Abstract
A topological group has the Approximate Fixed Point (AFP) property on a bounded convex subset of a locally convex space if every continuous affine action of on admits a net , , such that for all . In this work, we study the relationship between this property and amenability.
Key words
Amenable groups; approximate fixed point property; Følner property; Reiter property
INTRODUCTION
One of the most useful known characterizations of amenability is stated in terms of a fixed point property. A classical theorem of (Day 1961)r11 DAY M. 1961. Fixed-point theorems for compact sets. Illinois J Math 5: 585-590. says that a topological group is amenable if and only if every continuous affine action of on a compact convex subset of a locally convex space has a fixed point, that is, a point with for all . This result generalizes earlier theorems of (Kakutani 1938r24 KAKUTANI S. 1938. Fixed-point theorems concerning bicompact convex sets. Proc Imperial Acad Japan 14: 27-31.) and (Markov 1936r28 MARKOV A. 1936. Quelques théorèmes sur les ensembles abéliens. Dokl Akad Nauk SSSR (NS:) 10: 311-314.) obtained for abelian acting groups.
At the same time, an active branch of current research is devoted to the existence of approximate fixed points for single maps. Basically, given a bounded, closed convex set and a map , one wants to find a sequence such that . A sequence with this property will be called an approximate fixed point sequence.
The main motivation for this topic is purely mathematical and comes from several instances of the failure of the fixed point property in convex sets that are no longer assumed to be compact, cf. (Dobrowolski and Marciszewski 1997r12 DOBROWOLSKI T AND MARCISZEWSKI W. 1997. Rays and the fixed point property in noncompact spaces. Tsukuba J Math 21: 97-112., Edelstein and Kok-Keong 1994r13 EDELSTEIN M AND KOK-KEONG T. 1994. Fixed point theorems for affine operators on vector spaces. J Math Anal Appl 181: 181-187., Floret 1980r14 FLORET K. 1980. Weakly Compact Sets. Lecture Notes in Math. 801, Springer-Verlag, Berlin? Heidelberg? New York., Klee 1955r26 KLEE VL. 1955. Some topological properties of convex sets. Trans Amer Math Soc 78: 30-45., Nowak 1979r29 NOWAK B. 1979. On the Lipschitzian retraction of the unit ball in infinite dimensional Banach spaces onto its boundary. Bull Acad Polon Sci Sr Sci Math 27: 861-864.) and references therein. One of the most emblematic results on this matter states that if is a non-compact, bounded, closed convex subset of a normed space, then there exists a Lipschitz map such that (Lin and Sternfeld 1985r27 LIN PK AND STERNFELD Y. 1985. Convex sets with the Lipschitz fixed point property are compact. Proc Amer Math Soc 93: 33-39.). Notice in this case that there is no approximate fixed point sequence for . Previous results of topological flavour were discovered by many authors, including (Klee 1955r26 KLEE VL. 1955. Some topological properties of convex sets. Trans Amer Math Soc 78: 30-45.) who has characterized the fixed point property in terms of compactness in the framework of metrizable locally convex spaces.
Both results give rise to the natural question whether a given spacewithout the fixed point property might still have the approximate fixed point property. The firstthoughts on this subject were developed in (Barroso 2009r6 BARROSO CS. 2009. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete Cont Dyn Syst 25: 467-479., Barroso and Lin 2010r5 BARROSO CS AND LIN PK. 2010. On the weak approximate fixed point property. J Math Anal Appl 365: 171-175., Barroso et al. 2012r4 BARROSO CS, KALENDA OFK AND LIN PK. 2012. On the approximate fixed point property in abstract spaces. Math Z 271: 1271-1285., 2013r3 BARROSO CS, KALENDA OFK AND REBOUÇAS MP. 2013. Optimal approximate fixed point result in locally convex spaces. J Math Appl 401: 1-8.) in the context of weak topologies. Another mathematical motivation for the study of theapproximate fixed point property is the following open question.
Question 0.1. Let be a Hausdorff locally convex space. Assume is a sequentially compact convex set and is a sequentially continuous map. Does have a fixed point?
So far, the best answers for this question were delivered in (Barroso et al. 2012r4 BARROSO CS, KALENDA OFK AND LIN PK. 2012. On the approximate fixed point property in abstract spaces. Math Z 271: 1271-1285., 2013). Let us just summarize the results.
