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T. S. S. R. K. Rao

Publications and source records attributed to T. S. S. R. K. Rao.

12 recordsLinked to original sources
arXiv2026

Vector valued continuous function spaces as $C^\ast$-algebras

Let \(Ω\) be a compact Hausdorff space, and let \(\cA\) be a unital \(C^*\)-algebra. In this study, we continue our examination of the comparison between \(C^*\)-extreme points and linear extremal structures of the unit ball, in the vector-valued \(C^*\)-algebra \(C(Ω, \cA)\), building upon the work initiated in \cite{HR}. We first enlarge the class of $C^\ast$-algebras in which a $ C^\ast$-extreme point is an extreme point. We demonstrate that if \(\cA\) has a faithful tracial state, then any \(C^*\)-extreme point of the unit ball \(C(Ω, \cA)_1\) is a unitary. Additionally, we identify a classes of \(C^*\)-algebras where the concepts of \(C^*\)-extreme and pointwise \(C^*\)-extreme points in \(C(Ω, \cA)_1\) coincide. We show this holds if a von Neumann algebra has a separable predual with the Radon-Nikodým property.

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arXiv2026

Order-preserving unique Hahn-Banach extensions

Let $X$ be a real Banach lattice with a unit, and let $Y \subseteq X$ be a closed subspace containing the unit. In this paper, we study the order-theoretic (also isometric) structure of $Y$ that it may inherit from $X$ under some additional conditions. Specifically, we consider the case where every continuous positive linear functional in the unit sphere of $Y^\ast$ admits a unique positive norm-preserving extension in $X^\ast$. Our results depend on the specific nature of the embedding of $Y$ in $X$. For a compact convex set $K$ with closed extreme boundary $\partial_e K$, for the restriction isometry of $A(K)$ (which is also order-preserving) into $C(\partial_e K)$, uniqueness of extensions of positive functionals leads to $K$ being a simplex and the restriction embedding being onto. In contrast, for a Choquet simplex $K$, we show that under the canonical embedding in the bidual $A(K)^{\ast\ast}$ (which is an abstract $M$-space), positive linear functionals having unique positive linear extensions imply that $K$ is a finite-dimensional simplex.

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arXiv2026

Weakly Compact Operators Whose Adjoints Preserve $C^*$-Structure

Let $Ω$ be a compact Hausdorff space and $X$ be a complex Banach space such that $X^{**}$ is isometric to a $C^*$-algebra. Let $\mathcal{W}(X^*, C(Ω))$ denote the space of weakly compact operators. In this article, we show that the collection of operators $T$ in $\mathcal{W}(X^*, C(Ω))$ such that $T^*$ maps linear extreme points of the unit ball of $C(Ω)^*$ to $C^*$-extreme points of $X^{**}$ is a uniformly strongly extreme set. We also investigate this phenomenon when $X^{**}$ is isometric to a space of all operators on a Banach space.

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arXiv2026

Some Geometric Aspects Related to Lim's Condition

In their seminal work, Lau and Mah (1986) study $w^*$-normal structure in the space of operators $\mathcal{L}(H)$, on a Hilbert space $H$, using a geometric property of the dual unit ball called Lim's condition. In this paper, we study a weaker form of Lim's condition, which we call property ($\ddagger$), for $C^\ast$-algebras, uniform algebras, and $L^1$-predual spaces. In the case of a $C^\ast$-algebra, we prove that property $(\ddagger)$ is equivalent to Lim's condition and consequently, we obtain a geometric characterization of $C^*$-algebras which are $c_0$-direct sum of finite-dimensional operator spaces. For a uniform algebra, we extend a result of Lau and Mah to show that property $(\ddagger)$ implies that the space is finite-dimensional. In the case of an $L^1$-predual space, we show that this condition implies $k$-smoothness of the norm in the sense considered in Lin and Rao (2007).

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arXiv2025

Geometric Aspects of $C^*$-Extreme Points

We provide a characterization of the $C^*$-extreme points of the closed unit ball of a von Neumann algebra and demonstrate that $C^*$-extremality is equivalent to both linear extremality and strong extremality. As an application, we characterize certain classes of von Neumann algebras in terms of their $C^*$-extreme points.

