On some properties of the metric subalgebras ofℓ ^∞

Abstract

The objects studied are the subalgebras ofℓ ^∞ which contain c_o. These are isometrically isomorphic to the algebras C($\mathop N\limits^ \wedge$) where$\mathop N\limits^ \wedge$ is a compactification of a discrete denumerable set N . It is shown: 1) If$\mathop N\limits^ \wedge$ is metric then there is a projection of norm 1, P: C($\mathop N\limits^ \wedge$) → C($\mathop N\limits^ \wedge$) with kernel c_o defined by PF = f o ϕ where ϕ is a retraction of$\mathop N\limits^ \wedge$ onto$\mathop N\limits^ \vee$ =$\mathop N\limits^ \wedge$ − N . 2) If$\mathop N\limits^ \wedge$ is metric, then the group of homeomorphisms of$\mathop N\limits^ \wedge$ is isomorphic to a complete group of permutations of the natural numbers ℕ . 3) The group of homeomorphisms of a compact metric space is the homomorphic image of a complete group of permutations of ℕ ("complete" means "no outer automorphisms, trivial center").