Theorem 0.2 (Theorem 2.1, Proposition 2.5(i) in (Barroso et al. 2012r4 BARROSO CS, KALENDA OFK AND LIN PK. 2012. On the approximate fixed point property in abstract spaces. Math Z 271: 1271-1285., 2013)). Let be a topological vector space, a nonempty convex set, and let be a map.
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1. If is bounded, and is affine, then has an approximate fixed point sequence.
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2. If is locally convex and is sequentially continuous with totally bounded range, then . And indeed, has a fixed point provided that is metrizable.
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3. If is bounded and is -to- sequentially continuous, then , where is the original topology of and is a subspace of the topological dual . And, moreover, has a -approximate fixed point sequence provided that is separable under the strong topology induced by .
The idea of approximate fixed points is an old one. Apparently the first result on this kind was exploited in (Scarf 1967r34 SCARF H. 1967. The approximation of fixed points of a continuous mapping. SIAM J Applied Math 15: 1328-1343.), where a constructive method for computing fixed points of continuous mappings of a simplex into itself was described. Other important works along these lines can be found in Hazewinkel and Vel ( 1978r21 HAZEWINKEL M AND VAN DE VEL M. 1978. On almost-fixed-point theory. Canad J Math 30: 673-699.), Hadžić ( 1996r17 HADŽIĆ O. 1996. Almost fixed point and best approximations theorems in H-spaces. Bull Austral Math Soc 53: 447-454.), Idzik ( 1988r23 IDZIK A. 1988. Almost fixed point theorems. Proc Amer Math Soc 104: 779-784.), Park ( 1972r30 PARK S. 1972. Almost fixed points of multimaps having totally bounded ranges. Nonlinear Anal 51: 1-9.). Approximate fixed point property has a lot of applications in many interesting problems. In (Kalenda 2011r25 KALENDA OFK. 2011. Spaces not containing `1 have weak approximate fixed point property. J Math Anal Appl 373: 134-137.) it is proved that a Banach space has the weak-approximate fixed point property if and only if it does not contain any isomorphic copy of . As another instance, it can be used to study the existence of limiting-weak solutions for differential equations in reflexive Banach spaces (Barroso 2009r6 BARROSO CS. 2009. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete Cont Dyn Syst 25: 467-479.).
In this note, we study the existence of common approximate fixed points for a set of transformations forming a topological group. Not surprisingly, the approximate fixed point property for an acting group is also closely related to amenability of , however, the relationship appears to be more complex.
There is an extremely broad variety of known definitions of amenability of a topological group, which are typically equivalent in the context of locally compact groups yet may diverge beyond this class. One would expect the approximate fixed point property (or rather ‘‘properties,’’ for they depend on the class of convex sets allowed) to provide a new definition of amenability for some class of groups, and delineate a new class of topological groups in more general contexts. This indeed turns out to be the case.
We show that a discrete group is amenable if and only if every continuous affine action of on a bounded convex subset of a locally convex space (LCS) admits approximate fixed points. For a locally compact group, a similar result holds if we consider actions on bounded convex sets which are complete in the additive uniformity, while in general we can only prove that admits weakly approximate fixed points. This criterion of amenability is no longer true in the more general case of a Polish group, even if amenability of Polish groups can be expressed in terms of the approximate fixed point property on bounded convex subsets of the Hilbert space.
We view our investigation as only the first step in this direction, and so we close the article with a discussion of open problems for further research.
1 - AMENABILITY
Here is a brief reminder of some facts about amenable topological groups. For a more detailed treatment, see e.g. (Paterson 1988r31 PATERSON AT. 1988. Amenability. University Math. Surveys and Monographs 29, Amer Math Soc, Providence, RI.). All the topologies considered here are assumed to be Hausdorff.
Let be a topological group. The right uniform structure on has as basic entourages of the diagonal the sets of the form where is a neighbourhood of the identity in . This structure is invariant under right translations. Accordingly, a function is right uniformly continuous if for all , there exists a neighbourhood of in such that implies for every . Let denote the space of all right uniformly continuous functions equipped with the uniform norm. The group acts on on the left continuously by isometries: for all and where for all .
Definition 1.1. A topological group is amenable if it admits an invariant mean on , that is, a positive linear functional with , invariant under the left translations.
