Nonseparable Banach algebras whose squares are pathological

Abstract

There exists a nonseparable noncommutative Banach algebra A such that A 2 is nonclosed and of finite codimension in A. There exists a similar A with A 2 = A such that the natural mapping of the algebraic tensor product A ⊗ A into A is not open, and there is no bound on the number of summands needed to express an element as an element of A 2.