Let $\mathscr{F}$ be a free ultrafilter on $\mathbb{N}$. It is known that every bounded sequence $x$ is $\mathscr{F}$-convergent. Hence the space of bounded $\mathscr{F}$-convergent sequences is equal to $\ell_\infty$.
Is there something interesting about the space of bounded sequences which are $\mathscr{F}$-convergent to $0$?
I know it is a vague question, but I cannot find any interesting property.
Well, this space is a (non-unital) commutative C*-algebra so one way to understand it is using its spectrum. The spectrum of $\ell^\infty$ is $\beta\mathbb{N}$, the space of ultrafilters on $\mathbb{N}$ (considering each ultrafilter as the character $\ell^\infty\to\mathbb{C}$ sending a sequence to its limit with respect to the ultrafilter. So $\ell^\infty\cong C(\beta\mathbb{N})$, and then your subspace is just the ideal of elements which vanish at the ultrafilter $\mathscr{F}$. That is, its spectrum is $\beta\mathbb{N}\setminus\{\mathscr{F}\}$ and it is isomorphic to $C_0(\beta\mathbb{N}\setminus\{\mathscr{F}\})$, the algebra of continuous functions on the locally compact Hausdorff space $\beta\mathbb{N}\setminus\{\mathscr{F}\}$ which vanish at infinity.