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Few Bilinear Operators on Spaces of Continuous Functions

Leandro Candido

TL;DR

This work extends the few-operators phenomenon to bilinear mappings on $C_0(L)$, where $L$ is a locally compact, non-metrizable collapsing space constructed under Ostaszewski's $\clubsuit$-principle. Using a Riesz-type representation via $F$-2 measures on the compactification $K=L\cup\{\infty\}$ and a weak$^*$-continuous representing function $\nu:K\to\mathcal F_0(K,K)$, the authors prove that every bilinear operator $G:C_0(L)\times C_0(L)\to C_0(L)$ admits a unique decomposition into a scaled multiplication term $M$, a cross-interaction term $H$ determined by $\ell_1$-data on $L$, and a remainder $S$ with separable range. The decomposition $G(f,g)(t)= r\, f(t)g(t) + \sum_{w\in L}(a_w f(t)g(w) + b_w f(w)g(t)) + S(f,g)(t)$ is established with uniqueness, reflecting a bilinear analogue of the few-operators phenomenon under the set-theoretic hypothesis. The approach hinges on Fréchet measures, tail-vanishing properties, and the structural features of the collapsing space, yielding a precise, canonical form for bilinear operators on $C_0(L)$ in this exotic topological setting.

Abstract

Motivated by recent work exhibiting a locally compact scattered space $L$ constructed under Ostaszewski's $\clubsuit$-principle, which yielded a complete classification of linear operators on $C_0(L\times L)$, we extend the analysis to the bilinear setting. We show that, for this space $L$, every bilinear operator $G:C_0(L)\times C_0(L)\to C_0(L)$ admits a unique decomposition into the sum of trivially predictable components. This establishes a bilinear analogue of the few operators phenomenon.

Few Bilinear Operators on Spaces of Continuous Functions

TL;DR

This work extends the few-operators phenomenon to bilinear mappings on , where is a locally compact, non-metrizable collapsing space constructed under Ostaszewski's -principle. Using a Riesz-type representation via -2 measures on the compactification and a weak-continuous representing function , the authors prove that every bilinear operator admits a unique decomposition into a scaled multiplication term , a cross-interaction term determined by -data on , and a remainder with separable range. The decomposition is established with uniqueness, reflecting a bilinear analogue of the few-operators phenomenon under the set-theoretic hypothesis. The approach hinges on Fréchet measures, tail-vanishing properties, and the structural features of the collapsing space, yielding a precise, canonical form for bilinear operators on in this exotic topological setting.

Abstract

Motivated by recent work exhibiting a locally compact scattered space constructed under Ostaszewski's -principle, which yielded a complete classification of linear operators on , we extend the analysis to the bilinear setting. We show that, for this space , every bilinear operator admits a unique decomposition into the sum of trivially predictable components. This establishes a bilinear analogue of the few operators phenomenon.

Paper Structure

This paper contains 5 sections, 19 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Fréchet Measures and the Structure of Bilinear Forms on $C(K)$
  3. Dismantling of Bilinear Operators
  4. Proof of Main Theorem
  5. Acknowledgements

Key Result

Theorem 1.1

Assuming Ostaszewski's principle $\clubsuit$, there exists a non-metrizable locally compact scattered Hausdorff space $L$ such that, for every bilinear operator $G:C_0(L)\times C_0(L)\to C_0(L)$, there exist a unique $r\in \mathbb{R}$, unique $(a_w)_{w\in L}, (b_w)_{w\in L} \in \ell_1(L)$, and a uni for all $f,g \in C_0(L)$ and $t\in L$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 3.1
  • Definition 3.2
  • ...and 35 more