On the other hand, in  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
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.