HEISENBERG-LIE COMMUTATION RELATIONS IN BANACH ALGEBRAS

• Published in: Mathematical proceedings of the Royal Irish Academy 2009-10, Vol.109A (2), p.163-186
• Main Authors: Laustsen, Niels Jakob, Silvestrov, Sergei D.
• Format: Article
• Language: English
• Subjects: Adjoints; Algebra; Homomorphisms; Integers; Mathematical relations; Mathematical theorems; Semigroups; Sentence commutation; Uniqueness
• ISSN: 1393-7197; 2009-0021; 2009-0021
• DOI: 10.1353/mpr.2009.0009

Given qi, q2 Єℂ\ {0}, we construct a unital Banach algebra $B_{qt,q2} $ that contains a universal normalised solution to the (q1,q2)-deformed Heisenberg-Lie commutation relations in the following specific sense: (i) $B_{qt,q2} $ contains elements b₁, b₂ and b₃, which satisfy the (q1,q2)-deformed Heisenberg-Lie commutation relations (that is, b₁b₂ - q₁b₂b₁ = b₃, q₂b₁b₃ - b₃b₁ = 0 and b₂b₃ - q₂b₃b₂ = 0) and ║b₁║ = ║b₂║ = 1; (ii) whenever a unital Banach algebra A contains elements a₁, a₂ and a₃ satisfying the (q₁,q₂)-deformed Heisenberg-Lie commutation relations and ║a₁║, ║a₂║ ≤ 1, there is a unique bounded unital algebra homomorphism $\phi :B_{qt,q2} \to A$ such that $\phi (b_j ) = a_{j\,\,} for\,j = 1,\,\,2,\,\,3.$ For q₁q₂ Є ℝ\{0}, we obtain a counterpart of the above result for Banach *-algebras. In contrast, we show that if q₁, q₂ Є (— ∞, 0), q₁, q₂ Є (0,1), or q₁, q₂ Є (1, ∞), then a C*-algebra cannot contain a non-zero solution to the *-algebraic counterpart of the (q₁, q₂)-deformed Heisenberg-Lie commutation relations. However, for many other pairs q₁,q₂ Є ℝ\ {0}, an explicit construction based on a weighted shift operator on l₂ (ℤ) produces a non-zero solution to the *-algebraic counterpart of the (q₁, q₂)-deformed Heisenberg-Lie commutation relations. We determine all such pairs.