  We study the space $c_{0,\mathcal{I}}$ of all bounded sequences $(x_n)$ that
$\mathcal{I}$-converge to $0$, endowed with the sup norm, where $\mathcal{I}$
is an ideal of subsets of $\mathbb{N}$. We show that two such spaces,
$c_{0,\mathcal{I}}$ and $c_{0,\mathcal{J}}$, are isometric exactly when the
ideals $\mathcal{I}$ and $\mathcal{J}$ are isomorphic. Additionally, we analyze
the connection of the well-known Kat\v{e}tov pre-order $\leq_K$ on ideals with
some properties of the space $c_{0,\mathcal{I}}$. For instance, we show that
$\mathcal{I}\leq_K\mathcal{J}$ exactly when there is a (not necessarily onto)
Banach lattice isometry from $c_{0,\mathcal{I}}$ to $c_{0,\mathcal{J}}$,
satisfying some additional conditions.
  We present some lattice-theoretic properties of $c_{0,\mathcal{I}}$,
particularly demonstrating that every closed ideal of $\ell_\infty$ is equal to
$c_{0,\mathcal{I}}$ for some ideal $\mathcal{I}$ on $\mathbb{N}$. We also show
that certain classical Banach spaces are isometric to $c_{0,\mathcal{I}}$ for
some ideal $\mathcal{I}$, such as the spaces $\ell_\infty(c_0)$ and
$c_0(\ell_\infty)$. Finally, we provide several examples of ideals for which
$c_{0,\mathcal{I}}$ is not a Grothendieck space.
