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Anil Kumar Karn

Publications and source records attributed to Anil Kumar Karn.

At least 19 recordsLinked to original sources
arXiv2026

Bijections on the set of extreme points in a compact convex set

In a recent work, Roelands and Tiersma proved that, for a compact convex set $K$, the space $A(K)$ of all real-valued continuous affine functions on $K$, is a JB-algebra if and only if there is a gauge-reversing bijection on $A_c(K)$, the set of positive real-valued continuous affine functions on $K$. In this paper, we show that every such gauge-reversing bijection on $A_c(K)$ is completely determined by the induced bijection on the set of extreme points of $K$.

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arXiv2025

Position of $L(X, Y)$ in $Lip_0(X, Y)$

We prove that $L(X,Y)$ is complemented in $Lip_0(X, Y)$ (via a norm-one projection) provided that $Y$ is a dual space. Next, we introduce a vector-valued Lipschitz-free space $F_Y(X)$, a linear contraction $\beta_X^Y: F_Y(X) \to Y$ and prove that the quotient space ${Lip_{0}(X, Y)}/{L(X, Y) }$ is isometrically isomorphic to $L(\ker(\beta_X^Y), Y)$ whenever $Y$ is injective. We also consider a $Y$-valued duality pairing between $Lip_0(X, Y)$ and $F_Y(X)$ and obtain a necessary and sufficient condition for a Lipschitz map to be linear. As an application, we describe $\ker(\beta_{\mathbb{R}}^{\mathbb{R}})$ as the pre-dual of the quotient space $L^{\infty}(\mathbb{R})/\tilde{\mathbb{R}}$ where $\tilde{\mathbb{R}}$ is the set of all constant maps on $\mathbb{R}$.

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arXiv2024

Coexistence of Hilbert space effects and orthogonality

In this paper, we show that every pair of absolutely compatible Hilbert space effects are coexistent and exhibit a partial orthogonality property. We introduce the notion of partially ortho-coexistence. We generalize absolute compatibility to obtain more examples of partially ortho-coexistent pairs and introduce the notion of generalized compatibility. In the case of $\mathbb{M}_2$, we discuss a geometric behaviour of the generalized compatibility.

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arXiv2024

A generalization of Lipschitz mappings

Using the notion of modulus of continuity at a point of a mapping between metric spaces, we introduce the notion of extensively bounded mappings generalizing that of Lipschitz mappings. We also introduce a metric on it which becomes a norm if the codomain is a normed linear space. We study its basic properties. We also discuss a linearization of an extensively bounded mapping into a bounded linear mapping. As an application, we introduce the notion of extensively bounded operator ideals. We also discuss extensively bounded finite rank and extensively bounded compact mappings and their corresponding operator ideals.

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arXiv2024

Projections in an order unit space and orthogonality

We introduce the notion of order projections using the order unit property of a positive element in an order unit space and characterize them in terms of (geometric) orthogonality. We describe order projections of the order unit space obtained by adjoining an order unit to a normed linear space.

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arXiv2023

On the geometry of an order unit space

We introduce the notion of $\mathit{skeleton}$ with a head in a non-zero real vector space. We prove that skeletons with heads describe order unit spaces geometrically. Next, we consider the notion of $\mathit{periphery}$ corresponding to an order unit space which is a part of the skeleton. We note that periphery consists of boundary elements of the positive cone with unit norms. We discuss some elementary properties of the periphery. We also find a condition under which $V$ would contain a copy of $\ell_{\infty}^n$ for some $n \in \mathbb{N}$ as an order unit subspace.

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arXiv2023

Normed linear spaces which are isometric to order unit spaces

In this paper, we consider the linear direct sum of a real normed linear space with an order unit space and with a base normed space to obtain respectively a new order unit space and a new base normed space. As a consequence, we find that an $\ell_1$-space may be shown to be an order unit space. (However, not in the natural order.) Dually, an $\ell_{\infty}$-space may be shown to be a base normed space. To understand this aberration, we characterize the real normed linear spaces which are isometrically isomorphic to order unit spaces. We prove that other than the classical case of $\ell_{\infty}$, $\ell_1$ is also isometrically isomorphic to an order unit space whereas $\ell_2$ is not isometrically isomorphic to any order unit space.

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arXiv2022

Centre of a compact convex set

We introduce the notion centre of a convex set and study the space of continuous affine functions on a compact convex set with a centre. We show that these spaces are precisely the dual of a base normed space in which the underlying base has a (unique) centre. We also characterize the corresponding base norm space. We obtain a condition on a compact, balanced, convex subset of a locally convex space so that the corresponding space of continuous affine functions on the convex set is an absolute order unit space. Similarly, we characterize a condition on the base with a centre of a base normed space so that the latter becomes an absolutely base normed space.

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arXiv2022

A generalization of spin factors

Using a technique of adjoining an order unit to a normed linear space, we have characterized strictly convex spaces among normed linear spaces and Hilbert spaces among strictly convex Banach spaces respectively. This leads to a generalization of spin factors and provides a new class of absolute order unit spaces.

