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Let A be a commutative semisimple Banach algebra, ∆(A) its Gelfand spectrum, T a multiplier on A and T̂ its Gelfand transform. We study the following problems. (a) When is δ(T ) = inf{|T̂ (f)| : f ∈ ∆(A), T̂ (f) 6= 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 ∆(A)? Introduction. Let A be a commutative semisimple Banach algebra, M(A) be its multiplier algebra [La1] and T be a multiplier on A. We denote by ∆(A) the Gelfand spectrum of A and by T̂ the Gelfand transform of T ; the elements of ∆(A) are considered as multiplicative functionals on A. Let ∆(T̂ ) = {f ∈ ∆(A) : T̂ (f) 6= 0} and δ(T ) = inf{|T̂ (f)| : f ∈ ∆(T̂ )}. In this paper our aim is to study the following questions. (a) When is δ(T ) > 0? (b) When is the ideal T (A) closed in A? (c) When does the ideal T (A) have a BAI (= bounded approximate identity)? (d) How to characterize the idempotent elements of M(A) in terms of subsets of ∆(A)? The main motivation behind these questions is a well known problem of harmonic analysis first studied by I. Glicksberg [G] and later completely solved in [H-P] by B. Host and F. Parreau. To recall this problem, let G be a locally compact abelian group equipped with its Haar measure, L1(G) be its group algebra and M(G) be the algebra of regular complex Borel measures on G. The algebra M(G) is the multiplier algebra of L1(G). For a measure μ of the form μ = λ ∗ θ, where λ is an invertible element and θ an idempotent element of M(G), the ideal L1(G) ∗ μ is obviously closed in L1(G). The problem mentioned above, which was raised by E. Hewitt, was this: For a measure μ ∈ M(A), is the ideal L1(G) ∗ μ closed in L1(G) only when the measure μ is of the above form? That this is the case is the 2000 Mathematics Subject Classification: Primary 47B48, 46J10, 43A22.

Multipliers with closed range on commutative semisimple Banach algebras