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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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Algebras of fractions and strict Positivstellensätze for ∗-algebras

Konrad Schmüdgen1

1Universität Leipzig, Mathematisches Institut, Johannisgasse 26, 04103 Leipzig, Germany. e-mail:

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2010, Issue 647, Pages 57–86, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2010.073, October 2010

Publication History

Published Online:


In this paper we investigate a ∗-algebra 𝔛 of fractions associated with a unital complex ∗-algebra . The ∗-algebra 𝔛 and its Hilbert space representations are used to prove abstract noncommutative strict Positivstellensätze for . Multi-grading of are studied as technical tools to verify the assumptions of this theorem.

As applications we obtain new strict Positivstellensätze for the Weyl algebra and for the Lie algebra of the affine group of the real line. We characterize integrable representations of the Lie algebra in terms of resolvents of the generators and derive a new integrability criterion for representations of .

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Yurii Savchuk and Konrad Schmüdgen
Linear Algebra and its Applications, 2012, Volume 436, Number 3, Page 758

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