The closure of certain subspace of $\ell_\infty$

Question

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?

Answer 1

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 $.

Answer 2

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.