Multipliers with closed range on commutative semisimple Banach algebras

Abstract

Let A be a commutative semisimple Banach algebra, Delta(A) its Gelfand spectrum, T a multiplier on A and (T) over cap its Gelfand transform. We study the following problems. (a) When is delta(T) = inf {\(T) over cap (f)\ : f is an element of Delta(A), (T) over capT(f) not equal 0} < 0? (b) When is the range T (A) of T closed in A and does it have a bounded approximate identity? (c) How to characterize the idempotent multipliers in terms of subsets of Delta(A)?