Communications of the
Korean Mathematical Society CKMS

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.

Keywords: Zero divisor, topological divisor of zero

MSC numbers: Primary 13A70, 46H05

Supported by: The second author is supported by the Council of Scientific and Industrial Research (CSIR) NET-JRF, New Delhi, India, through grant 09/013(0891)/2019-EMR-I.