The states of a Banach algebra generate the dual

Abstract

In this paper we prove that the states of a unital Banach algebra generate the dual Banach space as a linear space (Theorem 2). This is a result of R. T. Moore (4, Theorem 1(a)) who uses a decomposition of measures in his proof. In the proof given here the measure theory is replaced by a Hahn-Banach separation argument. We shall let A denote a unital Banach algebra over the complex field, and D(1) denote {f ∈ A′: ‖f‖ = f(1) = 1} where A′ is the dual of A. The motivation of Moore's results is the theorem that in a C*-algebra every continuous linear functional is a linear combination of four states (the states are the elements of D(1)) (see (2, 2.6.4, 2.1.9, 1.1.10)).