Order and geometric properties of the set of Banach limits

Journal article, 2017

A positive functional  on the space of bounded sequences  is called a Banach limit if  and  for every . The set of all Banach limits is denoted by  and the set of its extreme points is denoted by . Various properties of these sets are studied. For instance, there exists  such that  if  and  The set  fails to possess the -property for an affine nonexpansive sequentially weak continuous mapping. A general result is proved, which implies that there is a wide class of subspaces of , defined in terms of Banach limits, that are not complemented in . In particular, this class includes the stabilizer  and the ideal stabilizer  of the subspace  of almost convergent sequences. In the second part of the paper, the object of study is the set  of all Banach limits invariant under the dilation operator ,  on  given byIf , then for all , , the inclusion  is proper; there exists  such that  for all ⧵ and  for all  if  for all . If , , then . Moreover, the cardinalities of the extreme point are estimated for some subsets of . In particular, .