Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semi denting points and related notions in Banach spaces

Sudeshna Basu, Priyanka Priyadarshini Behera, Susmita Seal

TL;DR

This work addresses the stability of localized geometric notions—semi denting, semi PC, and semi SCS points (and their $w^*$-versions)—in Banach spaces under $l_p$-sums, various ideals, and projective tensor products. It develops precise stability results for $\ell_p$-sums (notably $\ell_\infty$-sums) and provides necessary and sufficient conditions for componentwise semi-denting properties to lift to sums; it also establishes lifting principles for semi- properties through $M$-ideals, ai-ideals, and strict ideals, including $w^*$-versions. In the tensor-product setting, the paper shows that semi-denting and semi SCS points behave compatibly under the projective tensor product, with explicit constructions ensuring products like $a\otimes b$ inherit the semi-properties. Together, these results sharpen the understanding of the unit ball geometry under sums, ideals, and tensorial constructions and answer open questions on stability of semi notions in BS literature.

Abstract

In this work, we study the stability properties of semi denting, semi PC, and semi SCS points, as well as their $w^*$-analogues, in Banach spaces, with respect to $l_p$-sums ( $1\leq p \leq \infty),$ ideals, and projective tensor products.

Semi denting points and related notions in Banach spaces

TL;DR

This work addresses the stability of localized geometric notions—semi denting, semi PC, and semi SCS points (and their -versions)—in Banach spaces under -sums, various ideals, and projective tensor products. It develops precise stability results for -sums (notably -sums) and provides necessary and sufficient conditions for componentwise semi-denting properties to lift to sums; it also establishes lifting principles for semi- properties through -ideals, ai-ideals, and strict ideals, including -versions. In the tensor-product setting, the paper shows that semi-denting and semi SCS points behave compatibly under the projective tensor product, with explicit constructions ensuring products like inherit the semi-properties. Together, these results sharpen the understanding of the unit ball geometry under sums, ideals, and tensorial constructions and answer open questions on stability of semi notions in BS literature.

Abstract

In this work, we study the stability properties of semi denting, semi PC, and semi SCS points, as well as their -analogues, in Banach spaces, with respect to -sums ( ideals, and projective tensor products.
Paper Structure (8 sections, 33 theorems, 67 equations)

This paper contains 8 sections, 33 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Ideals of Banach spaces
  4. Projective tensor products
  5. Some results and examples
  6. Stability results with respect to $\ell_p$-sums of Banach spaces
  7. ideals of Banach spaces
  8. projective tensor products

Key Result

Theorem 1.3

DUGGMS A Banach space $X$ has RNP (resp. CPCP, SR) if and only if every closed bounded convex subset of $X$ is the closed convex hull of its denting points (resp. weak closure of its Point of Continuity, norm closure of its SCS points).

Theorems & Definitions (68)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 58 more