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Mathematics > Operator Algebras

arXiv:1805.02362 (math)
[Submitted on 7 May 2018 (v1), last revised 15 Jun 2019 (this version, v2)]

Title:A $q$-deformed Heisenberg algebra as a normed space

Authors:Rafael Reno S. Cantuba
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Abstract:Given a real number $q$ such that $0<q<1$, the natural setting for the mathematics of a $q$-oscillator is an infinite-dimensional, separable Hilbert space that is said to provide an interpolation between the Bargmann-Segal space of holomorphic functions and the Hardy-Lebesgue space of analytic functions. The traditional basis states are interrelated by the creation and annihilation operators. Since the commutation relation is $q$-deformed, the commutator algebra for the creation and annihilation operators is not a low-dimensional Lie algebra like that for the canonical commutation relation. In this study, a characterization of the elements of the said commutator algebra is obtained using spectral properties of the creation and annihilation operators as these faithfully represent the generators of a $q$-deformed Heisenberg algebra. The derived algebra of the commutator algebra is precisely the set of all compact operators, and the resulting Calkin algebra is algebraically isomorphic to the complex algebra of Laurent polynomials in one indeterminate. As for any operator that is not in the commutator algebra, the action of such an operator on an arbitrary basis state can be approximated by a Lie series of elements from the commutator algebra.
Subjects: Operator Algebras (math.OA)
MSC classes: 47B37, 47B47, 47L30, 46H70
Cite as: arXiv:1805.02362 [math.OA]
  (or arXiv:1805.02362v2 [math.OA] for this version)
  https://doi.org/10.48550/arXiv.1805.02362
Journal reference: published under a different title in: Letters in Mathematical Physics 110 (2020) 2639-2657
Related DOI: https://doi.org/10.1007/s11005-020-01304-x

Submission history

From: Rafael Reno Cantuba Ph.D. [view email]
[v1] Mon, 7 May 2018 06:32:18 UTC (17 KB)
[v2] Sat, 15 Jun 2019 08:05:44 UTC (36 KB)
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