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HEISENBERG–LIE COMMUTATION RELATIONS IN BANACH ALGEBRAS

HEISENBERG–LIE COMMUTATION RELATIONS IN BANACH ALGEBRAS Given q 1, q 2 ∈ ℂ \ {0}, we construct a unital Banach algebra B q 1 , q 2 that contains a universal normalised solution to the ( q 1, q 2)-deformed Heisenberg–Lie commutation relations in the following specific sense: (i) B q 1 , q 2 contains elements b 1 , b 2 and b 3 , which satisfy the ( q 1, q 2)-deformed Heisenberg–Lie commutation relations (that is, b 1 b 2 − q 1 b 2 b 1 = b 3 , q 2 b 1 b 3 − b 3 b 1 = 0 and b 2 b 3 − q 2 b 3 b 2 = 0) and ‖ b 1 ‖ = ‖ b 2 ‖ = 1; (ii) whenever a unital Banach algebra A contains elements a 1 , a 2 and a 3 satisfying the ( q 1, q 2)-deformed Heisenberg–Lie commutation relations and ‖ a 1 ‖, ‖ a 2 ‖ ≤ 1, there is a unique bounded unital algebra homomorphism φ: B q 1 , q 2 → A such that φ( b j ) = a j for j = 1, 2, 3. For q 1, q 2 ∈ ℝ\{0}, we obtain a counterpart of the above result for Banach *-algebras. In contrast, we show that if q 1, q 2 ∈ (−∞, 0), q 1, q 2 ∈ (0, 1), or q 1, q 2 ∈ (1,∞), then a C *-algebra cannot contain a non-zero solution to the *-algebraic counterpart of the ( q 1, q 2)-deformed Heisenberg–Lie commutation relations. However, for many other pairs q 1, q 2 ∈ ℝ \ {0}, an explicit construction based on a weighted shift operator on ℓ 2 (ℤ) produces a non-zero solution to the *-algebraic counterpart of the ( q 1, q 2)-deformed Heisenberg–Lie commutation relations. We determine all such pairs. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Proceedings of the Royal Irish Academy Royal Irish Academy

HEISENBERG–LIE COMMUTATION RELATIONS IN BANACH ALGEBRAS

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References (30)

Publisher
Royal Irish Academy
Copyright
Copyright © 2009 RIA
ISSN
1393-7197
eISSN
2009-0021
DOI
10.3318/PRIA.2009.109.2.163
Publisher site
See Article on Publisher Site

Abstract

Given q 1, q 2 ∈ ℂ \ {0}, we construct a unital Banach algebra B q 1 , q 2 that contains a universal normalised solution to the ( q 1, q 2)-deformed Heisenberg–Lie commutation relations in the following specific sense: (i) B q 1 , q 2 contains elements b 1 , b 2 and b 3 , which satisfy the ( q 1, q 2)-deformed Heisenberg–Lie commutation relations (that is, b 1 b 2 − q 1 b 2 b 1 = b 3 , q 2 b 1 b 3 − b 3 b 1 = 0 and b 2 b 3 − q 2 b 3 b 2 = 0) and ‖ b 1 ‖ = ‖ b 2 ‖ = 1; (ii) whenever a unital Banach algebra A contains elements a 1 , a 2 and a 3 satisfying the ( q 1, q 2)-deformed Heisenberg–Lie commutation relations and ‖ a 1 ‖, ‖ a 2 ‖ ≤ 1, there is a unique bounded unital algebra homomorphism φ: B q 1 , q 2 → A such that φ( b j ) = a j for j = 1, 2, 3. For q 1, q 2 ∈ ℝ\{0}, we obtain a counterpart of the above result for Banach *-algebras. In contrast, we show that if q 1, q 2 ∈ (−∞, 0), q 1, q 2 ∈ (0, 1), or q 1, q 2 ∈ (1,∞), then a C *-algebra cannot contain a non-zero solution to the *-algebraic counterpart of the ( q 1, q 2)-deformed Heisenberg–Lie commutation relations. However, for many other pairs q 1, q 2 ∈ ℝ \ {0}, an explicit construction based on a weighted shift operator on ℓ 2 (ℤ) produces a non-zero solution to the *-algebraic counterpart of the ( q 1, q 2)-deformed Heisenberg–Lie commutation relations. We determine all such pairs.

Journal

Mathematical Proceedings of the Royal Irish Academy – Royal Irish Academy

Published: Jul 1, 2009

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