I am interested in conditions in terms of standard scales of summation methods that guarantee the existence of an averaged limit for all [almost convergent sequences](http://en.wikipedia.org/wiki/Almost_convergent_sequence). For the Cesaro summation method $(C, 1)$ [this fails](https://mathoverflow.net/questions/93516/cesaro-means-and-banach-limits); is this true, e.g., for the [Cesaro methods](http://en.wikipedia.org/wiki/Cesaro_summation#.28C.2C_.CE.B1.29_summation) $(C, \alpha)$ with $\alpha<1$?



Cesaro means for $\alpha<1$ and Banach limits
The paper G.G. Lorentz: A contribution to the theory of divergent sequences; Acta mathematica, Volume 80, Number 1, 1960, 167-190; DOI: [10.1007/BF02393648](http://dx.doi.org/10.1007/BF02393648), contains several interesting results related to your questions on Banach limits.


A characterization of matrix methods that sum all almost convergent sequences is given (Theorem 7). In particular, each $C\_\alpha$ sums all almost convergent sequences.


However, it is shown that almost convergence cannot be represented by a regular matrix method. Also the following stronger result about a class of matrix methods is shown.


**Theorem 11.** For every sequence $\{A\_k\}$ of methods of the class $\mathfrak A$ there is
a bounded sequence $x = \{x\_n\}$ which is not almost convergent but is summable to the
value zero by every one of the methods $A\_k$.


The class $\mathfrak A$ in Lorentz's paper is the class of matrices fulfilling 
$$\lim\limits\_{m\to\infty} \max\limits\_n |a\_{mn}|=0.$$
I think it's not that hard to show that each matrix $C\_n$ belongs to $\mathfrak A$.


Some further references for almost convergence are mentioned e.g in the book Boos: [Classical and Modern Methods in Summability](http://books.google.com/books?id=kZ9cy6XyidEC).