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arXiv2025

Geometry of the unit ball of ${\mathcal L}(X,Y^*)$

In this work we study the geometry of the unit ball of the space of operators ${\mathcal L}(X,Y^*)$, by considering the projective tensor product $X\hat{\otimes}_π Y$ as a predual. We prove that if an elementary tensor (rank one operator) of the form $x_0^*\otimes y_0^* $ in the unit sphere $ S_{{\mathcal L}(X,Y^*)}$ is a weak$^*$-strongly extreme point of the unit ball, then $x_0^*$ is weak$^*$-strongly extreme point of unit ball of $X^*$ and $y_0^*$ is weak$^*$-strongly extreme point of the unit ball of $Y^*$. We show that a similar conclusion holds if the rank one operator is a Namioka point (equivalently, point of weak$^*$-weak continuity for the identity mapping) on the unit sphere of ${\mathcal L}(X,Y^*)$. We also study extremal phenomenon in the unit ball of ${\mathcal L}(X,Y^*)^*$. We partly solve the open problem, when does an elementary tensor, whose components are Namioka points is again a Namioka point? We show that if a point $z\in S_{{\mathcal L}(X,Y^*)^*}$ is a weak$^*$-strongly extreme point of the unit ball, then $z=x\otimes y$ for some weak$^*$-strongly extreme points $x\in S_X$ and $y\in S_Y$, provided the space of compact operators, $\mathcal{K}(X,Y^*)$ is separating for $X\hat{\otimes}_π Y$.

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arXiv2025

A study of weak$^*$-weak points of continuity in the unit ball of dual spaces

We study classes of Banach spaces where the points of weak$^*$-weak continuity for the identity mapping on the dual unit ball form a weak$^*$-dense and weak$^*$-$G_δ$ set. We also discuss how this property behaves in higher duals of Banach spaces. We prove in particular that if $\mathcal{A}$ is a von Neumann algebra and its predual has the Radon--Nikodým property, then there is no point of weak$^*$-weak continuity on the unit ball of $\mathcal{A}$.

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arXiv2020

Structure of sets of strong subdifferentiability in dual $L^1$-spaces

In this article, we analyse the structure of finite dimensional subspaces of the set of points of strong subdifferentiability in a dual space. In a dual $L_1(μ)$ space, such a subspace is in the discrete part of the Yoshida-Hewitt type decomposition. In this set up, any Banach space consisting of points of strong subdifferentiability is necessarily finite dimensional. Our results also lead to streamlined and new proofs of results from the study of strong proximinality for subspaces of finite co-dimension in a Banach space.

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arXiv2020

Uniqueness of Hahn-Banach extension and related norm-$1$ projections in dual spaces

In this paper we study two properties viz. property-$U$ and property-$SU$ of a subspace $Y$ of a Banach space which correspond to the uniqueness of the Hahn-Banach extension of each linear functional in $Y^*$ and in addition to that this association forms a linear operator of norm-1 from $Y^*$ to $X^*$. It is proved that, under certain geometric assumptions on $X, Y, Z$ these properties are stable with respect to the injective tensor product; $Y$ has property-$U$ ($SU$) in $Z$ if and only if $X\otimes_\e^\vee Y$ has property-$U$ ($SU$) in $X\otimes_\e^\vee Z$. We prove that when $X^*$ has the Radon-Nikod$\acute{y}$m Property for $1 3$ is a Hilbert space if and only if for any two subspaces $Y, Z$ with property-$SU$ in $X$, $Y+Z$ has property-$SU$ in $X$ whenever $Y+Z$ is closed. We characterize all hyperplanes in $c_0$ which have property-$SU$.

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arXiv2018

Approximate Birkhoff-James orthogonality and smoothness in the space of bounded linear operators

We study approximate Birkhoff-James orthogonality of bounded linear operators defined between normed linear spaces $\mathbb{X}$ and $\mathbb{Y}.$ As an application of the results obtained, we characterize smoothness of a bounded linear operator $T$ under the condition that $\mathbb{K}(\mathbb{X},\mathbb{Y}),$ the space of compact linear operators is an $M-$ideal in $\mathbb{L}(\mathbb{X},\mathbb{Y}),$ the space of bounded linear operators.

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arXiv2003

Very smooth points of spaces of operators

In this paper we study very smooth points of Banach spaces with special emphasis on spaces of operators. We show that when the space of compact operators is an $M$-ideal in the space of bounded operators, a very smooth operator $T$ attains its norm at a unique vector $x$ (up to a constant multiple) and $T(x)$ is a very smooth point of the range space. We show that if for every equivalent norm on a Banach space, the dual unit ball has a very smooth point then the space has the Radon--Nikodým property. We give an example of a smooth Banach space without any very smooth points.

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