Abstract Hardy Algebra Theory

Abstract

The present article is devoted to a portion of functional analysis which is an abstract version of a portion of classical analytic function theory as circumscribed by boundary value theory and Hardy spaces H P . The fascination of the field comes from the fact that famous classical theorems of typical complex-analytic flavor appear as instant outflows of an abstract theory the tools of which are standard real–analytic methods such as elementary functional analysis and measure theory. The abstract theory started in the fifties (Arens-Singer, Gleason, Helson-Lowdenslager, Bishop, Wermer,…) and went through several steps of abstraction (Dirichlet algebras, logmodular algebras,…). We present the ultimate step which has been under work for about ten years. The central concept is the abstract Hardy algebra situation. It can be looked upon as a local section of the abstract function algebra situation. To achieve the localization is the main business of the abstract F. and M. Riesz theorem and of the resultant Gleason part decomposition procedure. The abstract Hardy algebra situation as a comprehensive as well as pure and simple concept calls for methodic plainness and adequacy. It permits to build up a coherent theory of remarkable width and depth which even in its ultimate state of abstraction radiates back and illuminates the concrete classical theory. We start to concentrate on the abstract Hardy algebra situation and then turn to the localization procedure and to certain standard applications. The presentation follows the new Lecture Notes BARBEY-KONIG (1977) to which we also refer for detailed references. Compared with the actual Paderborn lecture in November 1976 some new results on the Marcel Riesz estimation for conjugate functions have been added. The author cannot conclude the Introduction without expression of his warmest thanks to Klaus Bierstedt and Benno Fuchssteiner who did the work and created the atmosphere of an unusually intense and pleasant mathematical conference.