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Panchugopal Bikram

Publications and source records attributed to Panchugopal Bikram.

At least 19 recordsLinked to original sources
arXiv2026

Non-Commutative Wiener-Wintner theorem for amenable group actions

Let $G$ be a locally compact, second countable, amenable group acting on a finite von Neumann algebra $(\mathcal{M},\tau)$ by trace-preserving automorphisms. In this article, we establish a Jacobs-de Leeuw-Glicksberg decomposition for this action, yielding a decomposition of $\mathcal{M}$ into its almost periodic and weakly mixing components. We also prove a noncommutative version of the van der Corput lemma. As an application, we establish a noncommutative Wiener-Wintner theorem for amenable group actions on finite von Neumann algebras.

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arXiv2025

Non-Commutative Maximal Inequalities for State-Preserving Actions of amenable groups

In this article, we establish maximal inequalities and deduce ergodic theorems for state-preserving actions of amenable, locally compact, second-countable groups on tracial non-commutative $L^1$-spaces. As a further consequence, in combination with the Neveu decomposition, we obtain a stochastic ergodic theorem for amenable group actions.

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arXiv2025

Maximal inequalities for square functions and quantitative mean ergodic theorems associated to group metric measure spaces

In this article, we establish weighted strong and weak type inequalities for non-commutative square functions that naturally arise in the analysis of differences between ball averages and martingale sequences within the framework of group metric measure spaces. Then we use these maximal inequalities to prove a quantitative mean ergodic theorem. Our study extends classical harmonic analysis techniques to the non-commutative setting, revealing intricate interactions between group structures, operator-valued functions, and associated filtration systems.

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arXiv2024

Convergence of noncommutative spherical averages for actions of free groups

In this article, we extend the Bufetov pointwise ergodic theorem for spherical averages of even radius for free group actions on noncommutative $L\log L$-space. Indeed, we extend it to more general Orlicz space $L^\Phi(M, \tau)$ (noncommutative/classical), where $M$ is the semifinite von Neumann algebra with faithful normal semifinite trace $\tau$ and $\Phi: [0, \infty ) $ is a Orlicz function such that $ [0, \infty ) \ni t \rightarrow \left({\Phi(t)}\right)^{ \frac{1}{p}}$ is convex for some $p >1$. To establish this convergence we follow similar approach as Bufetov and Anantharaman-Delaroche. Thus, additionally we obtain Rota theorem on the same noncommutative Orlicz space by extending the earlier work of Anantharaman-Delaroche. Anantharaman-Delaroche proved Rota theorem for noncommutative $L^p$-spaces for $p >1$, and mentioned as ``interesting open problem'' to extend it to noncommutative $L\log L$-space as classical case. In the end we also look at the convergence of averages of spherical averages associated to free group and free semigroup actions on noncommutative spaces.

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arXiv2024

Maximal Inequality Associated to Doubling Condition for State Preserving Actions

In this article, we prove maximal inequality and ergodic theorems for state preserving actions on von Neumann algebra by an amenable, locally compact, second countable group equipped with the metric satisfying the doubling condition. The key idea is to use Hardy-Littlewood maximal inequality, a version of the transference principle, and certain norm estimates of differences between ergodic averages and martingales.

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arXiv2023

On the non-commutative Neveu decomposition and stochastic ergodic theorems

In this article, we prove Neveu decomposition for the action of the locally compact amenable semigroup of positive contractions on semifinite von Neumann algebras and thus, it entirely resolves the problem for the actions of arbitrary amenable semigroup on semifinite von Neumann algebras. We also prove it for amenable group actions by Markov automorphisms on any $σ$-finite von Neumann algebras. As an application, we obtain stochastic ergodic theorem for actions of $ \mathbb{Z}_+^d$ and $\mathbb{R}_+^d$ for $ d \in \mathbb{N}$ by positive contractions on $L^1$-spaces associated with a finite von Neumann algebra. It yields the first ergodic theorem for positive contraction on non-commutative $L^1$-spaces beyond the Danford-Schwartz category.

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arXiv2023

On noncommutative ergodic theorems for semigroup and free group actions

In this article, we consider actions of \mathcal{Z}_+^d, \mathcal{R}_+^d and finitely generated free groups on a von Neumann algebras $M$ and prove a version of maximal ergodic inequality. Additionally, we establish non-commutative analogues of pointwise ergodic theorems for associated actions in the predual when M is finite.

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arXiv2023

Weighted Subsequential ergodic theorems on Orlicz spaces

For a semifinite von Neumann algebra M, individual convergence of subsequential, \mathcal{Z}(M) (center of M) valued weighted ergodic averages are studied in noncommutative Orlicz spaces. In the process, we also derive a maximal ergodic inequality corresponding to such averages in noncommutative L^p~ (1 \leq p < \infty) spaces using the weak (1,1) inequality obtained by Yeadon.

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arXiv2022

On the factoriality of q-deformed Araki-Woods von Neumann algebras

The $q$-deformed Araki-Woods von Neumann algebras $Γ_q(\mathcal{H}_\mathbb{R}, U_t)^{\prime \prime}$ are factors for all $q\in (-1,1)$ whenever $dim(\mathcal{H}_\mathbb{R})\geq 3$. When $dim(\mathcal{H}_\mathbb{R})=2$ they are factors as well for all $q$ so long as the parameter defining $(U_t)$ is `small' or $1$ $($trivial$)$ as the case may be.

