Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectrum of bidual uniform algebras

Marek Kosiek, Krzysztof Rudol

TL;DR

The paper addresses the problem of describing the spectrum $\operatorname{Sp}(A^{**})$ of the bidual $A^{**}$ of a uniform algebra $A$ on a compact space. It proves that $\operatorname{Sp}(A^{**})$ is precisely a quotient of the hyper-Stonean envelope $\operatorname{Sp}(C(X)^{**})$ by the relation induced by $A^{**}$, and it constructs an isometric embedding of $A^{**}$ into $C(\mathcal{S})$ for the resulting quotient space $\mathcal{S}$. The approach combines weak-* topology analysis, annihilator/pre-annihilator techniques, and convex-analytic arguments to connect measures, norming sets, and dual structures, yielding a concrete subquotient description of the bidual spectrum. This work clarifies the relationship between a uniform algebra and its bidual and provides a tangible link between the spectrum of $A$ and the hyper-Stonean envelope, with potential implications for related problems in function algebras.

Abstract

We obtain a description of the spectrum of bidual algebra $A^{**}$ of a uniform algebra $A$. This spectrum turns out to be a quotient space of the hyper-Stonean envelope of the spectrum of $A$.

Spectrum of bidual uniform algebras

TL;DR

The paper addresses the problem of describing the spectrum of the bidual of a uniform algebra on a compact space. It proves that is precisely a quotient of the hyper-Stonean envelope by the relation induced by , and it constructs an isometric embedding of into for the resulting quotient space . The approach combines weak-* topology analysis, annihilator/pre-annihilator techniques, and convex-analytic arguments to connect measures, norming sets, and dual structures, yielding a concrete subquotient description of the bidual spectrum. This work clarifies the relationship between a uniform algebra and its bidual and provides a tangible link between the spectrum of and the hyper-Stonean envelope, with potential implications for related problems in function algebras.

Abstract

We obtain a description of the spectrum of bidual algebra of a uniform algebra . This spectrum turns out to be a quotient space of the hyper-Stonean envelope of the spectrum of .
Paper Structure (4 sections, 7 theorems, 6 equations)

This paper contains 4 sections, 7 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Approximation in seminorm
  3. Weak-star topologies
  4. The spectrum of $A^{**}$

Key Result

Lemma 2.2

If a set $W\subset B_{[1]}$ is norming for $B^*$, then its norm-closed absolutely convex hull $\overline{\text{aco}}(W)$ contains $B_{[1]}$.

Theorems & Definitions (13)

  • Definition 2.1
  • Lemma 2.2
  • Corollary 2.3
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • ...and 3 more