Extending positive definite linear forms
Abstract: In the classical literature two properties, called here symmetry and bounded variation, of a positive definite linear form on an involutive Banach algebra are given. Together they form a necessary and sufficient condition that admit a positive definite linear extension to the involutive algebra obtained from by adjoining an identity. In this note we show that bounded variation alone suffices, in that it already implies symmetry.
-  Lynn H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Company, Inc., Toronto-New York-London, 1953. MR 0054173
-  M. A. Naĭmark, Normed rings, Reprinting of the revised English edition, Wolters-Noordhoff Publishing, Groningen, 1970. Translated from the first Russian edition by Leo F. Boron. MR 0355601
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46Kxx
Retrieve articles in all journals with MSC: 46Kxx
Keywords: Involutive Banach algebras, positive definite forms, symmetry, bounded variation, Gelfand-Segal representation, cyclic vectors, extensibility
Article copyright: © Copyright 1984 American Mathematical Society