The numerical range in the second dual of a Banach algebra

Abstract

An elementary consequence of the Hahn-Banach theorem is that every Banach space Y is ω*-dense in its second dual Y**, so that every element y ∈ Y** is the w*-limit of a net {ya}α ∈ Λ from Y. There is, of course, a great deal of choice in the selection of such a net, and so one may impose extra conditions on the net related to some special property of the limit point, and then ask for existence. The object of this paper is to present such a result in the context of a unital Banach algebra and its second dual , and then to give two applications to Banach algebra theory. The theorem to be proved is this: if the numerical range W(a) of an element in has non-empty interior then a is the ω*-limit of a net {aa}α ∈ Λ from whose numerical ranges are contained in W(a), while if W(a) has empty interior then the numerical ranges W(aα) are contained in a shrinking set of neighbourhoods of W(a).