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arXiv2021

Absolute compatibility and poincaré sphere

In this paper, we introduce the notion of strict projections and prove that an absolutely compatible pair of strict elements in a von Neumann algebra $\mathcal{M}$ unitarily equivalent to the elements $ \left((p - x_0) \otimes I_2 \right) P_0 + (x_0 \otimes I_2) P$, $\left((p - x_0) \otimes I_2 \right) P_0 + (x_0 \otimes I_2) P'$ of $M_2(\mathcal{M}_0)$ where $\mathcal{M}_0$ is an abelian von Neumann algebra, $x_0$ is a strict element of $\mathcal{M}_0^+$, $P_0 = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \in M_2(\mathcal{M}_0)$ and $P$ is a strict projection in $M_2(\mathcal{M}_0)$. We also discuss the geometric form of this representation when $\mathcal{M} = \mathbb{M}_2$.

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arXiv2021

$K_0$-group of absolute Matrix order unit spaces

In this paper, we describe the Grothendieck group $K_0(V)$ of an absolute matrix order unit space $V$. For this purpose, we discuss the direct limit of absolute matrix order unit spaces. We show that $K_0$ is a functor from category of absolute matrix order unit spaces with morphisms as unital completely $\vert \cdot \vert$-preserving maps to category of abelian groups. We study order structure on $K_0(V)$ and prove that under certain condition $K_0(V)$ is an ordered abelian group. We also show that the functor $K_0$ is additive on orthogonal unital completely $\vert \cdot \vert$-preserving maps.

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arXiv2019

Partial isometries in an absolute order unit space

In this paper, we extend the notion of orthogonality to the general elements of an absolute matrix order unit space and relate it to the orthogonality among positive elements. We introduce the notion of a partial isometry in an absolute matrix order unit space. As an application, we describe the comparison of order projections. We also discuss finiteness of order projections.

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arXiv2019

Absolutely compatible pair of elements in a von Neumann algebra-II

Let $A$ be a unital C$^*$-algebra with unity $1_A$. A pair of elements $0 \le a, b \le 1_A$ in $A$ is said to be \emph{absolutely compatible} if, $\vert a - b \vert + \vert 1_A - a - b \vert = 1_A.$ In this paper we provide a complete description of absolutely compatible pair of strict elements in a von Neumann algebra. The end form of such a pair has a striking resemblance with that of a `generic pair' of projections on a complex Hilbert space introduced by Halmos.

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arXiv2019

Isometries of absolute order unit spaces

We prove that for a bijective, unital, linear map between absolute order unit spaces is an isometry if, and only if, it is absolute value preserving. We deduce that, on (unital) $JB$-algebras, such maps are precisely Jordan isomorphisms. Next, we introduce the notions of absolutely matrix ordered spaces and absolute matrix order unit spaces and prove that for a bijective, unital, linear map between absolute matrix order unit spaces is a complete isometry if, and only if, it is completely absolute value preserving. We obtain that on (unital) C$^*$-algebras such maps are precisely C$^*$-algebra isomorphism.

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arXiv2018

Quantization of $A_{0}(K)$-Spaces

In this paper, we study $L^1$-matrix convex sets $\{K_{n}\}$ in $*$-locally convex spaces and show that every C$^*$-ordered operator space is complete isometrically, completely isomorphic to $\{A_{0}(K_{n}, M_{n}(V))\}$ for a suitable $L^1$-matrix convex set $\{K_{n}\}$. Further, we generalize the notion of regular embedding of a compact convex set to $L^{1}$-regular embedding of $L^{1}$-matrix convex set. Using $L^{1}$-regular embedding of $L^{1}$-convex set, we find conditions under which $A_{0}(K_{n}, M_{n}(V))$ is an abstract operator system.

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arXiv2018

$M$-ideals and split faces of the quasi state space of a non-unital ordered Banach space

We characterize $M$-ideals in order smooth $\infty$-normed spaces by extending the notion of split faces of the state space to those of the quasi-state space. We also characterize approximate order unit spaces as those order smooth $\infty$-normed spaces $V$ that are $M$-ideals in $\tilde{V}.$ Here $\tilde{V}$ is the order unit space obtained by adjoining an order unit to $V.$ To prove these results, we develop an order theoretic version of the "Alfsen-Efffros' cone decomposition theorem" for order smooth $1$-normed spaces. (As a quick application of this result, we sharpen a result on the extension of bounded positive linear functionals on subspaces of order smooth $\infty$-normed spaces.)

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arXiv2017

Algebraic orthogonality and commuting projections in operator algebras

We describe absolutely ordered $p$-normed spaces, for $1 \le p \le \infty$ which presents a model for "non-commutative" vector lattices and includes order theoretic orthogonality. To demonstrate its relevance, we introduce the notion of {\it absolute compatibility} among positive elements in absolute order unit spaces and relate it to symmetrized product in the case of a C$^{\ast}$-algebra. In the latter case, whenever one of the elements is a projection, the elements are absolutely compatible if and only if they commute. We develop an order theoretic prototype of the results. For this purpose, we introduce the notion of {\it order projections} and extend the results related to projections in a unital C$^{\ast}$-algebra to order projections in an absolute order unit space. As an application, we describe spectral decomposition theory for elements of an absolute order unit space.

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