We show that for a normal locally-
$\mathscr{P}$
space
$X$
(where
$\mathscr{P}$
is a topological property subject to some mild requirements) the subset
${C}_{\mathscr{P}} (X)$
of
${C}_{b} (X)$
consisting of those elements whose support has a neighborhood with
$\mathscr{P}$
, is a subalgebra of
${C}_{b} (X)$
isometrically isomorphic to
${C}_{c} (Y)$
for some unique (up to homeomorphism) locally compact Hausdorff space
$Y$
. The space
$Y$
is explicitly constructed as a subspace of the Stone–Čech compactification
$\beta X$
of
$X$
and contains
$X$
as a dense subspace. Under certain conditions,
${C}_{\mathscr{P}} (X)$
coincides with the set of those elements of
${C}_{b} (X)$
whose support has
$\mathscr{P}$
, it moreover becomes a Banach algebra, and simultaneously,
$Y$
satisfies
${C}_{c} (Y)= {C}_{0} (Y)$
. This includes the cases when
$\mathscr{P}$
is the Lindelöf property and
$X$
is either a locally compact paracompact space or a locally-
$\mathscr{P}$
metrizable space. In either of the latter cases, if
$X$
is non-
$\mathscr{P}$
, then
$Y$
is nonnormal and
${C}_{\mathscr{P}} (X)$
fits properly between
${C}_{0} (X)$
and
${C}_{b} (X)$
; even more, we can fit a chain of ideals of certain length between
${C}_{0} (X)$
and
${C}_{b} (X)$
. The known construction of
$Y$
enables us to derive a few further properties of either
${C}_{\mathscr{P}} (X)$
or
$Y$
. Specifically, when
$\mathscr{P}$
is the Lindelöf property and
$X$
is a locally-
$\mathscr{P}$
metrizable space, we show that
$$\begin{eqnarray*}\dim C_{\mathscr{P}}(X)= \ell \mathop{(X)}\nolimits ^{{\aleph }_{0} } ,\end{eqnarray*}$$
where
$\ell (X)$
is the Lindelöf number of
$X$
, and when
$\mathscr{P}$
is countable compactness and
$X$
is a normal space, we show that
$$\begin{eqnarray*}Y= {\mathrm{int} }_{\beta X} \upsilon X\end{eqnarray*}$$
where
$\upsilon X$
is the Hewitt realcompactification of
$X$
.