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.

That is, if [OMEGA] is a

compact Hausdorff space and E is a real Banach space, suppose that [V.

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.

For a

compact Hausdorff space K, we shall denote by C(K) the space of all continuous functions on K equipped with the supremum norm and it is known that its dual C(K)* is the space of all regular Borel measures on K.

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 C(X) be the algebra of all continuous complex-valued functions on a

compact Hausdorff space X.

Now, let M(X) be the Banach space of all complex Radon measures on the locally

compact Hausdorff space X with the total variation norm.

We denote by M(X) the set of all signed Radon measures on a locally

compact Hausdorff space X endowed with the norm ||[mu]|| = [[mu].

We observe that for a

compact Hausdorff space X, DP(X) is a complete lattice and we have characterized it by proving that for countably compact T3 spaces X and Y without isolated points, lattice DP(X) is isomorphic to lattice DP(Y) if and only if X and Y are homeomorphic.

is isometrically isomorphic to C(K), for some

compact Hausdorff space K that contains [sup.

We recall that a function algebra A on a

compact Hausdorff space X is natural if every nonzero complex homomorphism on A is an evaluation homomorphism at some point of X [6, 4.