# Hypo-Dirichlet algebra

Let $A$ be a uniform algebra on $X$ and $C ( X )$ the algebra of all continuous functions on $X$ (cf. also Algebra of functions). The algebra $A$ is called a hypo-Dirichlet algebra if the closure of $A + \overline{A}$ has finite codimension in $C ( X )$, and the linear span of $\operatorname { log } | A ^ { - 1 } |$ is dense in $\operatorname { Re } C ( X )$, where $A ^ { - 1 }$ is the family of invertible elements of $A$. Hypo-Dirichlet algebras were introduced by J. Wermer [a4].

Let $X$ be the boundary of a compact subset $Y$ in the complex plane whose complement has only finitely many components. Let $R ( X )$ be the algebra of all functions on $X$ that can be uniformly approximated by rational functions with poles off $Y$ (cf. also Padé approximation; Approximation of functions of a complex variable). Then $R ( X )$ is a hypo-Dirichlet algebra [a3].

Let $A$ be a hypo-Dirichlet algebra on $X$ and $\phi$ a non-zero complex homomorphism of $A$. If $m$ is a representing measure on $X$ such that $\operatorname { log } | \phi ( h ) | = \int \operatorname { log } | h | dm$ for $h$ in $A ^ { - 1 }$, then $m$ is unique. For $p \geq 1$, the abstract Hardy space $H ^ { p } ( m )$ is defined as the closure of $A$ in $L ^ { p } ( X , m )$ (cf. also Hardy spaces). Then a lot of theorems for the concrete Hardy space defined by $R ( X )$ are valid for abstract Hardy spaces [a2]. Using such a theory, J. Wermer [a4] showed that if the Gleason part $G ( \phi )$ of $\phi$ is non-trivial (cf. also Algebra of functions), then $G ( \phi )$ has an analytic structure.

 [a1] P. Ahern, D. Sarason, "On some hypodirichlet algebras of analytic functions" Amer. J. Math. , 89 (1967) pp. 932–941 [a2] P. Ahern, D. Sarason, "The $H ^ { p }$ spaces of a class of function algebras" Acta Math. , 117 (1967) pp. 123–163 [a3] H. Barbey, H.König, "Abstract analytic function theory and Hardy algebras" , Lecture Notes Math. : 593 , Springer (1977) [a4] J. Wermer, "Analytic disks in maximal ideal spaces" Amer. J. Math. , 86 (1964) pp. 161–170