WEAK CONTAINMENT AND HYPERGROUP ALGEBRAS
LILIANA PAVEL
LetK be a hypergroup with Haar measure. We consider the enveloping algebra C∗(K), the reduced algebra Cr∗(G), and an appropriate operator algebra gene- rated by the left regular representationC∗(λK), when the representations ofK satisfy certain weak containment conditions. We show thatC∗(K) andCr∗(K) are
*-isomorphic if and only ifK has the weak containment property. If the trivial representation is weakly contained in the left regular representation, then the existence of a nonzero multiplicative linear functional onC∗(λK) is proved.
AMS 2000 Subject Classification: Primary 43A62; Secondary 43A65, 43A07.
Key words: Hypergroup, weak containment, left regular representation, envelop- ing algebra, reduced algebra.
LetG be a locally compact group.The representation π of G is weakly contained in the representation τ if all positive definite functions associated withπare uniform limits on the compact subsets ofKof sums of positive func- tions associated withτ. Gis is said to have theweak containment property if any irreducible unitary representation of the group is weakly contained in the left regular representation. G is called amenable if there exists an invariant mean on the space of all bounded continuous functions onG. It is known that amenabilty, the fact that the trivial representation is weakly contained in the left regular representation, and the weak containment property are equivalent.
Based on this fact, characterizations of the amenability of a group have been given by means of the C∗-algebras naturally associated in the representation theory with a locally compact group: the enveloping algebra C∗(G), the re- duced algebra Cr∗(G) and the algebra C∗(λ(G)) generated by the operators of the left regular representation. More precisely, it is well known that G is amenable if and only ifC∗(G) and Cr∗(G) are *-isomophic (see for example [12]). Regarding C∗(λ(G)), in [1] it was proved that G is amenable if and only if there is a nonzero multiplicative linear functional on thisC∗-algebra.
Related results were obtained in [3].
REV. ROUMAINE MATH. PURES APPL.,52(2007),1, 87–93
Hypergroups are locally compact spaces whose bounded Radon measures form an algebra which has similar properties to the convolution measures al- gebra of a locally compact group. In particular, any locally compact group is a hypergroup. Consequently, many basic notions and results from the group theory have been transferred to hypergroups. The notion of representation weakly contained in another can be literally adapted from the group case.
Skantharajah [13] initiated the study of amenable hypergroups, extending the definition from groups. He discovered that despite the apparent analogy of this topic to groups, there are substantial differences, especially in connection with its relationship with the representation theory. One of the main differ- ence is that amenability is involved by the fact that the trivial representation is weakly contained in the left regular representation, but unlike the group case, these properties are not equivalent. As this equivalence is the main tool in the C∗-algebras characterizations of the amenablity of groups, it is not surprising that the study of the hypergroup algebras corresponding toC∗(G),Cr∗(G) and C∗(λ(G)) leads only to partial answers concerning the amenability of the hy- pergroup. We shall prove that the enveloping algebraC∗(K) is *-isomorphic to the reduced algebra Cr∗(K) if and only if the hypergroup K satisfies the weak containment property. This condition in the hypergroup setting is in general much stronger than amenability. We also shall prove that for hyper- groups with the property that the trivial representation is weakly contained in the left regular representation, there exists a multiplicative functional on an appropriate C∗-algebra generated by certain operators of the left regular representation, thus partially generalizing the results of [1], [3].
1. PRELIMINARIES
For basic definitions and results on hypergroups, we shall follow [9]. K allways stands for a hypergroup with a fixed left Haar measurem. ∆ is the modular function ofm. Symbols like
. . .dxwill allways denote the integra- tion with respect tom. The notation generally agrees with [9]. However, the following notation is different from [9]: x → x∨ denotes the involution on K and δx the Dirac measure concentrated at x.We recall thatM(K) is the al- gebra of all bounded regular (complex valued) Borel measures onK and that M+(K) is its part consisting of positive measures. In addition, we use the notationMPc(K) for the probability measures on K with compact support.
Iff is a Borel function onKandx,y∈K, then the left translatefx is defined on K by fx(y) =
fdδx∗δy =f(x∗y), if the integral exists. The function f∼ on K is given byf∼(x) =f(x∨). If µ∈ M(K) and f is a Borel function, then the convolution µ∗f is defined onK by (µ∗f)(x) =
f(y∨∗x)dµ(y).
