Sinclair Homomorphisms and Semigroups of Analytic Functions

In this paper we study homomorphisms from the convolution algebraL1(R+) into certain Banach algebras of functions on the closed unit disc Δ. For the algebra A+of absolutely convergent Taylor series on Δ, we prove that every homomorphismΦis a “Sinclair map,” that is, of the formΦ(h)=∫∞0 h(t) νt dt, h∈L1(R+)for some bounded, continuous semigroup (νt) in A+. A similar result holds for the algebra AC+of functions analytic inΔand absolutely continuous on the unit circle T. The result does not, however, hold in the disc algebra A(Δ), although we are able to represent homomorphisms into A(Δ) by means of semigroups in a certain weaker sense. Finally, we discuss the “Pisier algebra” P+defined in terms of random Taylor series on Δ. In particular, we prove that a homomorphism fromL1(R+) into P+need not be a Sinclair map, but that it can be represented by means of a semigroup which belongs to a certain larger Banach algebra of random Taylor series.