Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1811.01652

On some modifications of n-th von Neumann-Jordan constant for banach spaces

Maciej Ciesielski·Ryszard Płuciennik
View original source ↗Read source document ↗Export metadata ↓

Abstract

There are discussed some modifications of n-th von Neumann-Jordan constant considered by M. Kato, Y. Takahashi and K. Hashimoto. We study, among others, upper, lower, upper modified and lower modified n-th von Neumann-Jordan constant and relationships between them. There are characterized uniformly non-l_{n}^{1} Banach spaces in terms of the upper modified n-th von Neumann-Jordan constant. Moreover, this constant is calculated explicitly for Lebesgue spaces L^{p} and l^{p} (1\leq p\leq oo).

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Maciej Ciesielski, Ryszard Płuciennik. 2018-11-05. On some modifications of n-th von Neumann-Jordan constant for banach spaces. https://arxiv.org/abs/1811.01652

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Inverse-Limit Detection Envelopes as Calkin Algebras

For a unital Banach algebra $A$ with $\|1_A\|=1$ and a decreasing sequence $(I_r)$ of proper closed two-sided ideals of finite codimension, we introduce the bounded detection envelope $$\widehat A:=\operatorname{Env}^{b}_{(I_r)}(A)=\varprojlim^{b}(A/I_r,π_r^s),\qquad \|(a_r)_r\|=\sup_r\|a_r\|.$$ Combining vector-valued Bourgain--Delbaen extensions, oracle Turing-machine recursion, and Argyros--Haydon operator analysis, we construct a separable Banach space $X$ such that, isometrically as unital Banach algebras, $$\operatorname{Cal}(X)=\mathcal{B}(X)/\mathcal{K}(X)\simeq\widehat A. $$ We also characterize these envelopes as the unital Banach algebras with norm-one units admitting an isometric predual and countably many jointly faithful weak-star-to-norm continuous finite-dimensional unital representations. Applications include arbitrary finite-dimensional unital Banach algebras with norm-one units and their countable $\ell_\infty$-sums, as well as unitized coordinatewise algebras over normalized $1$-unconditional boundedly complete Schauder bases. These include incidence algebras, $\ell_\infty$, and the unitization of Tsirelson's original space. Further applications concern analytic operator algebras, Fourier--Stieltjes and Fourier multiplier algebras of countable discrete groups, and classical and quantum convolution algebras. In particular, we realize $H^\infty(\mathbb D)$, $\operatorname{Mult}(H_2^2)$, $\mathcal L_2$, $M_{\mathrm{cb}}A(\mathbb F_2)$, and $C^u(O_3^+)^*$ as Calkin algebras, with their canonical norms and products.

math.FA↗
arXiv2026

Duals of separable Banach Spaces as Calkin Algebras and Universal Ideal Quotients

For $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and every separable Banach space $V$, we use an oracle-relative two-sorted finite-extension construction, whose oracle records the local rational structure of $V$, to construct a separable Banach space $X_V$ such that $$ \operatorname{Cal}(X_V)=\mathcal{B}(X_V)/\mathcal{K}(X_V)\simeq \begin{pmatrix} \mathbb{K}&0\\ V^*&\mathbb{K} \end{pmatrix} $$ as Banach algebras. The identification of $V^*$ with the Jacobson radical is isometric. Consequently, every nonzero dual Banach space with a separable predual admits, after an equivalent renorming, a unital Banach-algebra structure isomorphic to the Calkin algebra of a separable Banach space. Moreover, there is a separable Banach space $X$ such that every separable Banach space is isometric to $\mathcal{J}/\mathcal{K}(X)$ for a closed two-sided ideal $\mathcal{J}$ of $\mathcal{B}(X)$, and the subspace--ideal correspondence preserves the canonical order.

math.FA↗
arXiv2026

On $e$-doubly stochastic matrices over hyperbolic (polynomial) systems

In the setting of a hyperbolic (polynomial) system $(\mathcal{V}, p, e)$ of degree $n$, an $n$-tuple $\mathbf{A} = \big[ a_1, a_2, \dots, a_n \big]$ is said to be an $e$-doubly stochastic $\mathcal{V}$-matrix if each $a_i$ belongs to the hyperbolicity cone, has trace $1$, and $a_1 + a_2 + \dots + a_n = e$. In this article, we characterize linear preservers of such $\mathcal{V}$-matrices, describe some connections between $e$-doubly stochasticity and majorization, and study extreme points of the set of all $e$-doubly stochastic $\mathcal{V}$-matrices. We show, for example, that $(i)$ when $n>1$, positive, unital, and trace-preserving transformations are (the only) linear transformations on $\mathcal{V}$ that preserve $e$-doubly stochasticity; $(ii)$ when $\mathbf{A}$ is $e$-doubly stochastic, the eigenvalue vector of the linear combination $\sum_{i=1}^{n} r_ia_i$ is majorized by the coefficient vector $(r_1, r_2, \dots, r_n)^{T} \in \mathbb{R}^n$; and $(iii)$ when $p$ is complete, $e$-doubly stochastic $\mathcal{V}$-matrices induced by (generalized) Jordan frames are extreme points of the compact convex set of all $e$-doubly stochastic $\mathcal{V}$-matrices.

math.FA↗