It is immediate thatδx∨∗f =fx for each x inK and f a Borel function.
Similarly to the locally compact group case, L1(K) equipped with the convolution, (f, g) → f ∗g, where (f ∗g)(x) =
f(x∗y)g(y∨)dy, ∀x ∈ K, and the involutionf →f∗, withf∗= (∆f)∼, is an involutive Banach algebra admitting approximate units.
Following [9], we define a representation of K as a norm increasing ∗- representation of the Banach∗-algebra M(K). Note that ifπ is a representa- tion ofK in some Hilbert space Hπand µ∈ M(K), then π(µ) is a bounded linear operator on Hπ with π(µ) ≤ µ. For notational convenience, we write πx forπ(δx), where x∈K. Just as for the locally compact group case, to any representation π of K and any vector ξ in Hπ, there corresponds a continuous positive definite function, necessarily bounded,x→ πxξ, ξ . The trivial representation and the left regular representation of K on L2(K) will be denoted byiKandλK, respectively; soiK(µ)(f) =f andλK(µ)(f) =µ∗f for eachf ∈L2(K) andµ∈ M(K). We will denote byB(H) the C∗-algebra of bounded linear operators on the Hilbert spaceH;B(L2(K)) will be of par- ticular interest.
2. THE RESULTS
Let Σ be the family of representations of K and S a part of Σ. For µ∈ M(K) define, as in the group case,
µS = sup
π∈Sπ(µ).
In the above formulaπ(µ) is the norm of the operatorπ(µ) in B(Hπ).
The next properties of · S are obvious:
(1) µ→ µS is a seminorm on M(K); · S is a norm if S contains a faithful representation ofK;
(2) for anyµ,ν ∈ M(K) we have
µ∗νS ≤ µS· νS, µ∗S ≤ µS, µ∗µ∗S ≤ µ2S, µS ≤ µ; (3) for anyf ∈ L1(K) andx∈K, we have
fxS ≤ fS.
We notice that in the particular case of locally compact group case, obviously, fxS =fS for anyf ∈ L1(G),x∈ Gand S ⊆Σ.
There are two situations we will be interested in: S =Σ andS ={λ}. WhenS =Σ, the completion of the Banach involutive algebra admitting approximate unitsL1(K) under the norm · Σ (see [4, 2.7]) has been widely used in the representation theory of hypergroups: it is the exact analogue of the enveloping algebra of a locally compact group, consequently denoted byC∗(K) and called the enveloping algebra of K. The (irreducible) representations
of K are in one-to-one correspondence with the non-degenerate (irreducible) representations of L1(K), hence of C∗(K). Thus, we are enabled to identify the dualK∧ withC∗(K)∧, each of the sets being endowed with the hull kernel topology [6]. Corresponding representations of the objectsK,C∗(K), will be denoted by the same letter. Forπ,τ ∈K∧ we say thatπ is weakly contained inτ if all positive functionals associated withπ can be weakly∗-approximated by sums of positive functionals associated with τ, or, equivalently, in terms of representations of K, if all positive definite functions associated with π are uniform limits on the compact subsets of K of sums of positive definite functions associated with τ. This means thatπ is in the closure with respect to the hull kernel topology of {τ}; in this case, we write π τ. Recall that suppτ ={π∈K∧ |πτ}.
When S={λK}, as the left regular representation is faithful, · {λK}is a norm, further denoted by · r. In analogy with groups we state
Definition 1. The completion of L1(K) under the norm · r is called thereduced algebraCr∗(K) of K.
Remark. From the point of view of representation theory, similarly to the locally compact group context, the algebra C∗(K) is the natural C∗-agebra associated with K; its norm is the greatest C∗-norm on L1(K). However, the reduced norm onL1(K), fr =λK(f), that isfr = sup{f ∗ϕ2 | ϕ2 = 1}, yields theC∗-algebra norm, which is easiest to compute.
The next theorem describes how the reduced algebra of a hypergroup can be obtained from its enveloping algebra, thus generalizing the group approach ([5, (1.15)]).
