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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On $ q$-normal operators and the quantum complex plane


Authors: Jaka Cimprič, Yurii Savchuk and Konrad Schmüdgen
Journal: Trans. Amer. Math. Soc. 366 (2014), 135-158
MSC (2010): Primary 14P99, 47L60; Secondary 14A22, 46L52, 11E25
Published electronically: September 4, 2013
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Abstract: For $ q>0$ let $ \mathcal {A}$ denote the unital $ *$-algebra with generator $ x$ and defining relation $ xx^*=qx^*x$. Based on this algebra we study $ q$-normal operators and the complex $ q$-moment problem. Among other things, we prove a spectral theorem for $ q$-normal operators, a variant of Haviland's theorem and a strict Positivstellensatz for $ \mathcal {A}.$ We also construct an example of a positive element of $ \mathcal {A}$ which is not a sum of squares. It is used to prove the existence of a formally $ q$-normal operator which is not extendable to a $ q$-normal one in a larger Hilbert space and of a positive functional on $ \mathcal {A}$ which is not strongly positive.


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Additional Information

Jaka Cimprič
Affiliation: Department of Mathematics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
Email: cimpric@fmf.uni-lj.si

Yurii Savchuk
Affiliation: Mathematisches Institut, Universität Leipzig, Augustusplatz 10/11, 04109 Leipzig, Germany
Email: savchuk@math.uni-leipzig.de, savchuk@math.fau.de

Konrad Schmüdgen
Affiliation: Mathematisches Institut, Universität Leipzig, Augustusplatz 10/11, 04109 Leipzig, Germany
Email: schmuedgen@math.uni-leipzig.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05733-9
PII: S 0002-9947(2013)05733-9
Keywords: $q$-normal operator, quantum complex plane, quantum moment problem, Positivstellensatz
Received by editor(s): February 18, 2011
Received by editor(s) in revised form: October 24, 2011, and October 28, 2011
Published electronically: September 4, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.