Examples of such groups include finite groups, solvable topological groups (including nilpotent, in particular abelian topological groups) and compact groups. Here are some more examples:
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The unitary group , equipped with the strong operator topology (Harpe 1973r19 DE LA HARPE P. 1973. Moyennabilité de quelques groupes topologiques de dimension infinie. CR Acad Sci Paris, Sér A 277: 1037-1040.).
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The infinite symmetric group with its unique Polish topology.
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The group of all formal power series in a variable that have the form , where is a commutative unital ring (Babenko and Bogatyi 2011r2 BABENKO I AND BOGATYI SA. 2011. The amenability of the substitution group of formal power series. Izv Math 75: 239-252.).
Let us also mention some examples of non-amenable groups:
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The free discrete group of two generators. More generally, every locally compact group containing as a closed subgroup.
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The unitary group , with the uniform operator topology (Harpe 1979r20 DE LA HARPE P. 1979. Moyennabilité du groupe unitaire et propriété de Schwartz des algèbres de von Neumann. Lecture Notes in Math 725: 220-227.).
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The group of all measure-preserving automorphisms of a standard Borel measure space , with the uniform topology, i.e. the topology determined by the metric (Giordano and Pestov 2002r15 GIORDANO T AND PESTOV V. 2002. Some extremely amenable groups. CR Acad Sci Paris Sér I 4: 273-278.).
The following is one of the main criteria of amenability in the locally compact case.
Theorem 1.2 (Følner’s condition). Let be a locally compact group and denote the left invariant Haar measure. Then is amenable if and only if satisfies the Følner condition: for every compact set and , there is a Borel set of positive finite Haar measure such that for each .
Recall that a Polish group is a topological group whose topology is Polish, i.e., separable and completely metrizable.
Proposition 1.3 (See e.g. (Al-Gadid et al. 2011r1 AL-GADID Y, MBOMBO B AND PESTOV V. 2011. Sur les espaces test pour la moyennabilité. CR Math Acad Sci Soc R Can 33: 65-77.), Proposition 3.7).
A Polish group is amenable if and only if every continuous affine action of on a convex, compact and metrizable subset of a locally convex space admits a fixed point.
For a most interesting recent survey about the history of amenable groups, see (Grigorchuk and de la Harpe in pressr15a GRIGORCHUK R AND DE LA HARPE P. IN PRESS. Amenability and Ergodic Properties of Topological groups: From Bogolyubov onwards..).
2 - GROUPS WITH APPROXIMATE FIXED POINT PROPERTY
Definition 2.1. Let be a convex bounded subset of a topological vector space . Say that a topological group has the approximate fixed point (AFP) property on if every continuous affine action of on admits an approximate fixed point net, that is, a net such that for every .
We will analyse the AFP property of various classes of amenable topological groups.
2.1 - CASE OF DISCRETE GROUPS
Theorem 2.2. The following properties are equivalent for a discrete group :
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1. is amenable,
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2. has the AFP property on every convex bounded subset of a locally convex space.
Proof.(1) (2). Let a discrete amenable group acting by continuous affine maps on a bounded convex subset of a locally convex space . Choose a Følner’s net, that is, a net of finite subsets of such that
Now, let , fix and define . Since is convex, for all . Notice that for all . Therefore we have
Thus and hence since is bounded.
(2) (1). Let be a discrete group acting continuously and by affine transformations on a nonempty compact and convex set in a locally convex space . By hypothesis, there is a net such that . By compactness of , this net has accumulation points in . Since , this insures invariance of accumulation points and shows the existence of a fixed point in . Therefore is an amenable group by Day’s fixed point theorem mentioned in the Introduction. ∎
2.2 - CASE OF LOCALLY COMPACT GROUPS
Recall from (Bourbaki 1963r7 BOURBAKI N. 1963. Intégration. Hermann, Paris.) the following notion of integration of functions with range in a locally convex space. Let be a locally convex vector space on or . denotes the dual space of and the algebraic dual of . We identify as usual (seen as a vector space without topology) with a subspace of by associating to any the linear form .
Let be a locally compact space and let a positive measure on . A map is essentially -integrable if for every element is essentially -integrable. If is essentially -integrable, then is a linear map on , i.e. an element of . The integral of is the element of denoted and defined by the condition: for every .
Note that, in general we don’t have . But we have the following.
Proposition 2.3 ((Bourbaki 1963r7 BOURBAKI N. 1963. Intégration. Hermann, Paris.), chap. 3, Proposition 7). Let be a locally compact space, a locally convex space and a function with compact support. If is contained in a complete (with regard to the additive uniformity) convex subset of , then .
