Bounded structures in locally A-convex algebras
Several bornologies (bounded structures) are examined in the frame of locally A-convex algebras. The least condition to ask for is the boundedness of every element. The latter with some completeness –in general, the Mackey completeness- allows a quiet good spectral theory. But it is much better to have a pseudo-Banach structure which, of course, is not always granted even if every element is bounded. By the way, this is still an open question even for complete m-convex algebras. For a general Mackey complete A-convex algebra, the study is reduced to the m-convex case and so the questions are exactly the same. Locally uniformly A-convex algebras behave very well. When unital and Mackey complete, the bound structure is the one of a Banach algebra norm. We introduce a larger class of algebras with the same property under the same conditions. These are named ‘Locally uniformly convex algebras’.