If A is a commutative
Banach algebra with identity, then the sets ℳ (all maximal ideals), ℳc (all
closed maximal ideals), ℳ1 (kernels of nonzero C-valued homomorphisms of
A), and ℳ0 (kernels of nonzero continuous C-valued hommorphisms of A)
coincide. If A is a commutative complete locally m-convex algebra, one has only
ℳc= ℳ0⊂ℳ1⊂ℳ, and the containments can be proper. Our goal is to
investigate ℳ and its relationship to ℳ0; specifically (1) to give a description of
ℳ(A) in terms of A and ℳ0(A) which is valid for at least the class of F-algebras,
(2) to determine when ℳ(A) is one of the standard compactifications (Wallman,
Stone-Čech) of ℳ0(A).
