The Duality of Compact Semigroups and C*-Bigebras
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K. H. Hofmann Tulane University, New Orleans, LA/USA
The Duality of Compact Semigroups and C*-Bigebras
Springer-Verlag Berlin' Heidelberg' New York 1970
This work is subject 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 privatesrse, 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 1970. Library of Congress Catalog Card Number 70-117194 Prinred in Germany. Tide No. 3286
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We propose a complete duality theory for the category of compact Hausdorff topological semigroups in terms of commutative C*-bigebras with identity. A bigebra (formerly called hyperalgebra or Hopf algebra) is. an algebra with a comultiplication. In a more expository introductory we discuss the concept of a bigebra over a which plays an important role in this context. The more technical part starts with a discussion of C*-algebras and their tensor products. after the establishment of the we show how it generalizes the Tannaka duality theory for compact and the Pontryagin duality for compact abelian groups.
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Introduction
One of the best known duality theories is the one between compact abelian groups and discrete abelian groups, discovered by PONTRYAGIN in the early thirties. A duality theory for not necessarily commutative compact groups was later described by TANNAKA, and its most modern forms are comparatively recent and go back to HOCHSCHILD [8]. All of these duality theories are closely tied up with the representation theory of these groups. At first sight this observation makes the outlook to possible generalizations for wider classes of compact objects (compact semigroups, say) seem gloomy. It is indeed well known and has been accepted with pessimistic fatalism that no useful representation theory is available for the category of all compact semigroups. The reason of this default is the absence in general compact semigroups of an invariant integral whose support is the entire semigroup. (In fact the only compact semigroups having a two sided invariant integral with full support are the groups.) On the other hand, if one proceeds to the end of the spectrum of compact objects, namely those objects having no additional structure whatsoever, then again we have a completely satisfactory duality theory: namely, the duality between compact spaces and commutative C*algebras which is based on the theorem of GELFAND and NAIMARK. It is therefore natural to look for dual objects of compact semigroups in the class of commutative C*algebras with an additional element of structure. And indeed we will be able to establish a complete duality between the category of compact semigroups and the category of C*algebras with a comultiplication. Algebras of certain
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