A fixed point theorem and the existence of a Haar measure for hypergroups satisfying conditions related to amenability
aa r X i v : . [ m a t h . F A ] O c t A FIXED POINT THEOREM AND THE EXISTENCE OF AHAAR MEASURE FOR HYPERGROUPS SATISFYINGCONDITIONS RELATED TO AMENABILITY
B. WILLSON
Abstract.
In this paper we present a fixed point property for amenable hy-pergroups which is analogous to Rickert’s fixed point theorem for semigroups.It equates the existence of a left invariant mean on the space of weakly rightuniformly continuous functions to the existence of a fixed point for any ac-tion of the hypergroup. Using this fixed point property, a certain class ofhypergroups are shown to have a left Haar measure. Introduction
Hypergroups arise as generalizations of the measure algebra of a locally compactgroup wherein the product of two points is a probability measure rather than a singlepoint. The formalization of hypergroups was introduced in the 1970s by Jewett [5],Dunkl [3], and Spector [10]. Actions of hypergroups have been considered in [8, 11].Amenable hypergroups have been considered in [9, 6, 12, 1]. As with groups, thereare connections between invariant means on function spaces and fixed points ofactions of the hypergroup.It is a longstanding problem whether or not every hypergroup has a left Haarmeasure. It is known that if a hypergroup is compact, discrete, or abelian then itdoes admit a Haar measure.Every locally compact group G admits a left Haar measure λ G . This Haarmeasure can be viewed as a positive, linear, (unbounded), left translation invariantfunctional on C C ( G ). It follows from the existence of this functional that for positive f ∈ C C ( G ) and positive µ, ν ∈ M ( G ) then(1.1) µ ∗ f ( x ) ≤ ν ∗ f ( x ) ∀ x ∈ G ⇒ k µ k ≤ k ν k . A similar statement can be made for amenable groups and positive, continuous,bounded functions with non-zero mean value. One might expect this statement tobe true more generally, but there are examples of unbounded functions and boundedfunctions on non-amenable groups for which (1.1) does not hold.The main result of this paper is to show that for amenable hypergroups, theexistence of a Haar measure is equivalent to this positivity of translation property(1.1). In section 2, we provide the neccessary preliminaries. In section 3, we prove afixed point theorem for the action of an amenable hypergroup. In the final section,we use property (1.1) to prove the existence of a measure which is invariant for all
Mathematics Subject Classification.
Primary 43A62, Secondary 43A05, 43A07.
Key words and phrases. invariant measure, hypergroup, amenable, function translations.The author gratefully acknowledges the financial support of Hanyang University. translations of f . We build on this, and use the fixed point theorem to prove thefinal result. 2. Background and Definitions
Definition 2.1.
A hypergroup, H , is a non-empty locally compact Hausdorff topo-logical space which satisfies the following conditions (see [2] for more details onhypergroups):(1) There is a binary operation ∗ , called convolution, on the vector space ofbounded Radon measures turning it into an algebra.(2) For x, y ∈ H , the convolution of the two point measures is a probabilitymeasure, and supp( δ x ∗ δ y ) is compact.(3) The map ( x, y ) δ x ∗ δ y from H × H to the compactly supported probabilitymeasures on H is continuous.(4) The map H × H ∋ ( x, y ) supp( δ x ∗ δ y ) is continuous with respect to theMichael topology on the space of compact subsets of H .(5) There is a unique element e ∈ H such that for every x ∈ H , δ x ∗ δ e = δ e ∗ δ x = δ x .(6) There exists a homeomorphism ˇ: H → H such that for all x ∈ H , ˇˇ x = x ,which can be extended to M ( H ) via ˇ µ ( A ) = µ ( { x ∈ H : ˇ x ∈ A } ), and suchthat ( µ ∗ ν )ˇ= ˇ ν ∗ ˇ µ .(7) For x, y ∈ H , e ∈ supp( δ x ∗ δ y ) if and only if y = ˇ x . Definition 2.2.
We denote the continuous and bounded complex-valued functionson H by C ( H ). For the compactly supported continuous functions we use C C ( H ).We use C ( H ) + , C C ( H ) + to denote the functions in those spaces which take onlynon-negative real values. We denote the complex space of bounded regular Borelmeasures on H by M ( H ), those that are compactly supported measures by M C ( H ),and those that are real valued measures by M ( H, R ). Definition 2.3.
A left Haar measure for H is a non-zero regular Borel measure(with values in [0 , ∞ ]), λ which is left-invariant in the sense that for any f ∈ C C ( H ),we have that λ ( δ x ∗ f ) = λ ( f ) for all x ∈ H . Remark 2.4.
It remains an open question whether every hypergroup admits a leftHaar measure. If H does admit a left Haar measure λ , however, it is unique upto a scalar multiple[5]. For hypergroups with a left Haar measures we are able todefine the standard L p ( H ) function spaces. Definition 2.5.
