Hardy spaces and Jensen measures

Abstract:

Suppose $A$ is a subalgebra of ${L^\infty }(m)$ on which $m$ is multiplicative. In this paper, we show that if $m$ is a Jensen measure and $A + \overline A$ is dense in ${L^2}(m)$, then $A + \overline A$ is weak-* dense in ${L^\infty }(m)$, that is, $A$ is a weak-* Dirichlet algebra. As a consequence of it, it follows that if $A + \overline A$ is dense in ${L^4}(m)$, then $A$ is a weak-* Dirichlet algebra. (It is known that even if $A + \overline A$ is dense in ${L^3}(m)$, $A$ is not a weak-* Dirichlet algebra.) As another consequence, it follows that if $B$ is a subalgebra of the classical Hardy space ${H^\infty }$ containing the constants and dense in ${H^2}$, then $B$ is weak-* dense in ${H^\infty }$.