On the positive linear functionals on the disc algebra (Q916057)

Let A be the disc algebra on the closed unit disc \(D=\{z\in {\mathbb{C}}:| z| \leq 1\}\), that is, the Banach algebra of all continuous functions on D which are analytic on the interior of D with the supremum norm. A linear functional F on A is said to be positive if \(F(f^*f)\geq 0\) for any \(f\in A\), where \(f^*(z)=\overline{f(\bar z)}\) for \(z\in D.\) The author gives an elementary proof of the following well-known result: Every positive linear functional F on A has the form \(F(f)=\int^{1}_{-1}f(t)d\mu (t)\) (f\(\in A)\), for some finite positive Borel measure \(\mu\) on [-1,1].