Theorem 2.4. The following are equivalent for a locally compact group :
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1. is amenable,
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2. has the AFP property on every complete, convex, and bounded subset of a locally convex space.
Proof. (1) (2). Let be a locally compact amenable group acting continuously by affine maps on a complete, bounded, convex subset of a locally convex space . Again, select a Følner net of compact subsets of such that . Fix and let be the corresponding orbit map. Define . By the above, this is an element of ; the barycenter of the push-forward measure on . We have, just like in the discrete case:
Now, let be any continuous seminorm on . We have:
where since is bounded. Thus .
(2) (1). Same argument as in the case of discrete groups. ∎
The assumption of completeness of does not look natural in the context of approximate fixed points, but we do not know if it can be removed. It depends on the answer to the following.
Question 2.5. Let be an affine homeomorphism of a bounded convex subset of a locally convex space . Can be extended to a continuous map (hence, a homeomorphism) of the closure of in ?
Nevertheless, we can prove the following.
Theorem 2.6. Every amenable locally compact group has a weak approximate fixed point property on each bounded convex subset of a locally convex space .
Proof.
In the notation of the proof of Theorem 2.4, let denote the push-forward of the measure along the orbit map . Let be the barycenter of . This time, need not belong to itself, but will belong to the completion , of (the closure of in the locally convex vector space completion ).
For every , denote the barycenter of the measure . Just like in the proof of Theorem 2.4., for every we have .
Now select a net of measures with finite support on , converging to in the vague topology (Bourbaki 1963r7 BOURBAKI N. 1963. Intégration. Hermann, Paris.). Denote the barycenter of the push-forward measure . Then in the vague topology on the space of finite measures on the compact space . Clearly, , and so is well-defined and for every . It follows that weakly converges to for every . ∎
Remark 2.7. Clearly the weak AFP property on each bounded convex subset implies amenability of as well. Recall that a topological group has the weak AFP property on if every weakly continuous affine action of on admits an approximate fixed point net.
2.3 - CASE OF POLISH GROUPS
The above criteria do not generalize beyond the locally compact case in the ways one might expect: not every amenable non-locally compact Polish group has the AFP property, even on a bounded convex subset of a Banach space.
Proposition 2.8. The infinite symmetric group equipped with its natural Polish topology does not have the AFP property on closed convex bounded subsets of .
If we think of as the group of all self-bijections of the natural numbers , then the natural (and only) Polish topology on is induced from the embedding of into the Tychonoff power , where carried the discrete topology.
We will use the following well-known criterion of amenability for locally compact groups.
Theorem 2.9 (Reiter’s condition). Let be any real number with . A locally compact group is amenable if and only if for any compact set and , there exists , , , such that: for all .
Proof of Proposition 2.8. Denote the set of all Borel probability measures on , in other words, the set of positive functions such that . This is the intersection of the unit sphere of with the cone of positive elements, a closed convex bounded subset of . The Polish group acts canonically on by permuting the coordinates:
Clearly, is invariant and the restricted action is affine and continuous. We will show that the action of on admits no approximate fixed point sequence.
Assume the contrary. Make the free group act on itself by left multiplication and identify with . In this way we embed into as a closed discrete subgroup. This means that the action of by left regular representation on also has almost fixed points, and , with regard to the left regular representation of , has almost invariant vectors. But this is the Reiter’s condition () for , a contradiction with non-amenability of this group. ∎
However, it is still possible to characterize amenability of Polish groups in terms of the AFP property.
Theorem 2.10. The following are equivalent for a Polish group :
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1. is amenable,
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2. has the AFP property on every bounded, closed and convex subset of the Hilbert space.
Proof. (1) (2). It is enough to show that a norm-continuous affine action of on a bounded closed convex subset of is continuous with regard to the weak topology, because then there will be a fixed point in by Day’s theorem.
Let and be any, and let be a weak neighbourhood of in . The weak topology on the weakly compact set coincides with the topology, hence one can choose and so that whenever for all .
Denote the diameter of . Because the action is norm-continuous, we can find in so that for all . The set is a neighbourhood of in .
As a weak neighbourhood of , take the set formed by all with for all . Equivalently, the condition on can be stated for all .
