Generalized open mapping theorems for bilinear maps, with an application to operator algebras

Abstract:

Cohen [4] gave an example of a surjective bilinear mapping between Banach spaces which was not open, and Horowitz [8] gave a much simpler example. We build on Horowitz’ example to produce a similar result for bilinear mappings such that every element of the target space is a linear combination of $n$ elements of the range. An immediate application is that Bercovici’s construction [1] of an operator algebra with property $({\mathbb {A}_1})$ but not $({\mathbb {A}_1}(r))$ can be extended to achieve property $({\mathbb {A}_{1/n}})$ without $({\mathbb {A}_{1/n}}(r))$.