives us the group C ∗ -algebra  Cr∗ (Γ). Similarly, when $M = C$, we obtain the group von Neumann algebra L(Γ). We  are interested in giving a complete description of the Γ-invariant sub-algebras of A⋊r Γ  or that of M ⋊ Γ.  In this talk, I shall discuss recent progress in this direction. We will focus on two  particular instances. Firstly, on intermediate algebras B of the form Cr∗ (Γ) ⊂ B ⊂  A ⋊r Γ or N of the form L(Γ) ⊂ N ⊂ M ⋊ Γ. Secondly, on invariant sub-algebras  N ≤ L(Γ). These are two particular cases of invariant sub-algebras. 

This talk is based on four recent joint works: [AJ23] (with Yongle Jiang), [AH24]  (with Yair Hartman), and [AGG24b, AGG24a] (with Eli Glasner and Yair Glasner). 
References 
[AGG24a] Tattwamasi Amrutam, Eli Glasner, and Yair Glasner, Crossed products of dynamical systems; rigidity vs. strong proximality, arXiv preprint  arXiv:2404.09803 (2024), 17pg. 
[AGG24b] ___, Non-abelian factors for actions of Z and other non-C*-simple  groups, Journal of Functional Analysis (2024), 110456. 
[AH24] Tattwamasi Amrutam and Yair Hartman, Subalgebras, subgroups, and singularity, Bulletin of the London Mathematical Society 56 (2024), no. 1,  380–395. 
[AJ23] Tattwamasi Amrutam and Yongle Jiang, On invariant von neumann subalgebras rigidity property, Journal of Functional Analysis 284 (2023), no. 5,  109804.