Theorem1. Let K be a hypergroup and I : (L1(K), · Σ) →(L1(K), · r) the identity mapping. Then
(i) I extends to a morphism I from C∗(K) onto Cr∗(K);
(ii) KerI={f ∈C∗(K)|λK(f) = 0};
(iii) I is injective if and only if K has the weak containment property.
Proof. (i) Clearly, sincefr ≤ fΣ for anyf ∈L1(K), the linear map- pingI : (L1(K), · Σ) →(L1(K), · r), I(f) = f, is continuous on L1(K).
AsL1(K) is dense in C∗(K), it follows that I can be extended by continuity to a bounded linear mapping on C∗(K), I. In addition, I is surjective be- cause its immage, containing L1(K), is dense in Cr∗(K), consequently, by [4, Corollary 1.8.3], it is closed.
(ii) Let f ∈ C∗(K) such that λK(f) = 0, equivalently, fr = 0. As f is in the completion of L1(K) under the norm · Σ, there exists a sequence (fn)ninL1(K) such thatf−fnΣ →n 0. It follows thatf−fnr →n 0. Then,
using the inequality
fnr≤ f−fnr+fr
we getfnr=I(fn)r →n 0. It follows that I(f) =I
limn fn
= lim
n I(fn) = 0,
which shows that{f ∈C∗(K)|λK(f) = 0} ⊆KerI. Conversely, suppose that I(f) = 0 withf ∈C∗(K). For some sequence (fn)n inL1(K) with f−fnΣ
→n 0 we have I(f)−I(fn)r →n 0 and, asI(f) = 0, fnr =I(fn)r →n 0.
Then, since
λK(f) ≤ λK(fn)−λK(f)+λK(fn)), we deduce thatλK(f) = 0.
(iii) This follows immediately from (ii). Indeed, by (ii), I is injective if and only if KerλK ={0}, so if and only if the kernel of each representation of K contains the kernel of the left regular representation, which means ([6], Theorem 1.2) thatK has the weak containment property.
Corollary1. The C∗-algebras C∗(K) and Cr∗(K) are *-isomorphic if and only if K has the weak containment property.
Proof. By Theorem 1, K has the weak containment property ⇔ I is injective, so, by [4, Proposition 1.8.1],I is an isometry.
Remark. This result shows that C∗(K) and Cr∗(K) can be identified for hypergroups possessing the weak containment property. We mention that this class is smaller than the class of amenable hypergroups. Skantharajah [13]
gives as example of amenable hypergroup with iK ∈/ suppλK the so-called Naimark example ([9, 9.5]). Moreover, as far as we know, it seems not to be known yet if the class of hypergroups satisfying iK λK coincides with the class of hypergroups having the weak containment property [15].
Corollary 2. Let K be a hypergroup with the weak containment pro- perty. Then
fr=fΣ=f1, ∀f ∈L1(K), f ≥0.
Proof. This is immediate, combining Corollary 1 with λK(µ) = µ for eachµ∈ M+(K) whenever ιK λK [13, Lemma 4.4].
For the group case a characterization of amenability (see [1] and [3] for a refinement of the result) is given by the existence of a nonzero multiplicative linear functional on C∗(λ(G)), the C∗-subalgebra of B(L2(G)) generated by λ(G), the image in B(L2(G)) of the left regular representation of the group.
C∗(λ(G)) is called the algebra generated by the left regular representation.
Further, we consider an appropriate generalization of this algebra to the con- text of hypergroups.
Definition 2. TheC∗-subalgebra ofB(L2(K) generated by {λK(µ)|µ∈ MPc(K)}is called the algebra generated by the left regular representation of K,C∗(λK).
Remark. Clearly, in the special case where the hypergroup is a lo- cally compact groupG, the above definedC∗(λG) contains C∗(λ(G)). Conse- quently, the next theorem partially extends the main result of [1, Theorem 1]
from amenable locally compact groups to hypergroups satisfying the condition ιK λK.
Theorem 2. Let K be a hypergroup such that ιK λK. Then there exists a nonzero multiplicative linear functional F on C∗(λK).
