Return to Revisions

If one can distill one central fact from the answers above and elsewhere, then it is that if one wants a good duality theory for algebras of continous functions, one has to move on from Banach spaces or algebras. This fact was recognised at least 50 years ago, and a suitable generalisation was discovered---the so-called strict topology on the space of bounded, continuous functions on a, say, completely regular space. Despite the eminence of its discoverers and advocates (Beurling, Herz, Buck), it seems to have become forgotten lore. A succinct argument on its behalf can be found in the 1960 Transactions paper by Herz, "The spectral theory of bounded functions" (as the title suggests, the motivation for its introduction comes from Harmonic Analysis). The original version used weighted seminorms and was valid for locally compact spaces. It was subsequently extended to completely regular spaces by several authors. The topology can be defined most succinctly as the finest locally convex one which agrees with compact convergence on the unit ball of $C^b(X)$. It is a complete locally convex spaces, its compact sets and convergent sequences can be simply characterised (the latter are the uniformly bounded sequences which converge uniformly on compacta). Its dual is the space of bounded Radon measures on $X$, the natural version of the Stone-Weierstra\ss theorem holds for it, its spectrum is naturally identifiable with $X$ and so one has a version of Gelfand-Naimark theory. One can also characterise the topological properties of $X$ in terms of those of $C^b(X)$, all in principle, many in practice. Importantly, one can characterise local compactness of $X$ in terms of the property of the algebra $C^b(X)$. A full account can be found in the book "Saks Spaces and Applications to Functional Analysis".