This paper combines two
extensions of the theory of Banach algebras. On the one hand, Arens and Michael
generalized the theory of Banach algebras by introducing the concept of a locally
multiplicatively-convex topological algebra (abbreviated “lmc” algebra). On the
other hand, Arens gave a procedure for defining on the bidual (the second
topological conjugate space) of a Banach algebra a multiplication which makes
the bidual also into a Banach algebra. We show that one can put an Arens
multiplication onto the bidual of an lmc algebra, and we study the algebraic
and topological properties of the bidual when it is endowed with such a
multiplication.