Proof. Assume that ιK λK. Then by [13, Lemma 4.4] for anyε > 0 and a compact subset E of K there exists a function ϕε,E ∈ L2(K), with ϕε,E2 = 1, such that δx ∗ϕε,E −ϕε,E2 < εfor everyx ∈ E. Define the functionalFε,E on C∗(λK) by
Fε,E(λ(µ)) =λK(µ)ϕε,E, ϕε,E =
Kδx∗ϕε,E−ϕε,E, ϕε,E dµ(x),
∀µ ∈ MPc(K). As |Fε,E(λK(µ))| ≤ λK(µ) = µ by [13, Lemma 4.4], and as µ = 1, it follows that Fε,E = 1, so that the functionals Fε,E are uniformly bounded. In addition, for each µ∈ MPc(K) we have
|Fε,E(λK(µ))−1|=|λK(µ)ϕε,E, ϕε,E − ϕε,E, ϕε,E |=
=|λK(µ)ϕε,E−ϕε,E, ϕε,E |=
Kδx∗ϕε,E−ϕε,E, ϕε,Ei dµ(x)≤
≤
K|δx∗ϕε,E−ϕε,E, ϕε,E |dµ(x)≤ε.
The family of the functionals Fε,E is a uniformly bounded net in C∗(λK)∗ (the subscripts (ε, E) are partially ordered by (ε, E) ≤ (ε, E) if ε ≤ ε and E⊆E). By the Banach-Alaoglu theorem, (Fε,E)ε,E is weak∗-compact, hence we may take a weak∗-limit point of this net, say, F. Clearly, F is linear and continuous. In addition, F(λK(µ)) = 1, ∀µ ∈ MPc(K). It follows that F(λK(µ)◦λK(ν)) =F(λK(µ∗ν)) =F(λK(µ))·F(λK(ν)),∀µ,ν ∈ MPc(K).
Consequently, as C∗(λK) is generated by {λK(µ) | µ ∈ MPc(K)}, F is a nonzero multiplicative linear functional onC∗(λK).
REFERENCES
[1] E. Bedos, On the C∗-algebra generated by the left regular representation of a locally compact group. Proc. Amer. Math. Soc.120(1994), 603–608.
[2] W.R. Bloom and H. Heyer,Harmonic Analysis of Probability Measures on Hypergroups.
DeGruyter, Berlin, 1965.
[3] C. Ching and X. Guangwu, The weak closure o the set of left translations operators.
Proc. Amer. Math. Soc.127(1999), 465–471.
[4] J. Dixmier, Les C∗-algebres et leurs representations. Gauthier-Villars, Paris, 1964.
[5] P. Eymard, L’algebre de Fourier d’un groupe localement compact. Bull. Soc. Math.
France92(1964), 181–236.
[6] J.M.G. Fell, The dual spaces of C∗-algebras. Trans. Amer. Math. Soc. 94 (1960), 365–403.
[7] E. Hewitt and K.A. Ross,Abstract Harmonic Analysis, II. Springer, Berlin–Heidelberg–
New York, 1970.
[8] A. Hulanicki,Groups whose regular representation weakly contains all unitary represen- tations. Studia Math.XXIV(1964), 37–59.
[9] R.I. Jewett, Spaces with an abstract convolution of measures. Advances in Math. 18 (1975), 1–101.
[10] R. Lasser, On the character space of commutative hypergroups. Jahresber. Deutsch.
Math.-Verein.104(2002), 3–16.
[11] L. Pavel,Reiter’s condition (P2)and hypergroup representations. C.R. Math. Acad. Sci.
Paris341(2005), 475–480.
[12] J.P. Pier,Amenable Locally Compact Groups. Willey, New York, 1984.
[13] M. Skantharajah,Amenable hypergroups. Illinois J. Math.36(1992), 15–46.
[14] R. Spector, Aper¸cu de la th´eorie des hypergroupes. In: Lecture Notes in Math. 497, pp. 643–673. Springer, Berlin–Heidelberg–New York, 1975.
[15] M. Voit, On the dual space of a commutative hypergroups. Arch. Math. 56 (1991), 360–385.
Received 20 January 2006 University of Bucharest
Faculty of Mathematics and Computer Science Str. Academiei 14
010014 Bucharest, Romania lpavel@fmi.unibuc.ro
