Heisenberg–Lie commutation relations in Banach algebras

Given q1, q2 ∈ C \ {0}, we construct a unital Banach algebra Bq1,q2 which contains a universal normalized solution to the (q1, q2)-deformed Heisenberg–Lie commutation relations in the following specific sense: (i) Bq1,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‖ 6 1, there is a unique bounded unital algebra homomorphism φ : Bq1,q2 → A such that φ(bj) = aj for j = 1, 2, 3. For q1, q2 ∈ R\{0}, we obtain a counterpart of the above result for Banach ∗-algebras. In contrast, we show that if q1, q2 ∈ (−∞, 0), q1, q2 ∈ (0, 1), or q1, q2 ∈ (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 ∈ R \ {0}, an explicit construction based on a weighted shift operator on `2(Z) produces a non-zero solution to the ∗-algebraic counterpart of the (q1, q2)-deformed Heisenberg–Lie commutation relations; we determine all such pairs. 2000 Mathematics Subject Classification: primary 46H15, 46K05; secondary 47B37, 47B47, 43A20.