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N. Shravan Kumar

Publications and source records attributed to N. Shravan Kumar.

At least 19 recordsLinked to original sources
arXiv2026

Weak containment of representation on topological groupoids

Let $G$ be a second-countable, locally compact Hausdorff groupoid equipped with a Haar system. This paper investigates the weak containment of continuous unitary representations of groupoids. We show that both induction and inner tensor product of representations preserve weak containment. Additionally, we introduce the notion of a topological invariant mean on $G/H$ and explore its connection to amenability. With that, we establish a groupoid analogue of Greenleaf's theorem. Finally, we provide independent results concerning the restriction of induced representations for continuous unitary representations of relatively clopen wide subgroupoids $H\subseteq G$ with discrete unit space and closed transitive wide subgroupoids of compact transitive groupoids.

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arXiv2026

Amenable unitary representations of locally compact groupoids

Let $G$ be a second countable locally compact groupoid equipped with a Haar system $λ$.In this work, we introduce and develop the notion of amenability for continuous unitary representations of $G$, formulated in terms of Hilbert bundles over the unit space $G^{0}$. We prove that $G$ is amenable if and only if its left regular representation is amenable, thereby extending Bekka's characterisation of amenable unitary representations from groups to groupoids. We further investigate the amenability of induced representations of $G$ and also study the representation of properly amenable groupoids. We define a topological invariant mean associated with a representation, constructed by utilising the theory of operator-valued vector measures on the unit space $G^{0}$, to characterise amenability. Finally, we investigate the relationship between the amenability of a representation and the amenability of its restriction to isotropy subgroups with an emphasis on HLS and certain transitive groupoids.

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arXiv2026

Continuous field of Orlicz space on locally compact groupoids and related results

Let $G$ be a locally compact second countable groupoid with a fixed Haar system $λ=\{λ^{u}\}_{u\in G^{0}}$ and $(Φ,Ψ)$ be a complementary pair of $N$-functions satisfying $Δ_{2}$-condition. In this article, we introduce the continuous field of Orlicz space $(L^Φ_{0},Δ_{1})$ and provide a sufficient condition for the space of continuous sections vanishing at infinity, denoted $E^Φ_{0}$, to be an Banach algebra under a suitable convolution. Further, the condition for a closed $C_{b}(G^{0})$-submodule $I$ of $E^Φ_{0}$ to be a left ideal is established. Moreover, we provide a groupoid analogue of the characterization of the space of convolutors of Morse-Transue space for locally compact groups.

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arXiv2026

Induced Representation of Topological groupoids

Let $G$ be a locally compact second countable groupoid with a Haar system. In this article, we introduce the induced representation of $G$ from a continuous unitary representation of a closed wide subgroupoid $H$ with a Haarsystem provided there exists a full equivariant system of measures $μ=\{μ^{u}\}_{u\in G^{0}}$ on $G/H$. We prove some basic properties of induced representation and a theorem on induction in stages. A groupoid version of Mackey's tensor product theorem is also provided. We also prove a groupoid version of Frobenius Reciprocity theorem on compact transitive groupoids.

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arXiv2026

Invariant Means on $VN^n(G)$

Let $G$ be a locally compact group, and $VN^n(G)$ is the dual of the multidimensional Fourier algebra $A^n(G)$. In this article, we define invariant means on $VN^n(G)$ and prove that the set of all invariant means on $VN^n(G)$ is non-empty. Further, we investigated the invariant means on $VN^n(G)$ for discrete and non-discrete cases of $G$. Also, we show that if $H$ is an open subgroup of $G$, then the number of invariant means on $VN^n(H)$ is the same as that of $VN^n(G)$. Finally, we study invariant means on the dual of the algebra $A_0^n(G)$, the closure of Fourier algebra $A^n(G)$ in the cb-multiplier norm.

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arXiv2025

Topological Center of the Double Dual of the Orlicz Figà-Talamanca Herz Algebra

Let $G$ a locally compact group and $(Φ,Ψ)$ be a complementary pair of Young functions. Let $A_Φ(G)$ be the Orlicz analogue of the classical Figà-Talamanca Herz algebra $A_p(G).$ In this article, we establish a necessary and sufficient condition for the equality $Λ(A_Φ(G)^{\ast\ast}) = A_Φ(G)$ to hold, where $Λ(A_Φ(G)^{\ast\ast})$ denotes the topological center of the double dual of $A_Φ(G)$ when equipped with the first Arens product. Furthermore, we prove several results concerning the semi-simplicity of the Banach algebras $A_Φ(G)^{\ast\ast}$ and $UCB_Ψ(\widehat{G})^\ast.$

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arXiv2025

Isomorphism Theorems for the Algebras of $Φ-$Pseudofunctions and $Φ-$Pseudomeasures

In this article, we study the isomorphism problem for the algebras of $Φ-$Pseudofunctions and $Φ-$Pseudomeasures, denoted by $PF_Φ(G)$ and $PM_Φ(G),$ respectively. More precisely, for a certain class of Young functions $Φ,$ we prove that if there exists an isometric isomorphism between $PF_Φ(G_1)$ and $PF_Φ(G_2),$ or between $PM_Φ(G_1)$ and $PM_Φ(G_2),$ then $G_1$ and $G_2$ are isomorphic as topological groups. In addition, we present an Orlicz version of Parrott's theorem.

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arXiv2025

Spectral synthesis in multidimensional Fourier algebras

Let $G$ be a locally compact group and let $A^n(G)$ denote the $n$-dimensional Fourier algebra, introduced by Todorov and Turowska. We investigate spectral synthesis properties of the multidimensional Fourier algebra $A^n(G).$ In particular, we prove versions of the subgroup lemma, injection, and inverse projection theorems for both spectral sets and Ditkin sets. Additionally, we provide a result on the parallel synthesis between $A^n(G)$ and $A^{n+1}(G)$ and finally prove Malliavin's theorem.

