A very proper Heisenberg-Lie Banach *-algebra.

For each pair of non-zero real numbers q_1 and q_2, Laustsen and Silvestrov have constructed a unital Banach *-algebra C_{q_1,q_2} which contains a universal normalized solution to the *-algebraic (q_1,q_2)-deformed Heisenberg-Lie commutation relations. We show that in the case where (q_1,q_2) = (1,-1) or (q_1,q_2) = (-1,1), this Banach *-algebra is very proper; that is, if M is a natural number and a_1,..., a_M are elements of either C_{1,-1} or C_{-1,1} such that a_1^*a_1 + a_2^*a_2 + ... + a_M^*a_M = 0, then necessarily a_1 = a_2 = ... = a_M = 0