We say that a continuous function f ∈ C ( H ) is right uniformlycontinuous [weakly right uniformly continuous] if the map H ∋ x δ x ∗ f is continuous in norm [weakly]. We denote the collection of right uniformly continu-ous functions [weakly right uniformly continuous] on H by U CB r ( H ) [ W U CB r ( H )]. Remark 2.6.
Skantharajah [9] showed that for hypergroups with left Haar mea-sure,
U CB r ( H ) = L ( H ) ∗ L ∞ ( H ). IXED POINT THEOREM AND HAAR MEASURE FOR CERTAIN HYPERGROUPS 3 Fixed point property
Let H be a hypergroup. Let E be a Hausdorff locally convex vector space and let K ⊂ E be a compact, convex subset. Suppose that there is a separately continuousmapping · : H × K → K . Then for x, y ∈ H and ξ ∈ K the weak integral Z H ( t · ξ ) d ( δ x ∗ δ y )( t )exists uniquely in K . Definition 3.1. A separately continuous [jointly continuous] action of H on K isa separately continuous [jointly continuous] mapping · : H × K → K such that(1) e · ξ = ξ for all ξ ∈ K ;(2) x · ( y · ξ ) = R H ( t · ξ ) d ( δ x ∗ δ y )( t ).Furthermore, the action is called affine if, for each x ∈ H ξ x · ξ is affine.The following theorem is similar to a result of Rickert for amenable semigroupsas presented in [7]. Theorem 3.2.
Suppose that
U CB r ( H ) has a left invariant mean m . Then foreach jointly continuous affine action of H on some K , a compact convex subset ofa Hausdorff locally convex vector space, there is a point ξ ∈ K such that x · ξ = ξ for all x ∈ H . Furthermore, the result holds if U CB r ( H ) is replaced by W U CB r ( H ) and jointly continuous is replaced by separately continuous.Proof. Suppose that there is a hypergroup action of H on K . We denote the set ofaffine functions on K by Af f ( K ). It is clear that for each point ξ ∈ K , evaluationat ξ is a mean on Af f ( K ). Indeed, K can be identified with the collection of all means on Af f ( K ) and this identification is an affine homeomorphism. (See forinstance [7] 2.20)Given this identification, we see that the existence of a fixed point in K isequivalent to the existence of a mean on Af f ( K ) which is invariant under theaction of H .Suppose that φ ∈ Af f ( K ) and ξ ∈ K . Let ˆ ξ ( φ ) ∈ C ( H ) be defined byˆ ξ ( φ )( x ) = φ ( x · ξ ). It is clear that ˆ ξ ( φ ) is a continuous function provided thatthat action of H is (separately) continuous. We further claim that if the action isseparately continuous then ˆ ξ ( φ ) is in W U CB r ( H ) and if it is jointly continuousthen ˆ ξ ( φ ) is in U CB r ( H ).To show the first claim, consider some mean F on C ( H ) and some net x α → x in H . Then F ◦ ˆ ξ : Af f ( K ) → C is a mean on Af f ( K ). It follows then, that thereis some ξ ∈ K such that F ◦ ˆ ξ ( φ ) = φ ( ξ ). Therefore, h F, δ ˇ x α ∗ ˆ ξ ( φ ) i = h F ◦ ˆ ξ , δ ˇ x α ∗ φ i = δ ˇ x α ∗ φ ( ξ )= φ ( x α · ξ ) → φ ( x · ξ )= h F, δ ˇ x ∗ ˆ ξ ( φ ) i From this we conclude that for any F ∈ C ( H ) ∗ the same holds. That is, thatˆ ξ ( φ ) is in W U CB r ( H ). Now since there is a left invariant mean M on W U CB r ( H ) B. WILLSON it follows that M ◦ ˆ ξ is a mean on Af f ( K ) which corresponds to evaluation atsome point ξ ∈ K . It is apparent that ξ is a fixed point of the action of H .For the second claim, observe that k δ ˇ x α ∗ φ − δ ˇ x ∗ φ k → x α → x since theaction is jointly continuous. Furthermore, since ˆ ξ is contractive for each ξ ∈ K , k δ ˇ x α ∗ ˆ ξ ( φ ) − δ ˇ x ∗ ˆ ξ ( φ ) k → x α → x . But this shows that ˆ ξ ( φ ) is in U CB r ( H )as required. By a similar argument as above, the mean on U CB r ( H ) generates afixed point in K . (cid:3) Existence of a left Haar measure
The existence of a Haar measure on an arbitrary hypergroup is still an openquestion. In this section we present an approach motivated in part by a result ofIzzo[4] which uses the Markov-Kakutani fixed point theorem to prove the existenceof a Haar measure on abelian groups. We use the fixed point theorem (3.2) fromthe previous section to show that amenable hypergroups which satisfy an additionalpositivity property of translations have a left Haar measure. This property is relatedto amenability in the following sense.
