  Let $X$ be a (topological) space and let ${\mathscr I}$ be an ideal in $X$,
that is, a collection of subsets of $X$ which contains all subsets of its
elements and is closed under finite unions. The elements of ${\mathscr I}$ are
called null. The space $X$ is locally null if each $x$ in $X$ has a null
neighborhood in $X$. Let $C_b(X)$ denote the normed algebra of all continuous
bounded real-valued mappings on $X$ equipped with the supremum norm, $C_0(X)$
denote the subalgebra of $C_b(X)$ consisting of elements vanishing at infinity
and $C_{00}(X)$ the subalgebra of $C_b(X)$ consisting of elements with compact
support.
  We study the normed subalgebra $C^{\mathscr I}_{00}(X)$ of $C_b(X)$
consisting of all $f$ in $C_b(X)$ whose support has a null neighborhood in $X$,
and the Banach subalgebra $C^{\mathscr I}_0(X)$ of $C_b(X)$ consisting of all
$f$ in $C_b(X)$ such that $|f|^{-1}([1/n,\infty))$ has a null neighborhood in
$X$ for all positive integer $n$. We prove that if $X$ is a normal locally null
space then $C^{\mathscr I}_{00}(X)$ and $C^{\mathscr I}_0(X)$ are respectively
isometrically isomorphic to $C_{00}(Y)$ and $C_0(Y)$ for a unique locally
compact Hausdorff space $Y$; furthermore, $C^{\mathscr I}_{00}(X)$ is dense in
$C^{\mathscr I}_0(X)$. We construct $Y$ explicitly as a subspace of the
Stone--\v{C}ech compactification $\beta X$ of $X$. The space $Y$ is locally
compact (and countably compact, in certain cases), contains $X$ densely, and in
specific cases turns out to be familiar subspaces of $\beta X$. The known
topological structure of $Y$ enables us to establish several commutative
Gelfand--Naimark type theorems and derive results not generally expected to be
deducible from the standard Gelfand theory.