If now and , one has
This shows that , and so the action of on is continuous with regard to the weak topology.
(2) (1). Suppose that acts continuously and by affine transformations on a compact convex and metrizable subset of a LCS . If is equipped with the usual norm topology, then acts continuously by affine transformations on the subspace of consisting of affine continuous functions on . Since is a metrizable compact set, the space is separable, so is the space . Fix a dense countable subgroup of , and let be a dense subset of the closed unit ball of which is -invariant. The map is an affine homeomorphism of onto a convex compact subset of . The subgroup acts continuously and by affine transformations on the affine topological copy of by, the obvious rule . The action of extends by continuity to a continuous affine action of on . By hypothesis, admits an approximative fixed point sequence in , and every accumulation point of this sequence is a fixed point since is compact. Therefore is amenable.
∎
3 - DISCUSSION AND CONCLUSION
3.1 - APPROXIMATELY FIXED SEQUENCES
As we have already noted, we do not know if every locally compact amenable group has the AFP property on all convex bounded subsets of locally convex spaces. Another interesting problem is to determine when does an acting group possess not merely an approximately fixed net, but an approximate fixed sequence.
Recall that a topological group is -compact if it is a union of countably many compact subsets. It is easy to see that if an amenable locally compact group is -compact, then it admits an approximate fixed sequence for every continuous action by affine maps on a closed bounded convex set.
Question 3.1. Let be a metrizable separable group acting continuously and affinely on a convex bounded subset of a metrizable and locally convex space. If the action has an approximate fixed net, does there necessarily exist an approximate fixed sequence?
This is the case, for example if is the union of a countable chain of amenable locally compact (in particular, compact) subgroups, and the convex set is complete.
Recall in this connection that amenability (and thus Day’s fixed point property) is preserved by passing to the completion of a topological group. At the same time, the AFP property is not preserved by completions. Indeed, the group of all permutations of integers with finite support is amenable as a discrete group, and so, equipped with any group topology, will have the AFP property on every bounded convex subset of a locally convex space. However, its completion with regard to the pointwise topology is the Polish group which, as we have seen, fails the AFP property on a bounded convex subset of .
Question 3.2. Does every amenable group whose left and right uniformities are equivalent (a SIN group) have the AFP property on complete convex sets?
3.2 - DISTAL ACTIONS
Let be a topological group acting by homeomorphisms on a compact set . The flow is called distal if whenever for some net in , then . A particular class of distal flows is given by equicontinuous flows, for which the collection of all maps forms an equicontinuous family on the compact space . We have the following fixed point theorem:
Theorem 3.3 (Hahn ( 1967r18 HAHN F. 1967. A fixed-point theorem. Math Systems Theory 1: 55-57.)). If a compact affine flow is distal, then there is a -fixed point.
An earlier result by (Kakutani 1938r24 KAKUTANI S. 1938. Fixed-point theorems concerning bicompact convex sets. Proc Imperial Acad Japan 14: 27-31.) established the same for the class of equicontinuous flows.
Question 3.4. Is there any approximate fixed point analogue of the above results for distal or equicontinuous actions by a topological group on a (non-compact) bounded convex set ?
3.3 - NON-AFFINE MAPS
Historically, Day’s theorem (and before that, the theorem of Markov and Kakutani) was inspired by the classical Brouwer fixed point theorem (Brouwer 1911r9 BROUWER LEJ. 1911. Uber Abbildung von Mannigfaltigkeiten. Math Ann 71: 97-115.) and its later more general versions, first for Banach spaces (Schauder 1930r35 SCHAUDER J. 1930. Der Fixpunktsatz in Functionalraumen. Studia Math 02: 171-180.) andlater for locally convex linear Hausdorff topological spaces (Tychonoff 1935r36 TYCHONOFF A. 1935. Ein Fixpunktsatz. Math Ann 03: 767-776.).(Recently it was extended to topological vector spaces (Cauty 2001r10 CAUTY R. 2001. Solution du problème de point fixe de Schauder. Fund Math 170: 231-246.)). The Tychonoff fixed point theorem states the following. Let be a nonvoid compact convex subset of a locally convex space and let be a continuous map. Then has a fixed point in . The map is not assumed to be affine here.
