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April 10, 2015 By

Some algebras of bounded functions on the disc

C. R. Math. Rep. Acad. Sci. Canada Vol. 27, (3), 2005 pp. 72–75

September 30, 2005

Alexander J. Izzo, Department of Mathematics and Statistics, Bowling Green State, University, Bowling Green, OH 43403 USA; email: aizzo@math.bgsu.edu

Abstract/Résumé:

Let \(B\) be a uniformly closed algebra of functions on the unit circle \(\partial D\) between \(H^\infty\) and \(L^\infty\), and let \(C_B\) be the \(C^*\)-algebra generated by those Blaschke products that are invertible in \(B\). Let \(A\) be the algebra of bounded holomorphic functions on the open unit disc \(D\) whose boundary value functions are in \(C_B\). It is shown that if \(f\) is a bounded harmonic nonholomorphic function on \(D\) whose boundary value function is also in \(C_B\), then the uniformly closed algebra \(A[f]\) generated by \(A\) and \(f\) contains \(C(\overline D)\). This generalizes an earlier result of the author, which in turn contains as special cases a result on the disc algebra due to Čirca and a result on \(h^\infty (d)\) due to Axler and Shields.

Soit \(B\) une algèbre uniformément fermée de fonctions sur le cercle unité \(\partial D\) entre \(H^\infty\) et \(L^\infty\), et soit \(C_B\) la \(C^*\)-algèbre générée par les produits de Blaschke qui soient invertibles dans \(B\). Soit \(A\) l’algèbre des fonctions holomorphiques bornées sur le disque unité ouvert \(D\) dont les fonctions de valeur à la borne se trouvent dans \(C_B\). Il est démontré que, si \(f\) est une fonction non-holomorphique harmonique bornée sur \(D\) dont la fonction de valeur à la borne est aussi dans \(C_B\), alors l’algèbre uniformément fermeé \(A[f]\) générée par \(A\) et \(f\) renferme \(C(\overline D)\). Ceci généralise un résultat antérieur de l’auteur qui, à son tour, contient, en tant que cas particuliers, un résultat concernant l’algèbre disc dû à Čirca et un résultat sur \(h^\infty (d)\) dû à Axler et Shields.

Keywords:
AMS Subject Classification: Banach algebras of continuous functions; function algebras 46J10

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