arXiv subjects

Tattwamasi Amrutam

Publications and source records attributed to Tattwamasi Amrutam.

32 records · Page 2Linked to original sources

arXiv2025

Non-commutative Intermediate Factor theorem associated with $W^*$-dynamics of product groups

Let $G = G_{1} \times G_{2}$ be a product of two locally compact, second countable groups and $μ\in \mathrm{Prob}(G)$ be of the form $μ= μ_{1} \times μ_{2}$, where $μ_{i} \in \mathrm{Prob}(G_{i})$. Let $(B,ν_B)$ be the associated Poisson boundary. We show that every intermediate $G$-von Neumann algebra $\mathcal{M}$ with \[ \mathcal{N} \subseteq \mathcal{M} \subseteq \mathcal{N} \,\bar{\otimes}\, L^{\infty}(B,ν) \] splits as a tensor product of the form $\mathcal{N}\bar{\otimes}L^{\infty}(C,ν_C)$, where $(C,ν_C)$ is a $(G,μ)$-boundary. Here, $\mathcal{N}$ is a tracial von Neumann algebra on which $G$ acts trace-preservingly. This generalizes the Intermediate Factor Theorem proved by Bader--Shalom (\cite[Theorem~1.9]{BS06}) in the measurable setup. In addition, we give various other examples of the splitting phenomenon associated with $W^{*}$-dynamics. We also show that certain assumptions are necessary for the intermediate algebras to split, and ideals in the ambient tensor product algebra obstruct the splitting phenomenon. We also use the Master theorem from \cite{glasner2023intermediate} to resolve the second part of \cite[Problem~5.2]{jiangskalski} in the affirmative.

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arXiv2025

Invariant subalgebras rigidity for von Neumann algebras of groups arising as certain semidirect products

We study the ISR (von Neumann invariant subalgebra rigidity) property for certain discrete groups arising as semidirect products from algebraic actions on certain 2-torsion groups, mostly arising as direct products of $\mathbb{Z}_2$. We present, in particular, the first example of an amenable group with the ISR property that admits a non-trivial abelian normal subgroup. Several other examples are discussed, notably including an infinite amenable group whose von Neumann algebra admits precisely one invariant von Neumann subalgebra which does not come from a normal subgroup. We also investigate the form of invariant subalgebras of the group von Neumann algebra of the standard lamplighter group.

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arXiv2025

Splitting of Tensor Products and Intermediate Factor Theorem: Continuous Version

Let $G$ be a discrete group. Given unital $G$-$C^*$-algebras $\mathcal{A}$ and $\mathcal{B}$, we give an abstract condition under which every $G$-subalgebra $\mathcal{C}$ of the form $\mathcal{A}\subset \mathcal{C}\subset \mathcal{A}\otimes_{\text{min}}\mathcal{B}$ is a tensor product. This generalizes the well-known splitting results in the context of $C^*$-algebras by Zacharias and Zsido. As an application, we prove a topological version of the Intermediate Factor theorem. When a product group $G=Γ_1\timesΓ_2$ acts (by a product action) on the product of corresponding $Γ_i$-boundaries $\partialΓ_i$, using the abstract condition, we show that every intermediate subalgebra $C(X)\subset\mathcal{C}\subset C(X)\otimes_{\text{min}}C(\partialΓ_1\times \partialΓ_2)$ is a tensor product (under some additional assumptions on $X$). This can be considered as a topological version of the Intermediate Factor theorem. We prove that our assumptions are necessary and cannot generally be relaxed. We also introduce the notion of a uniformly rigid action for $C^*$-algebras and use it to give various classes of inclusions $\mathcal{A}\subset \mathcal{A}\otimes_{\text{min}}\mathcal{B}$ for which every invariant intermediate algebra is a tensor product.

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arXiv2024

On the amenable subalgebras of group von Neumann algebras

We approach the study of sub-von Neumann algebras of the group von Neumann algebra $LΓ$ for countable groups $Γ$ from a dynamical perspective. It is shown that $L(Γ)$ admits a maximal invariant amenable subalgebra. The notion of invariant probability measures (IRAs) on the space of sub-algebras is introduced, analogous to the concept of Invariant Random Subgroups. And it is shown that amenable IRAs are supported on the maximal amenable invariant sub-algebra.