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arXiv2025

Approximate identity and approximation properties of multidimensional Fourier algebras

For a locally compact group $G$, let $A^n(G)$ denote the multidimensional Fourier algebra given by $ \otimes_{n}^{eh} A(G).$ This work explores the approximation identity and operator amenability of the algebra $A^n(G)$. Further, we study the approximation properties (AP) and the concept of weak amenability of the multidimensional Fourier algebra.

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arXiv2025

Amenability and Invariant subspaces of the algebra of pseudomeasures

Let $G$ be a locally compact group and $(Φ,Ψ)$ a complimentary pair of Young functions. In this article, we consider the Banach algebra of $Ψ$-pseudomeasures $PM_Ψ(G)$ and the Orlicz Figà-Talamanca Herz algebra $A_Φ(G).$ We prove sufficient conditions for a group $G$ to be amenable in terms of the norm closed topologically invariant subspaces of $PM_Ψ(G).$ Further, for an amenable group $G$ with the Young function $Φ$ satisfying the MA condition, we establish a one-to-one correspondence between certain topologically invariant subalgebras of $PM_Ψ(G)$ and the class of closed subgroups of $G.$ Moreover, we prove a similar result for the predual $A_Φ(G)$ and derive a bijection between certain topologically invariant subalgebras of $A_Φ(G)$ and the set of compact subgroups of $G.$

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arXiv2024

Vector-valued properties of the Weyl transform

In this paper, we introduce and study the Weyl transform of functions which are integrable with respect to a vector measure on a phase space associated to a locally compact abelian group. We also study the Weyl transform of vector measures. Later, we also introduce and study the convolution of functions from $L^p$-spaces associated to a vector measure. We also study the Weyl transform of vector-valued functions and prove a vector-valued analogue of the Hausdorff-Young inequality.

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arXiv2024

Ideals in the dual of introverted subspaces of $Ψ$-pseudomeasures

Let $G$ be a locally compact group and $(Φ,Ψ)$ a complementary pair of Young functions satisfying the $Δ_2$-condition. Let $A_Φ(G)$ be the Orlicz analogue of the Figà-Talamanca Herz algebra $A_p(G).$ The dual of the algebra $A_Φ(G)$ is the space of $Ψ$-pseudomeasures, denoted by $PM_Ψ(G).$ For certain topologically introverted subspaces $\mathcal{A}$ of $PM_Ψ(G)$ and the Banach algebras $W_Φ(G)$ or $B_Φ(G),$ denoted by $\mathcal{B},$ we characterise the maximal regular left/right/two-sided ideals of the Banach algebras $\mathcal{A}^{'}$ and $\mathcal{B}^{''}$ considered with the Arens product. We further characterise the minimal left ideals of $\mathcal{A}^{'}$ and prove the necessary and sufficient conditions for the existence of minimal ideals in the algebras $A_Φ(G)$ and $\mathcal{B}.$

math.FA↗
arXiv2024

Restriction theorems for the $p$-analog of the Fourier-Stieltjes algebra

For a locally compact group $G$ and $1 < p < \infty,$ let $B_p(G)$ denote the $p$-analog of the Fourier-Stieltjes algebra $B(G) \, (\text{or} \, B_2(G))$. Let $r: B_p(G) \to B_p(H)$ be the restriction map given by $r(u) = u|_H$ for any closed subgroup $H$ of $G.$ In this article, we prove that the restriction map $r$ is a surjective isometry for any open subgroup $H$ of $G.$ Further, we show that the range of the map $r$ is dense in $B_p(H)$ when $H$ is either a compact normal subgroup of $G$ or compact subgroup of an [SIN]$_H$-group.

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arXiv2024

Arens regularity of the Orlicz Figà-Talamanca Herz Algebra

Let G be a locally compact group and let $A_Φ(G)$ be the Orlicz-version of the Figà-Talamanca Herz algebra of G associated with a Young function $Φ.$ We show that if $A_Φ(G)$ is Arens regular, then $G$ is discrete. We further explore the Arens regularity of $A_Φ(G)$ when the underlying group $G$ is discrete. In the running, we also show that $A_Φ(G)$ is finite-dimensional if and only if $G$ is finite. Further, for amenable groups, we show that $A_Φ(G)$ is reflexive if and only if $G$ is finite, under the assumption that the associated Young function $Φ$ satisfies the MA-condition.

math.FA↗
arXiv2023

Spectral Synthesis in the Multidimensional Fourier Algebra and the Varopolous Algebra for Compact Groups

Let $G$ be a compact group and let $A^n(G)$ denote the multidimensional Fourier algebra introduced by Todorov and Turowska. In this note, we first define the multidimensional version of the Varopolous algebra and show that the multidimensional Fourier algebra can be embedded into the multidimensional Varopolous algebra. Using this embedding, we also prove a result on parallel synthesis, subsuming the earlier results of Varopolous, Spronk-Turowska and Parthsarathy-Prakash.

math.FA↗
arXiv2023

Paley inequality for the Weyl transform and its applications

In this paper, we prove several versions of the classical Paley inequality for the Weyl transform. As an application, we discuss $L^p$-$L^q$ boundedness of the Weyl multipliers and prove a version of the Hörmander's multiplier theorem. We also prove Hardy-Littlewood inequality. Finally, we study vector-valued versions of these inequalities. In particular, we consider the inequalities of Paley, Hausdorff-Young, and Hardy-Littlewood and their relations.

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