In [10], some similar results (Theorem 1 on page 98, Corollaries 1 and 2 on page 99) are presented for a

compact Hausdorff space X and a topological linear space Y.

Let Xbea

compact Hausdorff space, let G be a finite group acting freely on [S.sup.n] and let H be a cyclic subgroup of G of order prime p.

It follows that the spectrum [bar.S] of A is a

compact Hausdorff space. Furthermore, since points of S determine complex homomorphisms of A, there is a continuous map [alpha] : S [right arrow] [bar.S], with dense image, such that f [??] fo[alpha] : C([bar.S]) [right arrow] A is an isometric isomorphism of C([bar.S]) onto A, when [alpha] is injective.

Let X [not equal to] 0 be a

compact Hausdorff space and K(X) a set of nonempty compact subsets of X.Let I = {1, ..., N} be a finite set of positive integers, [I.sup.[infinity]] a set of infinite sequences of numbers from I, and [f.sub.i] : X [right arrow] X, i [member of] I, continuous mappings.

Liu also proved the isometric extension problem in the C([OMEGA])-space, where [OMEGA] is a

compact Hausdorff space, by showing a geometric representation of the isometries.

We always assume that X is a

compact Hausdorff space and most of the time the group T is countable (with the discrete topology).

-- A topological vector lattice (E,[tau]) is homeomorphic and lattice isomorphic to a dense subspace V of (C(Y),k) for some locally

compact Hausdorff space Y such that V [intersection] [C.sub.[infinity]](Y) is dense in ([C.sub.o](Y), ||*||) if, and only if, (E,[tau]) is an M-partition space.

We propose a unified approach to the study of isometries with respect to the sum norm on Banach algebras Lip(K,C(Y)), [lip.sub.[alpha]](FT,C(Y)), and [C.sup.1](K,C(y)), where K is a compact metric space, [0,1], or T (T denotes the unit circle on the complex plane), and Y is a

compact Hausdorff space. We study isometries without assuming that they preserve unit.

On the other hand, in [17] it was shown that for a real Banach space X and a

compact Hausdorff space K, if T : X [right arrow] [C.sub.R] (K) is an isometry with T(0) = 0, then there exists a closed nonempty subset L [subset or equal to] K, such that the superposition Q [omicron] T : X [right arrow] [C.sub.R](L) is a linear map.

Let X be a locally

compact Hausdorff space, and let w : X [right arrow] R be an upper semi-continuous function such that w(t) [greater than or equal to] 1 for every t [member of] X.

Let X be a locally

compact Hausdorff space. If [C.sub.0](X) has the weak fixed point property, then X is dispersed.

Let K be a

compact Hausdorff space and let Y be a finite co-dimensional subspace of C(K).