25 - Multipliers on commutative Banach algebras

Summary

Multipliers

In this chapter I want to give an impression of what sort of information becomes available via local spectral theory when it is applied to a particular class of operators, namely the multipliers on a commutative Banach algebra. Throughout this chapter, the letter A will denote a commutative, complex Banach algebra.

Definition 25.1.1A linear map T: A → A is a multiplier if aT (b) = T (a)b for all a, b ∈ A. The set of multipliers on A is denoted by M(A).

The most obvious example, given A, is the multiplication operator L_a induced by a fixed element a ∈ A, that is, the operator L_a(b):= ab for all b ∈ A. If A has a unit e (A is unital) then every multiplier T is a multiplication operator. In this case (Exercise 25.3.2).

If the map a → L_a: A → B(A) is injective (faithful), then A is said to be a faithful algebra. Every unital algebra is faithful, as is every semisimple, and also every semiprime algebra – the latter term means, in our commutative case, that the algebra contains no non-zero nilpotent elements.

Example 25.1.2 Let A:= C₀(Ω) be the Banach algebra of all continuous complex-valued functions vanishing at ∞ on the locally compact Hausdorff space Ω, and let f ∈ C^b(Ω) be a bounded continuous function on Ω. Then T := L_f (notation self-explanatory by now) is a multiplier. Conversely, it can be shown that a multiplier gives rise to a bounded continuous function on Ω with respect to which the multiplier acts by pointwise multiplication (Exercise 25.3.4).