Abstract Analytic Function Theory and Hardy Algebras
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593 IIIIII IIIIIIIIIIIII
Klaus Barbey Heinz KSnig
Abstract Analytic Function Theory and Hardy Algebras III II
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Springer-Verlag Berlin-Heidelberq • New York 1977
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II III I
Authors Klaus Barbey Fachbereich Mathematik UniversitAt Regensburg U niversit~tsstraBe 31 8400 Regensburg/BRD Heinz K6nig Fachbereich Mathematik Universit~t des Saarlandes 6 6 0 0 SaarbdJcken/BRD
AMS Subject Classifications (1970): 46J 10, 46J 15 ISBN 3-540-08252-2 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-38?-08252-2 Springer-Verlag New York • Heidelberg • Berlin This work is subiect to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Preface
The p r e s e n t work w a n t s tional-analytic
theory.
classical analytic
It is an a b s t r a c t v e r s i o n of those parts of
f u n c t i o n t h e o r y w h i c h can be c i r c u m s c r i b e d by b o u n d a r y
v a l u e t h e o r y and Hardy the
to be the s y s t e m a t i c p r e s e n t a t i o n of a func-
spaces H p. The f a s c i n a t i o n of the field comes from
fact that famous c l a s s i c a l
theorems
of t y p i c a l
vor a p p e a r as i n s t a n t o u t f l o w s of an a b s t r a c t are s t a n d a r d r e a l - a n a l y t i c m e t h o d s and m e a s u r e
theory.
such as e l e m e n t a r y
in papers of Arens
and Singer,
and went t h r o u g h
c h l e t algebras, and i l l u m i n a t e
Gleason,
functional analysis
H e l s o n and L o w d e n s l a g e r ,
several steps of a b s t r a c t i o n
logmodular algebras,...). the c o n c r e t e c l a s s i c a l
concept
theory.
We p r e s e n t
the u l t i m a t e
is the a b s t r a c t Hardy a l g e b r a situation.
IV-IX.
It is
to b u i l d up a cohe-
rent t h e o r y of r e m a r k a b l e and p l e a s a n t w i d t h and depth. in C h a p t e r s
(Diri-
for about ten years.
c o m p r e h e n s i v e as w e l l as pure a n d s i m p l e and p e r m i t s present a systematic account
Bochner,
It never c e a s e d to r a d i a t e back
step of a b s t r a c t i o n w h i c h has b e e n u n d e r w o r k The c e n t r a l
fla-
The a b s t r a c t t h e o r y s t a r t e d a b o u t t w e n t y y e a r s ago
Bishop,
Wermer,...
complex-analytic
theory the tools of w h i c h
We a t t e m p t
to
The a b s t r a c t Hardy alge-
bra s i t u a t i o n can be looked upon as a local s e c t i o n of the a b s t r a c t function a l g e b r a situation.
To a c h i e v e the l o c a l i z a t i o n is the m
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