# Banach limit

Banach limits originated in [a1], Chapt. II, Sect. 3. Denoting the positive integers by , the set is the real vector space of all bounded sequences of real numbers. For any element , one defines by for all . S. Banach showed that there exists an element in the dual , called , such that

1) for all ;

2) for all non-negative sequences ;

3) for all ;

4) for all convergent sequences . Banach proved the existence of this generalized limit by using the Hahn–Banach theorem. Today (1996), Banach limits are studied via the notion of amenability.

For a semi-group one defines to be the real vector space of all real bounded functions on . For an element one denotes the left (respectively, right) shift by (respectively, ). Thus, for all and for all . An element is called a left- (respectively right-) invariant mean if

1) ;

2) (respectively, ), where, e.g., is the adjoint of . itself is called left (respectively, right) amenable if there exists a left- (respectively, right-) invariant mean in . The existence of Banach limits above is a special case of an invariant mean, where equals the semi-group of natural numbers. Banach also proved that the real numbers are amenable (left and right). M.M. Day has proved that every Abelian semi-group is left and right amenable. On the other hand, , the free group on two generators, is not amenable.

Another approach to amenability is the measure-theoretic point of view. In fact, the prehistory of amenability starts with the following question by H. Lebesgue in the classic "Leçons sur L'Intégration et la Recherche des Fonctions Primitives" ([a5], pp. 114–115): Can countable additivity of the Lebesgue measure be replaced by finite additivity? Banach answered the question in the negative, constructing a finitely additive measure on all subsets of the real numbers, invariant under translation, again using the Hahn–Banach theorem. More generally, if a group is acting on a set , a finitely additive probability measure on the collection of all subsets of , invariant under , is sometimes also called an invariant mean. If is the isometry group of , one can ask for a finitely additive measure invariant under . Such a measure does exist for , but not for . For this leads to so-called paradoxical decompositions or the Banach–Tarski paradox (see Tarski problem; for a survey, see [a8]). For all , the group contains the non-amenable as a subgroup. It has been proved that the Banach–Tarski paradox is effectively (i.e., in ZF set theory) implied by the Hahn–Banach theorem (see [a7]).

For a survey of results of the role of amenability, see [a6] and for a survey of the Hahn–Banach theorem, see [a2]. For the early history of Banach limits and invariant means, including many important results, see [a3] and [a4].

How to Cite This Entry:
Banach limit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach_limit&oldid=39373
This article was adapted from an original article by G. Buskes (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article