However, for a common fixed point of more than one function, the situation is completely different. Papers (Boyce 1969r8 BOYCE WM. 1969. Commuting functions with no common fixed point. Trans Amer Math Soc 137: 77-92.) and (Huneke 1969r22 HUNEKE JP. 1969. On common fixed points of commuting continuous functions on an interval. Trans Amer Math Soc 139: 371-381.) contain independent examples of two commuting maps without a common fixed point. Hence if a common fixed point theorem were to hold, there should be further restrictions on the nature of transformations beyond amenability, and for Day’s theorem, this restriction is that the transformations are affine.
The Tychonoff fixed point theorem is being extended in the context of approximate fixed points. For instance, here is one recent elegant result.
Theorem 3.5 (Kalenda ( 2011r25 KALENDA OFK. 2011. Spaces not containing `1 have weak approximate fixed point property. J Math Anal Appl 373: 134-137.)). Let be a Banach space. Then every nonempty, bounded, closed, convex subset has the weak AFP property with regard to each continuous map if and only if does not contains an isomorphic copy of .
We do not know if a similar program can be pursued for topological groups.
Question 3.6. Does there exist a non-trivial topological group which has the approximate fixed point property with regard to every continuous action (not necessarily affine) on a bounded, closed convex subset of a locally convex space? of a Banach space?
Question 3.7. The same, for the weak AFP property.
Natural candidates are the extremely amenable groups, see e.g. Pestov ( 2006r32 PESTOV V. 2006. Dynamics of Infinite-dimensional Groups: The Ramsey-Dvoretzky-Milman phenomenon. University Lecture Series, vol. 40, Amer Math Soc, Providence, RI.). A topological group is extremely amenable if every continuous action of on a compact space has a common fixed point. The action does not have to be affine, and is not supposed to be convex. This is a very strong nonlinear fixed point property.
Some of the most basic examples of extremely amenable Polish groups are:
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The unitary group with the strong operator topology (Gromov and Milman 1983r16 GROMOV M AND MILMAN VD. 1983. A topological application of the isoperimetric inequality. Amer J Math 105(4): 843-854.).
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The group of order-preserving bijections of the rational numbers with the topology induced from (Pestov 1998r33 PESTOV V. 1998. On free actions, minimal flows, and a problem by Ellis. Trans of the American Mathematical Society 350: 4149-4165.).
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The group of measure preserving transformations of a standard Lebesgue measure space equipped with the weak topology. (Giordano and Pestov 2002r15 GIORDANO T AND PESTOV V. 2002. Some extremely amenable groups. CR Acad Sci Paris Sér I 4: 273-278.).
However, at least the group does not have the AFP property with regard to continuous actions on the Hilbert space. To see this, one can use the same construction as in Proposition 2.8, together with the well-known fact that contains a closed discrete copy of .
ACKNOWLEDGMENTS
Cleon S. Barroso is currently as a visiting researcher scholar at the Texas A&M University. He takes the opportunity to express his gratitude to Prof. Thomas Schlumprecht for his support and friendship. Also, he acknowledges Financial Support form Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) by the Science Without Bordes Program, PDE 232883/2014-9. Brice R. Mbombo was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) postdoctoral grant, processo 12/20084-1. Vladimir G. Pestov is a Special Visiting Researcher of the program Science Without Borders of CAPES, processo 085/2012.
REFERENCES
-
r1AL-GADID Y, MBOMBO B AND PESTOV V. 2011. Sur les espaces test pour la moyennabilité. CR Math Acad Sci Soc R Can 33: 65-77.
-
r2BABENKO I AND BOGATYI SA. 2011. The amenability of the substitution group of formal power series. Izv Math 75: 239-252.
-
r3BARROSO CS, KALENDA OFK AND REBOUÇAS MP. 2013. Optimal approximate fixed point result in locally convex spaces. J Math Appl 401: 1-8.
-
r4BARROSO CS, KALENDA OFK AND LIN PK. 2012. On the approximate fixed point property in abstract spaces. Math Z 271: 1271-1285.
-
r5BARROSO CS AND LIN PK. 2010. On the weak approximate fixed point property. J Math Anal Appl 365: 171-175.
-
r6BARROSO CS. 2009. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete Cont Dyn Syst 25: 467-479.
-
r7BOURBAKI N. 1963. Intégration. Hermann, Paris.
-
r8BOYCE WM. 1969. Commuting functions with no common fixed point. Trans Amer Math Soc 137: 77-92.
-
r9BROUWER LEJ. 1911. Uber Abbildung von Mannigfaltigkeiten. Math Ann 71: 97-115.
