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A note on values of noncommutative polynomials


Authors: Matej Bresar and Igor Klep
Journal: Proc. Amer. Math. Soc. 138 (2010), 2375-2379
MSC (2010): Primary 08B20, 16R99, 47L30
DOI: https://doi.org/10.1090/S0002-9939-10-10324-4
Published electronically: March 15, 2010
MathSciNet review: 2607866
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Abstract | References | Similar Articles | Additional Information

Abstract: We find a class of algebras $ \mathcal{A}$ satisfying the following property: for every nontrivial noncommutative polynomial $ f(X_1,\ldots,X_n)$, the linear span of all its values $ f(a_1,\ldots,a_n)$, $ a_i\in \mathcal{A}$, equals $ \mathcal{A}$. This class includes the algebras of all bounded and all compact operators on an infinite dimensional Hilbert space.


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Additional Information

Matej Bresar
Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 19, SI-1000 Ljubljana, Slovenia – and – Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, SI-2000 Maribor, Slovenia
Email: matej.bresar@fmf.uni-lj.si

Igor Klep
Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 19, SI-1000 Ljubljana, Slovenia – and – Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, SI-2000 Maribor, Slovenia
Email: igor.klep@fmf.uni-lj.si

DOI: https://doi.org/10.1090/S0002-9939-10-10324-4
Keywords: Noncommutative polynomial, Lie ideal, Hilbert space, bounded operator, compact operator
Received by editor(s): September 30, 2009
Received by editor(s) in revised form: December 2, 2009
Published electronically: March 15, 2010
Additional Notes: The first author was supported by the Slovenian Research Agency (program No. P1-0288).
The second author was supported by the Slovenian Research Agency (program No. P1-0222).
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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