The notion of a [Banach limit][1] is usually defined for the space of bounded sequences, but one can define it for more general spaces (see "[What is a generalized limit?][2]" and "[Do multiplicative Banach limits exist?][3]" and references therein).

I am interested in the Banach space of Bounded functions $f: \mathbb{R}_+ \to \mathbb{C}$ with respect to the uniform norm  $ \|f\| = \sup_{t \in \mathbb{R}_+} |f(t)|$ and Banach limits on it. Consider a sequence $f_n(t)$ ,$n\in \mathbb{N}$, of such functions and a Banach limit denoted by $L$. Suppose that for all $t\in \mathbb{R}_+$: $\sum_{n=1}^\infty f_n(t)$ exists, and consider its Banach limit 
$$L\Big(   \sum_{n=1}^{\infty} f_n(t) \Big) $$
Is it true that
$$L\Big(  \sum_{n=1}^{\infty} f_n(t) \Big) =  \sum_{n=1}^{\infty}L\big( f_n(t) \big) $$

for all Banach limits $L$?

With usual limits, i.e. $\lim_{t\to \infty} \lim_N \sum_{n=1}^N f_n(t) $, it's not always the case that we can commute the two limits.

So, does the dominated convergence theorem extend to Banach limits, so that the Banach limit $L$ can be moved inside the summation/commute the limits? Under what assumptions?




I'd appreciate references.


  [1]: https://en.wikipedia.org/wiki/Banach_limit
  [2]: https://mathoverflow.net/questions/242307/what-is-a-generalized-limit
  [3]: https://mathoverflow.net/questions/356806/do-multiplicative-banach-limits-exist