Sinclair homomorphisms and semigroups of analytic functions (Q1355079)

The author studies homomorphisms from the convolution algebra \(L^1(\mathbb{R}^+)\) into certain Banach algebras of functions on the closed unit disc \(\mathbb{D}\). It is shown that for the algebra \(A^+\) of absolutely convergent Taylor series on \(\mathbb{D}\), every homomorphism \(\Phi\) is of the form \(\Phi(h)= \int^\infty_0 h(t)\nu^t dt\), \(\forall h\in L^1(\mathbb{R}^+)\), where \(\nu^t\) is a bounded, continuous semigroup in \(A^+\). A similar result holds for the algebra of functions analytic on the open disc and absolutely continuous on the unit circle. For the disc algebra and the Pisier algebra this is not true, but certain weaker representations are obtained.