The closure of certain subspace of $\ell_\infty$ What is the  closure of the following subspace $A$ of $\ell_\infty$  with the standard sup norm of $\ell_\infty$:
$$A=\{(a_n)\in \ell_\infty\mid (A_n)=a_1+a_2+\ldots+a_n\; \text{belongs to} \;\ell_\infty \}$$
Is it true to say that $\bar{A}$ is the space of  all $(a_n)\in \ell_\infty$ such that the  Cesaro sum $\frac{a_1+a_2 +\ldots +a_n}{n}$  goes to zero?
If it is the case, what is a proof?
 A: Consider the linear map
$
  T :\ell ^\infty \to \ell ^\infty ,
  $
given by $T ((s_n)_{n\geq 1}) = (a_n)_{n\geq 1}$,  where
$$
  a_n =
  \left\{\matrix{
    s_1, & \text{if } n=1,\hfill\cr
    s_n-s_{n-1}, & \text{otherwise },
    }\right.
  $$
and observe that $A$ is precisely the range of $T $.
Moreover, observe that $T =I-S$, where $S$ is the right shift on $\ell ^\infty $.
For a continuous linear functional $f$ on $\ell ^\infty $ one then has that $f$ vanishes on $A$ if and only if $f =f\circ S$.
The answer is then that $\bar A$ coincides with the intersection of the kernels of all right-invariant linear functionals
on $\ell ^\infty $.
A: The closure is the space of sequences that are almost convergent to zero. See Theorem 3(i) of Bennett, G.; Kalton, N. J., Consistency theorems for almost convergence, Trans. Am. Math. Soc. 198, 23-43 (1974). ZBL0301.46005. It is not sufficient that the Cesàro sum to is zero: a random $\pm1$-valued sequence has zero Cesàro sum but is not almost convergent.
