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 $C[a,b].$ 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.