On $q$-normal operators and the quantum complex plane
HTML articles powered by AMS MathViewer
- by Jaka Cimprič, Yurii Savchuk and Konrad Schmüdgen PDF
- Trans. Amer. Math. Soc. 366 (2014), 135-158 Request permission
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.References
- C.-G. Ambrozie and F.-H. Vasilescu, Operator-theoretic positivstellensätze, Z. Anal. Anwendungen 22 (2003), no. 2, 299–314. MR 2000269, DOI 10.4171/ZAA/1147
- J. Cimprič, Maximal quadratic modules on $\ast$-rings, Algebr. Represent. Theory 11 (2008), no. 1, 83–91. MR 2369102, DOI 10.1007/s10468-007-9076-z
- Jaka Cimprič, A method for computing lowest eigenvalues of symmetric polynomial differential operators by semidefinite programming, J. Math. Anal. Appl. 369 (2010), no. 2, 443–452. MR 2651693, DOI 10.1016/j.jmaa.2010.03.045
- Earl A. Coddington, Formally normal operators having no normal extensions, Canadian J. Math. 17 (1965), 1030–1040. MR 200719, DOI 10.4153/CJM-1965-098-8
- David Handelman, Rings with involution as partially ordered abelian groups, Rocky Mountain J. Math. 11 (1981), no. 3, 337–381. MR 722571, DOI 10.1216/RMJ-1981-11-3-337
- J. William Helton, “Positive” noncommutative polynomials are sums of squares, Ann. of Math. (2) 156 (2002), no. 2, 675–694. MR 1933721, DOI 10.2307/3597203
- J. William Helton, Scott McCullough, and Mihai Putinar, Strong majorization in a free $\ast$-algebra, Math. Z. 255 (2007), no. 3, 579–596. MR 2270289, DOI 10.1007/s00209-006-0032-0
- Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
- Gottfried Köthe, Topological vector spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York, Inc., New York, 1969. Translated from the German by D. J. H. Garling. MR 0248498
- Murray Marshall, $*$-orderings on a ring with involution, Comm. Algebra 28 (2000), no. 3, 1157–1173. MR 1742648, DOI 10.1080/00927870008826887
- Murray Marshall, Positive polynomials and sums of squares, Mathematical Surveys and Monographs, vol. 146, American Mathematical Society, Providence, RI, 2008. MR 2383959, DOI 10.1090/surv/146
- Murray Marshall and Yufei Zhang, Orderings, real places, and valuations on noncommutative integral domains, J. Algebra 212 (1999), no. 1, 190–207. MR 1670638, DOI 10.1006/jabr.1998.7630
- V. Ostrovskyi and Yu. Samoilenko, Introduction to the theory of representations of finitely presented *-algebras. I. Representations by bounded operators, Reviews in Mathematics and Mathematical Physics, vol. 11, Harwood Academic Publishers, Amsterdam, 1999. MR 1997101
- Schôichi Ôta, Some classes of $q$-deformed operators, J. Operator Theory 48 (2002), no. 1, 151–186. MR 1926049
- Schôichi Ôta and Franciszek Hugon Szafraniec, Notes on $q$-deformed operators, Studia Math. 165 (2004), no. 3, 295–301. MR 2110154, DOI 10.4064/sm165-3-7
- Schôichi Ôta and Franciszek Hugon Szafraniec, $q$-positive definiteness and related operators, J. Math. Anal. Appl. 329 (2007), no. 2, 987–997. MR 2296901, DOI 10.1016/j.jmaa.2006.07.006
- Bruce Reznick, Uniform denominators in Hilbert’s seventeenth problem, Math. Z. 220 (1995), no. 1, 75–97. MR 1347159, DOI 10.1007/BF02572604
- Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
- Konrad Schmüdgen, Unbounded operator algebras and representation theory, Operator Theory: Advances and Applications, vol. 37, Birkhäuser Verlag, Basel, 1990. MR 1056697, DOI 10.1007/978-3-0348-7469-4
- Konrad Schmüdgen, Noncommutative moment problems, Math. Z. 206 (1991), no. 4, 623–649. MR 1100846, DOI 10.1007/BF02571369
- Konrad Schmüdgen, A strict Positivstellensatz for the Weyl algebra, Math. Ann. 331 (2005), no. 4, 779–794. MR 2148796, DOI 10.1007/s00208-004-0604-4
- Konrad Schmüdgen, A strict positivstellensatz for enveloping algebras, Math. Z. 254 (2006), no. 3, 641–653. MR 2244371, DOI 10.1007/s00209-006-0965-3
- Konrad Schmüdgen, Noncommutative real algebraic geometry—some basic concepts and first ideas, Emerging applications of algebraic geometry, IMA Vol. Math. Appl., vol. 149, Springer, New York, 2009, pp. 325–350. MR 2500470, DOI 10.1007/978-0-387-09686-5_{9}
- Naum Z. Shor, Nondifferentiable optimization and polynomial problems, Nonconvex Optimization and its Applications, vol. 24, Kluwer Academic Publishers, Dordrecht, 1998. MR 1620179, DOI 10.1007/978-1-4757-6015-6
- Yu. Savchuk and K. Schmüdgen, Unbounded induced $*$-representations, Algebr. Represent. Theory, doi: 10.1007/s10468-011-9310-6.
- Ivan Vidav, On some $^*$regular rings, Acad. Serbe Sci. Publ. Inst. Math. 13 (1959), 73–80. MR 126735
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
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 135-158
- MSC (2010): Primary 14P99, 47L60; Secondary 14A22, 46L52, 11E25
- DOI: https://doi.org/10.1090/S0002-9947-2013-05733-9
- MathSciNet review: 3118394