**Bounded structures in locally A-convex
algebras**

M. Oudadess

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’.