If S is a locally compact and
Hausdorff space and A is a continuous linear operator from C0(S) into the space
C(T) with the supremum norm topology then the Riesz Representation Theorem
yields the formula [Af](αj) =∫sf(y)λ(x,dy), where for each x ∈ Tλ(x,⋅) is a
complex-valued regular Borel measure on S. More generally a study is made of
kernel functions λ such that ∫sf(y)λ(⋅,dy) ∈ C(T) for f of compact support
on S. It is shown that λ(⋅,E) is measurable for each Borel set E and that
μ(E) =∫Tλ(x,E)ν(dαj) is a regular measure on S yielding the adjoint
formula A∗ν = μ. Necessary and sufficient conditions are given on λ so that
A∗∗(C(S)) ⊂ C(T) and that A∗∗ be continuous from C(S)β to C(T)β when S is
paracompact. Furthermore, kernel representations of β-continuous operators are
studied with applications to semi-groups of operators in C0(S) and C(S)β when S is
locally compact.
