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arXiv · 2402.09909

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

Harish Chandra·Anurag Kumar Patel
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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.

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Harish Chandra, Anurag Kumar Patel. 2024-02-15. A Characterization of Zero Divisors and Topological Divisors of Zero in $C[a, b]$ and $\ell^\infty$. https://arxiv.org/abs/2402.09909

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