Commutativity criteria in Banach algebras (Q2793546)

Let \(A\) be a complex unital Banach algebra. It is shown that \(A\) is commutative if and only if either \((ab)^2=a^2b^2\) or \((ab)^3=a^3b^3\) for all \(a,b\in A\) and \(A\) is almost commutative if \((ab)^k=a^kb^k\) with \(k\geqslant 2\) for all \(a,b\in A\). Examples are given which show that, in the case of nonunital Banach algebras, these statements cannot be true. The author mentions that similar results hold also for complete locally multiplicatively convex algebras.