A CHARACTERIZATION OF ZERO DIVISORS AND TOPOLOGICAL DIVISORS OF ZERO IN C[a, b] AND ℓ∞

Abstract

We give a characterization of zero divisors of the ring C[a, b]. Using theWeierstrass 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 ℓ∞. Further, we show that zero is the only zero divisor in the disk algebra A (D) and that the class of singular elements in A (D) properly contains the class of topological divisors of zero. Lastly, we construct a class of topological divisors of zero of A (D) which are not zero divisors © 2023 Korean Mathematical Society