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arXiv2024

On relative commutants of subalgebras in group and tracial crossed product von Neumann algebras

Let $Γ$ be a discrete group acting on a compact Hausdorff space $X$. Given $x\in X$, and $μ\in\text{Prob}(X)$, we introduce the notion of contraction of $μ$ towards $x$ with respect to unitary elements of a group von Neumann algebra not necessarily coming from group elements. Using this notion, we study relative commutants of subalgebras in tracial crossed product von Neumann algebras. The results are applied to negatively curved groups and $\text{SL}(d,\mathbb{Z})$, $d \geq 2$.

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arXiv2024

Crossed products of dynamical systems; rigidity Vs. strong proximality

Given a dynamical system $(X, Γ)$, the corresponding crossed product $C^*$-algebra $C(X)\rtimes_{r}Γ$ is called reflecting, when every intermediate $C^*$-algebra $C^*_r(Γ)<\mathcal{A} < C(X)\rtimes_{r}Γ$ is of the form $\mathcal{A}=C(Y)\rtimes_{r}Γ$, corresponding to a dynamical factor $X \rightarrow Y$. It is called almost reflecting if $\mathbb{E}(\mathcal{A}) \subset \mathcal{A}$ for every such $\mathcal{A}$. These two notions coincide for groups admitting the approximation property (AP). Let $Γ$ be a non-elementary convergence group or a lattice in $\text{SL}_d(\mathbb{R})$ for some $d \ge 2$. We show that any uniformly rigid system $(X,Γ)$ is almost reflecting. In particular, this holds for any equicontinuous action. In the von Neumann setting, for the same groups $Γ$ and any uniformly rigid system $(X,\mathcal{B},μ, Γ)$ the crossed product algebra $L^{\infty}(X,μ)\rtimesΓ$ is reflecting. An inclusion of algebras $\mathcal{A}\subset\mathcal{B}$ is called $\textit{minimal ambient}$ if there are no intermediate algebras. As a demonstration of our methods, we construct examples of minimal ambient inclusions with various interesting properties in the $C^*$ and the von Neumann settings.

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arXiv2024

Non-Abelian Factors for Actions of $\mathbb{Z}$ and Other Non-$C^*$-Simple Groups

Let $Γ$ be a countable group and $(X, Γ)$ a compact topological dynamical system. We study the question of the existence of an intermediate $C^*$-subalgebra $\mathcal{A}$ $$C^{*}_{r}(Γ)<\mathcal{A}<C(X)\rtimes_rΓ,$$ which is not of the form $\mathcal{A} = C(Y) \rtimes_r Γ$, corresponding to a factor map $(X,Γ) \to (Y,Γ)$. Here $ C^{*}_{r} (Γ)$ and $C(X) \rtimes_r Γ$ are the reduced $C^*$-algebras of $Γ$ and $(X,Γ)$ respectively. Our main results are (1) For $Γ$, which is not $C^*$-simple, if $(X,Γ)$ admits a $Γ$-invariant probability measure, then such a sub-algebra always exists. (2) For $Γ= \mathbb{Z}$ and $(X, Γ)$ an irrational rotation of the circle $X = S^1$, we give a full description of all these non-crossed-product subalgebras.

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arXiv2024

Powers averaging for actions on $C(X)$-algebras

Given a unital $C(X)$-algebra $A$ discrete group $Γ$ and an action $α: Γ\to \text{aut}(A)$ which leaves $C(X)$ invariant and such that $C(X)\rtimes_{α,r} Γ$ is simple, and a $2$-cocycle $ω$, we obtain a bijective correspondence between maximal $Γ$-invariant ideals of $A$ and maximal ideals in $A\rtimes_{α,ω,r} Γ$. In particular, $A\rtimes_{α,ω,r} Γ$ is simple if and only if $A$ has no $Γ$-invariant ideals.

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arXiv2023

Intermediate crossed product $C^*$-algebras

Let $B$ be a separable $C^*$-algebra, let $Γ$ be a discrete countable group, let $α: Γ\to \text{Aut}(B)$ be an action, and let $A$ be an invariant subalgebra. We find certain freeness conditions which guarantee that any intermediate $C^*$-algebra $A \rtimes_{α,r} Γ\subseteq C \subseteq B \rtimes_{α,r} Γ$ is a crossed product of an intermediate invariant subalgebra $A \subseteq C_0 \subseteq B$ by $Γ$. Those are used to generalize related results by Suzuki.