-
r10CAUTY R. 2001. Solution du problème de point fixe de Schauder. Fund Math 170: 231-246.
-
r11DAY M. 1961. Fixed-point theorems for compact sets. Illinois J Math 5: 585-590.
-
r12DOBROWOLSKI T AND MARCISZEWSKI W. 1997. Rays and the fixed point property in noncompact spaces. Tsukuba J Math 21: 97-112.
-
r13EDELSTEIN M AND KOK-KEONG T. 1994. Fixed point theorems for affine operators on vector spaces. J Math Anal Appl 181: 181-187.
-
r14FLORET K. 1980. Weakly Compact Sets. Lecture Notes in Math. 801, Springer-Verlag, Berlin? Heidelberg? New York.
-
r15GIORDANO T AND PESTOV V. 2002. Some extremely amenable groups. CR Acad Sci Paris Sér I 4: 273-278.
-
r15aGRIGORCHUK R AND DE LA HARPE P. IN PRESS. Amenability and Ergodic Properties of Topological groups: From Bogolyubov onwards.
-
r16GROMOV M AND MILMAN VD. 1983. A topological application of the isoperimetric inequality. Amer J Math 105(4): 843-854.
-
r17HADŽIĆ O. 1996. Almost fixed point and best approximations theorems in H-spaces. Bull Austral Math Soc 53: 447-454.
-
r18HAHN F. 1967. A fixed-point theorem. Math Systems Theory 1: 55-57.
-
r19DE LA HARPE P. 1973. Moyennabilité de quelques groupes topologiques de dimension infinie. CR Acad Sci Paris, Sér A 277: 1037-1040.
-
r20DE LA HARPE P. 1979. Moyennabilité du groupe unitaire et propriété de Schwartz des algèbres de von Neumann. Lecture Notes in Math 725: 220-227.
-
r21HAZEWINKEL M AND VAN DE VEL M. 1978. On almost-fixed-point theory. Canad J Math 30: 673-699.
-
r22HUNEKE JP. 1969. On common fixed points of commuting continuous functions on an interval. Trans Amer Math Soc 139: 371-381.
-
r23IDZIK A. 1988. Almost fixed point theorems. Proc Amer Math Soc 104: 779-784.
-
r24KAKUTANI S. 1938. Fixed-point theorems concerning bicompact convex sets. Proc Imperial Acad Japan 14: 27-31.
-
r25KALENDA OFK. 2011. Spaces not containing `1 have weak approximate fixed point property. J Math Anal Appl 373: 134-137.
-
r26KLEE VL. 1955. Some topological properties of convex sets. Trans Amer Math Soc 78: 30-45.
-
r27LIN PK AND STERNFELD Y. 1985. Convex sets with the Lipschitz fixed point property are compact. Proc Amer Math Soc 93: 33-39.
-
r28MARKOV A. 1936. Quelques théorèmes sur les ensembles abéliens. Dokl Akad Nauk SSSR (NS:) 10: 311-314.
-
r29NOWAK B. 1979. On the Lipschitzian retraction of the unit ball in infinite dimensional Banach spaces onto its boundary. Bull Acad Polon Sci Sr Sci Math 27: 861-864.
-
r30PARK S. 1972. Almost fixed points of multimaps having totally bounded ranges. Nonlinear Anal 51: 1-9.
-
r31PATERSON AT. 1988. Amenability. University Math. Surveys and Monographs 29, Amer Math Soc, Providence, RI.
-
r32PESTOV V. 2006. Dynamics of Infinite-dimensional Groups: The Ramsey-Dvoretzky-Milman phenomenon. University Lecture Series, vol. 40, Amer Math Soc, Providence, RI.
-
r33PESTOV V. 1998. On free actions, minimal flows, and a problem by Ellis. Trans of the American Mathematical Society 350: 4149-4165.
-
r34SCARF H. 1967. The approximation of fixed points of a continuous mapping. SIAM J Applied Math 15: 1328-1343.
-
r35SCHAUDER J. 1930. Der Fixpunktsatz in Functionalraumen. Studia Math 02: 171-180.
-
r36TYCHONOFF A. 1935. Ein Fixpunktsatz. Math Ann 03: 767-776.
Publication Dates
-
Publication in this collection
Mar 2017
History
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Received
25 June 2015 -
Accepted
10 Oct 2016