Remark 4.1.
Every locally compact group G admits a left Haar measure. Someof the key properties of the Haar measure are as follows. For any f, g ∈ C C ( G ) and µ ∈ M ( G ) we have f ( x ) ≤ g ( x ) ∀ x ∈ G ⇒ Z G f ( x ) dλ ( x ) ≤ Z G g ( x ) dλ ( x ) , and Z G ( µ ∗ f )( x ) dλ ( x ) = µ ( H ) Z G f ( x ) dλ ( x ) . Hence for every non-zero f ∈ C C ( G ) + and µ, ν ∈ M ( G ) + if µ ∗ f ≤ ν ∗ f then k µ k ≤ k ν k .Relatedly, if G is amenable with left invariant mean m ∈ C ( G ) ∗ then for every f ∈ C ( G ) + with m ( f ) > µ, ν ∈ M ( G ) + if µ ∗ f ≤ ν ∗ f then k µ k ≤ k ν k . Definition 4.2.
Let H be a hypergroup. Fix some non-zero f ∈ C C ( H ) + . We saythat H has the positivity property of translations of f if for every µ, ν ∈ M + ( H )(4.1) µ ∗ f ≤ ν ∗ f ⇒ k µ k ≤ k ν k . Lemma 4.3.
Suppose that H is a hypergroup with property (4.1) for some non-zero f ∈ C C ( H ) + .Then there is a positive, linear functional Γ from the real-vector space C C ( H, R ) to R such that Γ( f ) = 1 and Γ( ρ ∗ f ) = ρ ( H ) for ρ ∈ M C ( H, R ) .Proof. Let V f := { µ ∗ f : µ ∈ M C ( H, R ) } . Since M C ( H, R ) is a vector space, itfollows that V f is also a vector space. Define the linear functional Γ f on V f byΓ f ( µ ∗ f ) = µ ( H ) . Claim: This map is well defined.If µ ∗ f = ν ∗ f then consider a Jordan decomposition, ( µ + − µ − ) ∗ f = ( ν + − ν − ) ∗ f .Then ( µ + µ − + ν − ) ∗ f = ( ν + ν − + µ − ) ∗ f and these are both positive with equalnorm by property (4.1) hence µ ( H ) = ν ( H ). Therefore the map Γ f is well defined.Claim: This map is positive. IXED POINT THEOREM AND HAAR MEASURE FOR CERTAIN HYPERGROUPS 5
By a similar argument, if 0 ≤ ν ∗ f then ν − ∗ f ≤ ν + ∗ f hence ν − ( H ) ≤ ν + ( H )and so ν ( H ) = ν + ( H ) − ν − ( H ) is positive.By [2][Lemma 1.2.22] for any g ∈ C C ( H, R ) there is a µ ∈ M ( H, R ) + C such that g ≤ µ ∗ f . Hence we can apply M. Riesz’s extenstion theorem to extend Γ f to apositive linear functional on all of C C ( H, R ). (cid:3) The above lemma shows that the collection of positive linear functionals on C C ( H ) sending all translations of the function f by elements of H to 1 is non-empty.By applying the fixed point theorem to this collection, we prove the existence ofa non-zero positive linear functional on C C ( H ) which is fixed under the action oftranslation by elements of the hypergroup. This non-zero positive linear functionalis precisely a Haar measure. Corollary 4.4.
Suppose that H has property (4.1) for some f as above. Then thecollection K of all positive linear functionals Λ on C C ( H ) satisfying: Λ( µ ∗ f ) = µ ( H ) for every µ ∈ M C ( H ) . is non-empty.Proof. It suffices to use the complexification of Γ from lemma (4.3). (cid:3)
Theorem 4.5.