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arXiv2023

Subalgebras, subgroups, and singularity

This paper concerns the non-commutative analog of the Normal Subgroup Theorem for certain groups. Inspired by Kalantar-Panagopoulos, we show that all $Γ$-invariant subalgebras of $LΓ$ and $C^*_r(Γ)$ are ($Γ$-) co-amenable. The groups we work with satisfy a singularity phenomenon described in Bader-Boutonnet-Houdayer-Peterson. The setup of singularity allows us to obtain a description of $Γ$-invariant intermediate von Neumann subalgebras $L^{\infty}(X,ξ)\subset\mathcal{M}\subset L^{\infty}(X,ξ)\rtimesΓ$ in terms of the normal subgroups of $Γ$.

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arXiv2022

On invariant von Neumann subalgebras rigidity property

We say that a countable discrete group $Γ$ satisfies the invariant von Neumann subalgebras rigidity (ISR) property if every $Γ$- invariant von Neumann subalgebra $\mathcal{M}$ in $L(Γ)$ is of the form $L(Λ)$ for some normal subgroup $Λ\lhd Γ$. We show many ``negatively curved" groups, including all torsion free non-amenable hyperbolic groups and torsion free groups with positive first $L^2$-Betti number under a mild assumption, and certain finite direct product of them have this property. We also discuss whether the torsion-free assumption can be relaxed.

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arXiv2021

A generalized Powers averaging property for commutative crossed products

We prove a generalized version of Powers' averaging property that characterizes simplicity of reduced crossed products $C(X) \rtimes_λG$, where $G$ is a countable discrete group, and $X$ is a compact Hausdorff space which $G$ acts on minimally by homeomorphisms. As a consequence, we generalize results of Hartman and Kalantar on unique stationarity to the state space of $C(X) \rtimes_λG$ and to Kawabe's generalized space of amenable subgroups $\operatorname{Sub}_a(X,G)$. This further lets us generalize a result of the first named author and Kalantar on simplicity of intermediate C*-algebras. We prove that if $C(Y) \subseteq C(X)$ is an inclusion of unital commutative $G$-C*-algebras with $X$ minimal and $C(Y) \rtimes_λG$ simple, then any intermediate C*-algebra $A$ satisfying $C(Y) \rtimes_λG \subseteq A \subseteq C(X) \rtimes_λG$ is simple.

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arXiv2019

On Intermediate $C^$-subalgebras of $C^$-simple Group Actions

We show that for a large class of actions $Γ\curvearrowright \mathcal{A}$ of $C^*$-simple groups $Γ$ on unital $C^*$-algebras $\mathcal{A}$, including any non-faithful action of a hyperbolic group with trivial amenable radical, every intermediate $C^*$-subalgebra $\mathcal{B}$, $C_λ^*(Γ)\subseteq \mathcal{B} \subseteq \mathcal{A}\rtimes_{r}Γ$, is of the form $\mathcal{A}_1\rtimes_{r}Γ$, where $\mathcal{A}_1$ is a unital $Γ$-$C^*$-subalgebra of $\mathcal{A}$.

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arXiv2018

On simplicity of intermediate C*-algebras

We prove simplicity of all intermediate $C^*$-algebras $C^*_{r}(Γ)\subseteq \mathcal{B} \subseteq Γ\ltimes_r C(X)$ in the case of minimal actions of $C^*$-simple groups $Γ$ on compact spaces $X$. For this, we use the notion of stationary states, recently introduced by Hartman and Kalantar (arXiv:1712.10133v2). We show that the Powers' averaging property holds for the reduced crossed product $Γ\ltimes_r \mathcal{A}$ for any action $Γ\curvearrowright \mathcal{A}$ of a $C^*$-simple group $Γ$ on a unital $C^*$-algebra $\mathcal{A}$, and use it to prove a one-to-one correspondence between stationary states on $\mathcal{A}$ and those on $Γ\ltimes_r \mathcal{A}$.

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