1. A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings
- Author
-
Jaka Cimpric
- Subjects
Pure mathematics ,Representation theorem ,Mathematics::Operator Algebras ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Mathematics - Rings and Algebras ,010103 numerical & computational mathematics ,Commutative ring ,01 natural sciences ,Noncommutative geometry ,Representation theory ,Quadratic equation ,16W80, 46L05, 46L89, 14P99 ,Rings and Algebras (math.RA) ,FOS: Mathematics ,Real algebraic geometry ,0101 mathematics ,Commutative property ,Associative property ,Mathematics - Abstract
We present a new approach to noncommutative real algebraic geometry based on the representation theory of $C^\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings, \cite[Theorem 5]{jacobi}. We show that this theorem is a consequence of the Gelfand-Naimark representation theorem for commutative $C^\ast$-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., 12 pages
- Published
- 2009
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