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A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings

Published online by Cambridge University Press:  20 November 2018

Jakob Cimprič
Jakob Cimprič*
Affiliation:
University of Ljubljana, Faculty of Mathematics and Physics, Department of Mathematics, Jadranska 19 SI-1000, Ljubljana, Slovenija e-mail: jaka.cimpric@fmf.uni-lj.si
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Abstract

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We present a new approach to noncommutative real algebraic geometry based on the representation theory of ${{C}^{*}}$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand–Naimark representation theorem for commutative ${{C}^{*}}$-algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

Keywords

16W8046L0546L8914P99Ordered rings with involutionC*-algebras and their representationsnoncommutative convexity theoryreal algebraic geometry
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 1 , 01 March 2009 , pp. 39 - 52
Copyright
Copyright © Canadian Mathematical Society 2009

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