arXiv subjects

Lav Kumar Singh

Publications and source records attributed to Lav Kumar Singh.

7 recordsLinked to original sources

arXiv2026

Sherman-Takeda type theorems for locally C*-algebras

In this article, we will first establish some density results for a locally $C^*$-algebra $\mathcal A$ and then identify a property, called Kaplansky density property (KDP). We then give a induced faithful continuous $*$-representation $\varphi$ of $\mathcal A^{**}$ (equipped with unique Arens product) on the space $B_{loc}(\mathcal H)$ such that $\varphi(\mathcal A^{**})\subset \overline{\pi(\mathcal A)}^{WOT}$, where $\pi:\mathcal A\to B_{loc}(\mathcal H)$ is the associated universal $*$-representation and $\mathcal H$ is the associated locally Hilbert space. Finally we show that for a Fr\'echet locally $C^*$-algebra $\mathcal A$ possessing KDP, the second strong dual is algebraically and topologically $*$-isomorphic to $ \overline{\pi(\mathcal A)}^{WOT}$, which is a direct analogue of the classical Sherman-Takeda theorem for $C^*$-algebras. We shall also observe the joint continuity of some associated bilinear maps in the running.

math.OA↗

arXiv2025

On some Fr\'echet spaces associated to the functions satisfying Mulholland inequality

In this article we explore a new growth condition on Young functions, which we call Mulholland condition, pertaining to the mathematician H.P Mulholland, who studied these functions for the first time, albeit in a different context. We construct a non-trivial Young function $\Omega$ which satisfies Mulholland condition and $\Delta_2$-condition. We then associate exotic $F$-norms to the vector space $X_1\oplus X_2$, where $X_1$ and $X_2$ are Banach spaces, using the function $\Omega$. This $F$-spaces contains the Banach space $X_1$ and $X_2$ as a maximal Banach subspace. Further, the Banach envelope $(X_1\oplus X_2,||.||_{\Omega_o})$ of this $F$-space corresponds to the Young function $\Omega_o$ who characteristic function is an asymptotic line to the characteristic function of the Young function $\Omega$. Thus these $F$-spaces serves as "interpolation space" for Banach spaces $X_1$ and $(X_1\oplus X_2, ||.||_{\Omega_o})$ in some sense. These $F$-space are well behaved in regards to Hahn-Banach extension property, which is lacking in classical $F$-spaces like $L^p$ and $H^p$ for $0<p<1$. Towards the end, some direct sums for Orlicz spaces are discussed.

math.FA↗

arXiv2024

Annihilators in the bidual of generalized group algebra of a discrete group

In this short note, the second dual of generalized group algebra $(\ell^1(G,\mathcal A),\ast)$ equipped with both Arens products is investigated, where $G$ is any discrete group and $\mathcal A$ is a Banach algebra containing a complemented algebraic copy of $(\ell^1(\mathbb N),\bullet)$. We give an explicit family of annihilators(w.r.t both the Arens product) in the algebra $\ell^1(G,\mathcal A)^{**}$, arising from non-principal ultrafilters on $\mathbb N$ and which are not in the topological center. As a consequence, we also deduce the fact that $\ell^1(G,\mathcal A)$ is not Strongly Arens irregular.

math.FA↗

arXiv2023

Geometry of Banach algebra $\mA$ and the bidual of $L^1(G,\mA)$

This article is intended towards the study of the bidual of generalized group algebra $L^1(G,\mA)$ equipped with two Arens product, where $G$ is any locally compact group and $\mA$ is a Banach algebra. We show that the left topological center of $(L^1(G)\hat\otimes\mA)^{**}$ is a Banach $L^1(G)$-module if $G$ is abelian. Further it also holds permanance property with respect to the unitization of $\mA$. We then use this fact to extend the remarkable result of A.M Lau and V. Losert\cite{Lau-losert}, about the topological center of $L^1(G)^{**}$ being just $L^1(G)$, to the reflexive Banach algebra valued case using the theory of vector measures. We further explore pseudo-center of $L^1(G,\mA)$ for non-reflexive Banach algebras $\mA$ and give a partial characterization for elements of pseudo-center using the Cohen's factorization theorem. In the running we also observe few consequences when $\mA$ holds the Radon-Nikodym property and weak sequential completeness.

math.FA↗

arXiv2021

On strong Arens irregularity of projective tensor product of Hilbert-Schmidt space

It was shown in [16] that the Banach algebra $A:=S_2(\ell^2)\otimes^{\gamma} S_2(\ell^2)$ is not Arens regular, where $S_2(\ell^2)$ denotes the Banach algebra of the Hilbert-Schmidt operators on $\ell^2$. In this article, employing the notion of limits along ultrafilters, we prove that the irregularity of $S_2(\ell^2)\otimes^{\gamma} S_2(\ell^2)$ is not strong. Along the way, we provide a class of functionals in $A^{**}$ which lie in the topological center but are not in $A$; and, as a consequence, we deduce that $A^{**}$ is not an annihilator Banach algebra with respect to any of the two Arens products.

math.FA↗

arXiv2020

On arens regularity of projective tensor product of Schatten p-class operators

In this paper we discuss the Arens regularity of projective tensor product of Schatten p-class operators. We use the biregularity condition given by Ülger to prove that $S_p(\mathcal H)\otimes^γS_q(\mathcal H)$ is not Arens regular. We further prove that $B(S_2(\mathcal H))\otimes^γS_2(\mathcal H)$ is not Arens regular(with respect to usual multiplication) while it is regular with respect to Schur product. Thus we demonstrate the importance of biregularity condition given in \cite{Ulger} and the convenience of its use to prove Arens regularity or irregularity through some concrete examples.

math.FA↗

arXiv2019

Some functorial properties of Schatten classes

In this paper, we begin with the study of elements in $C^*$-algebras which are mapped to Schatten class ideals through faithful left regular representation. We further give some functorial properties of Schatten classes on the category of representations of a $C^*$-algebra and category of unitary representations of a group.

math.FA↗