Suppose H has property (4.1) . Suppose also that W U CB r ( H ) hasa left invariant mean. Then H has a left Haar measure.Proof. Let K be the collection of all positive linear functionals on C C ( H ) satisfying:Λ( µ ∗ f ) = µ ( H ) for every µ ∈ M C ( H ) . We equip K with the weak* topology induced by C C ( H ). If Λ α → Λ and eachΛ α ∈ K then Λ is in K and so K is closed. For g ∈ C C ( H ) there is a µ g ∈ M ( H ) + such that | g | ≤ µ g ∗ f . Subsequently for any Λ ∈ K , | Λ( g ) | ≤ Λ( | g | ) ≤ Λ( µ g ∗ f ) ≤k µ g k . Hence K is compact.To see that K is convex, suppose that Λ , Λ ∈ K and 0 ≤ c ≤
1. Then( c Λ + (1 − c )Λ )( µ ∗ f ) = c + 1 − c = 1 so ( c Λ + (1 − c )Λ ) is an element of K .It now suffices to show that left translation is an action of H on K and thatthis action is separately continuous. From this we can apply Theorem 3.2 to find apoint in K which is fixed under left translation.For x ∈ H , g ∈ C C ( H ) and Λ ∈ K we define x · Λ( g ) = Λ( δ ˇ x ∗ g ). Then e · Λ = Λ and x · Λ( µ ∗ f ) = Λ (( δ ˇ x ∗ µ ) ∗ f ) = 1 so K is closed under the action of H . Furthermore x · ( y · Λ)( g ) = y · Λ( δ ˇ x ∗ g )= Λ( δ ˇ y ∗ δ ˇ x ∗ g )= Z Λ( δ ˇ t ∗ g ) d ( δ x ∗ δ y )( t ))= Z t · Λ( g ) d ( δ x ∗ δ y )( t ) . To see that this action is separately continuous, consider a net x α → x ∈ H .Eventually x α will stay within a compact neighbourhood N of x . Then for Λ ∈ K and g ∈ C C ( H ) it suffices to consider the restriction Λ | ˇ N ∗ supp( g ) of Λ which is a B. WILLSON finite measure on the compact set ˇ N ∗ supp( g ). This allows us to consider the righttranslate of g by this measure (which is a continuous function) and we get x α · Λ( g ) = Λ( δ ˇ x α ∗ g )= Λ | ˇ N ∗ supp( g ) ( δ ˇ x α ∗ g )= δ x α ( g ∗ ˇΛ | ˇ N ∗ supp( g ) ) → δ x ( g ∗ ˇΛ | ˇ N ∗ supp( g ) )= x · Λ( g ) . Now suppose there is a net Λ β → Λ ∈ K . So for x ∈ H and f ∈ C C ( H ) x · Λ β ( f ) = Λ β ( δ ˇ x ∗ f ) → Λ( δ ˇ x ∗ f )= x · Λ( f ) . Therefore the action is separately continuous.By Theorem 3.2 there is a fixed point Λ in K under the action of H . This fixedpoint is a positive linear functional on C C ( H ) satisfyingΛ ( g ) = Λ ( δ x ∗ g ) ∀ g ∈ C C ( H ) , x ∈ H. It is non-zero since Λ ( f ) = 1 so Λ is a left Haar measure for H . (cid:3) Corollary 4.6.
Let H be a hypergroup with a left translation invariant mean on W U CB r ( H ) . The following are equivalent:(1) There is a left Haar measure on H .(2) The property of positivity of translations (4.1) holds for every f ∈ C C ( H ) + .(3) The property of positivity of translations (4.1) holds for some nonzero f ∈ C C ( H ) + .Acknowledgements. Thanks to Hanyang University for providing funding in theform of a research fund for new professors.
References
1. M. Alaghmandan,
Amenability notions of hypergroups and some applications to locally com-pact groups , ArXiv e-prints (2014).2. W. R. Bloom and H. Heyer,
Harmonic analysis of probability measures on hypergroups , deGruyter Studies in Mathematics, vol. 20, Walter de Gruyter & Co., Berlin, 1995.3. C. F. Dunkl,
The measure algebra of a locally compact hypergroup , Trans. Amer. Math. Soc. (1973), 331–348.4. A. J. Izzo,
A functional analysis proof of the existence of Haar measure on locally compactabelian groups , Proc. Amer. Math. Soc. (1992), no. 2, 581–583.5. R. I. Jewett,
Spaces with an abstract convolution of measures , Advances in Math. (1975),no. 1, 1–101.6. R. Lasser and M. Skantharajah, Reiter’s condition for amenable hypergroups , Monatsh Math(2010), 1–12.7. A. L. T. Paterson,
Amenability , Mathematical Surveys and Monographs, vol. 29, AmericanMathematical Society, Providence, RI, 1988.8. Ajit Iqbal Singh,
Completely positive hypergroup actions , vol. 593, American MathematicalSoc., 1996.9. M. Skantharajah,
Amenable hypergroups , Illinois J. Math. (1992), no. 1, 15–46.10. R. Spector, Mesures invariantes sur les hypergroupes , Trans. Amer. Math. Soc. (1978),147–165.
IXED POINT THEOREM AND HAAR MEASURE FOR CERTAIN HYPERGROUPS 7
11. Nazanin Tahmasebi,
Hypergroups and invariant complemented subspaces , Journal of Mathe-matical Analysis and Applications (2014), no. 2, 641–655.12. Benjamin Willson,
Configurations and invariant nets for amenable hypergroups and relatedalgebras , Trans. Amer. Math. Soc. (2014), no. 10, 5087–5112. MR 3240918
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