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arXiv2022

Poisson boundary on full Fock space

This article is devoted to studying the non-commutative Poisson boundary associated with $\Big(B\big(\mathcal{F}(\mathcal{H})\big), P_ω\Big)$ where $\mathcal{H}$ is a separable Hilbert space (finite or infinite-dimensional), $\dim \mathcal{H} > 1$, with an orthonormal basis $\mathcal{E}$, $B\big(\mathcal{F}(\mathcal{H})\big)$ is the algebra of bounded linear operators on the full Fock space $\mathcal{F}(\mathcal{H})$ defined over $\mathcal{H}$, $ω= \{ω_e : e \in \mathcal{E} \}$ is a sequence of positive real numbers such that $\sum_e ω_e = 1$ and $P_ω$ is the Markov operator on $B\big(\mathcal{F}(\mathcal{H})\big)$ defined by \begin{align*} P_ω(x) = \sum_{e \in \mathcal{E}} ω_e l_e^* x l_e, \ x \in B\big(\mathcal{F}(\mathcal{H})\big), \end{align*} where, for $e \in \mathcal{E}$, $l_e$ denotes the left creation operator associated with $e$. The non-commutative Poisson boundary associated with $\Big(B\big(\mathcal{F}(\mathcal{H})\big), P_ω\Big)$ turns out to be an injective factor of type $III$ for any choice of $ω$. Moreover, if $\mathcal{H}$ is finite-dimensional, we completely classify the Poisson boundary in terms of its Connes $S$-invarinat and curiously they are type $III _{λ}$ factors with $λ$ belonging to a certain small class of algebraic numbers.

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arXiv2016

Generator masas in $q$-deformed Araki-Woods von Neumann algebras and factoriality

To any strongly continuous orthogonal representation of $\R$ on a real Hilbert space $\CH_\R$, Hiai constructed $q$-deformed Araki-Woods von Neumann algebras for $-1< q< 1$, which are $W^{\ast}$-algebras arising from non tracial representations of the $q$-commutation relations, the latter yielding an interpolation between the Bosonic and Fermionic statistics. We prove that if the orthogonal representation is not ergodic then these von Neumann algebras are factors whenever $dim(\CH_\R)\geq 2$ and $q\in (-1,1)$. In such case, the centralizer of the $q$-quasi free state has trivial relative commutant. In the process, we study `generator masas' in these factors and establish that they are strongly mixing. The analysis is inspired by a previous work of É. Ricard on Bo$\overset{.}{\text{z}}$ejko-Speicher's factors \cite{ER} and measure-multiplicity invariant of masas introduced by K. Dykema, A. Sinclair and R. Smith in \cite{DSS06}.

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arXiv2014

On the classification and modular extendability of E$_0$-semigroups on factors

In this paper we study modular extendability and equimodularity of endomorphisms and E$_0$-semigroups on factors with respect to f.n.s. weights. We show that modular extendability is a property that does not depend on the choice of weights, it is a cocycle conjugacy invariant and it is preserved under tensoring. We say that a modularly extendable E$_0$-semigroup is of type EI, EII or EIII if its modular extension is of type I, II or III, respectively. We prove that all types exist on properly infinite factors. We also compute the coupling index and the relative commutant index for the CAR flows and $q$-CCR flows. As an application, by considering repeated tensors of the CAR flows we show that there are infinitely many non cocycle conjugate non-extendable $E_0$-semigroups on the hyperfinite factors of types II$_1$, II$_{\infty}$ and III$_λ$, for $λ\in (0,1)$.

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arXiv2014

CAR flows on type III factors and its extendability

In this paper using one of the necessary conditions obtained for extendability in [BISSar], we prove that the CAR flows ([Amo01]) on type III factors arising from most quasi-free states are not extendable. As a consequence we find the super product system of CAR flows. We know from [Arv03] that CCR flows and CAR flows on type I factors with the same Arveson index are cocycle conjugate. But our result together with [BISSar] will show that CCR flows and CAR flows on type III factors are not cocycle conjugate.

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arXiv2013

Extendable endomorphisms on factors

We begin this note with a von Neumann algebraic version of the elementary but extremely useful fact about being able to extend inner-product preserving maps from a total set of the domain Hilbert space to an isometry defined on the entire domain. This leads us to the notion of when `good' endomorphisms of a factorial probability space $(M,ϕ)$ (which we call equi-modular) admit a natural extension to endomorphisms of $L^2(M,ϕ)$. We exhibit examples of such extendable endomorphisms. We then pass to $E_0$-semigroups $α= {α_t: t \geq 0}$ of factors, and observe that extendability of this semigroup (i.e., extendability of each $α_t$) is a cocycle-conjugacy invariant of the semigroup. We identify a necessary condition for extendability of such an $E_0$-semigroup, which we then use to show that the Clifford flow on the hyperfinite $II_1$ factor is not extendable.

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