Multipliers with closed range on regular commutative Banach algebras

Abstract

Conditions equivalent with closure of the range of a multiplier T, defined on a commutative semisimple Banach algebra A, are studied. A main result is that if A is regular then T 2 A {T^2}A is closed if and only if T is a product of an idempotent and an invertible. This has as a consequence that if A is also Tauberian then a multiplier with closed range is injective if and only if it is surjective. Several aspects of Fredholm theory and Kato theory are covered. Applications to group algebras are included.