Let $ces_0:=\{\xi\in\ell^\infty : \lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^{n}\xi_k=0\}$ and $q:\ell^\infty \to \ell^\infty/ces_0$ be the usual quotient map. The space $ces_0$ is closed in $\ell^\infty$, but is not an order ideal with respect to the standard order in $\ell^\infty$. Therefore the quotient $\ell^\infty/ces_0$ need not necessarily be a Banach lattice. Still, one may equip it with the quotient cone $q(\ell^\infty_+)$ making $\ell^\infty/ces_0$ into an ordered Banach space.

Banach spaces ordered by closed cones tend to be a bit nicer, so a question I'm struggling with is:

Is the quotient cone $q(\ell^\infty_+)$ (norm) closed in $\ell^\infty/ces_0$? Or equivalently: is $ces_0 + \ell^\infty_+$ (norm) closed in $\ell^\infty$?


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