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].
References
[a1] | S. Banach, "Théorie des opérations linéaires" , PWN (1932) |
[a2] | G. Buskes, "The Hahn–Banach theorem surveyed" Dissertationes Mathematicae , CCCXXVII (1993) |
[a3] | M.M. Day, "Normed linear spaces" , Ergebnisse der Mathematik und ihrer Grenzgebiete , 21 , Springer (1973) |
[a4] | Greenleaf, F.P, "Invariant means on topological groups and their applications" , v. Nostrand (1969) |
[a5] | H. Lebesgue, "Oeuvres Scientifiques" , L'Enseign. Math. , II , Inst. Math. Univ. Genæve (1972) |
[a6] | A.L.T. Paterson, "Amenibility" , Mathematical Surveys and Monographs , 29 , Amer. Math. Soc. (1988) |
[a7] | J. Pawlikowski, "The Hahn–Banach theorem implies the Banach–Tarski paradox" Fundam. Math. , 138 (1991) pp. 20–22 |
[a8] | S. Wagon, "The Banach–Tarski paradox" , Cambridge Univ. Press (1986) |
Banach limit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach_limit&oldid=13656