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A Characterization of Zero Divisors and Topological Divisors of Zero in $C[a, b]$ and $\ell^\infty$

Harish Chandra, Anurag Kumar Patel

Abstract

We give a characterization of zero divisors of the ring $C[a,b].$ Using the Weierstrass approximation theorem, we completely characterize topological divisors of zero of the Banach algebra $C[a,b].$ We also characterize the zero divisors and topological divisors of zero in $\ell^\infty.$ Further, we show that zero is the only zero divisor in the disk algebra $\mathscr{A}(\mathbb{D})$ and that the class of singular elements in $\mathscr{A}(\mathbb{D})$ properly contains the class of topological divisors of zero. Lastly, we construct a class of topological divisors of zero of $\mathscr{A}(\mathbb{D})$ which are not zero divisors.

A Characterization of Zero Divisors and Topological Divisors of Zero in $C[a, b]$ and $\ell^\infty$

Abstract

We give a characterization of zero divisors of the ring Using the Weierstrass approximation theorem, we completely characterize topological divisors of zero of the Banach algebra We also characterize the zero divisors and topological divisors of zero in Further, we show that zero is the only zero divisor in the disk algebra and that the class of singular elements in properly contains the class of topological divisors of zero. Lastly, we construct a class of topological divisors of zero of which are not zero divisors.
Paper Structure (4 sections, 16 theorems, 23 equations, 2 figures)

This paper contains 4 sections, 16 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A characterization of Zero divisors and Topological divisors of zero in the Banach algebra $C[a,b]$
  3. A characterization of Zero divisors and Topological divisors of zero in the Banach algebra $\ell^\infty$
  4. Zero divisors and Topological divisors of zero in the disk algebra $\mathscr{A}(\mathbb{D})$

Key Result

Lemma 1

Let $f \in C[0,1]$. Then the zero set of $f$ is a closed set.

Figures (2)

  • Figure 1: Graph of the function $g$
  • Figure 2: Bounds for $|f(z)|$

Theorems & Definitions (37)

  • Definition 1: Zero Set
  • Lemma 1
  • Definition 2
  • Definition 3
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1.1
  • Definition 4
  • Definition 5
  • ...and 27 more