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Hypergroups with Invariant MetricAuthor(s): Michael VoitSource: Proceedings of the American Mathematical Society, Vol. 126, No. 9 (Sep., 1998), pp.2635-2640

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PROCEEDINGS OF THEAMERICAN MATHEMATICAL SOCIETYVoluime 126, Number 9, September 1998, Pages 2635-2640S 0002-9939(98)04366-4

HYPERGROUPS WITH INVARIANT METRICMICHAEL VOIT

(Communicated by Palle E. T. Jorgensen)

ABSTRACT. The purpose of this note is to extend the following classical resultfrom groups to hypergroups in the sense of C.F. Dunkl, R.I. Jewett, and R.Spector: If a hypergroup has a countable neighborhood base of its identity,then K admits a left- or a right-invariant metric. Moreover, it admits aninvariant metric if and only if there exists a countable conjugation-invariantneighborhood base of the identity.

A classical result due to Birkhoff [1], Kakutani [5] and others states that eachsecond countable locally compact group G admits a left-invariant metric d, i.e., onehas(1) d(x, y) = d(zx, zy) for all x, y, z E G.The purpose of this note is to prove a result of this kind for hypergroups in the senseof Dunkl, Jewett, and Spector. In this case the convolution of two points x, y is acompactum {x} * {y} and usually no longer a single point. In this case one usuallyfinds elements x, y in the hypergroup with x $& and ({z} * {x}) n ({z} * {y}) $& .This shows that one cannot expect to obtain an invariant metric d satisfying a stronginvariance property like d(x, y) = d(a, b) for all x, y, z, a, b E K with a E {z} * {x}and b E {z} * {y}. However, the following weaker condition can be achieved:Definition 1. Let K be a hypergroup with identity e such that the topology of Kis generated by some metric d. Then d is called left-invariant, if for all x E K andc > 0 the e-balls U,(x) := {y E K: d(x,y) < e} satisfy U,(x) = {x} * U,(e).Right-invariance is defined in the same way, and d is called invariant if it is bothleft- and right-invariant.

If the hypergroup is a group G, then it can easily seen that our left-invariancejust means that d(e, y) = d(z, zy) for all z, y E G. As this implies thatd(x, y) = d(e, x-1y) = d(zx, (zx)(x1-y)) = d(zx, zy) for all x, y, z E G,

it follows that for groups our definition agrees with the classical one stated in Eq.(1).The following theorem is the main result of this note:Theorem 1. If a hypergrouphas a countableneighborhoodbase of its identity, thenit admits a right-invariant metric.

Received by the editors December 26, 1996 and, in revised form, January 27, 1997.1991 Mathematics Subject Classification. Primary 43A62; Secondary 20N20, 54E35.Key words and phrases. Hypergroups, invariant metric.(?)1998 American Mathematical Society

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2636 MICHAEL VOIT

Assume that a hypergroup K admits an invariant metric d. Then all e-ballsU6(e) around e satisfy {x} * U6(e) = U6(e) * {x} for all c > 0 and x E K, whichmeans that there exists a countable conjugation-invariant neighborhood base ofthe identity e E K. Therefore, one direction of the following theorem is clear:Theorem 2. A hypergroupadmits an invariant metric if and only if there existsa countable conjugation-invariant neighborhoodbase of its identity.Remarks. (1) If a commutative hypergroup K has a countable neighborhood baseof its identity, then these neighborhoods trivially are conjugation-invariant,and K admits an invariant metric.(2) For any discrete hypergroup, the metric d with d(x, y) =1 for x 7 y isinvariant.(3) We presently do not know whether each second countable compact hypergroup

admits a conjugation-invariant neighborhood base of e and hence an invariantmetric.Before we prove the theorems, we recapitulate some notation and facts abouthypergroups; for further details see [2] and [4].

Definition 2. A hypergroup (K, *) consists of a locally compact Hausdorff spaceK together with a bilinear, associative, weakly continuous convolution on the Ba-nach space Mb(K) of all bounded regular Borel measures on K with the followingproperties:(1) For all x, y E K, the convolution of the point measures 6x* by is a probabilitymeasure with compact support.(2) The mapping K x K > C(K), (x, y) supp(6x * 6y), is continuous withrespect to the Michael topology on the space C(K) of all nonvoid compactsubsets of K, where this topology is generated by the sets

Uv,w := {L E C(K): L n v 4 0, L c W} with V,W open in K.(3) There is an identity e E K with 6x * 6e = 6e * 6x = 6x forall x E K.(4) There is a continuous involution x X >x on K such that (6x * by)- =b * 6xand e E supp(6x * by) 4 x = y for x, y E K.We use the common abbreviation A * B := UXCA,YCBUPP(6X 6Y)for setsA, B C K. Moreover, A is the image of A c K under the involution on K. Theclosure of A is denoted by cl A.The following facts from Sections 2.5, 3.2, and 4.1 of Jewett [4] will be neededin the proof of the theorems:

(A) ForallA,B,CcK,wehave(A*B)nC= 0 -=* An(C*B)=0.(B) If U is a neighborhood of the identity e in K, then for all A c K, cl A c A *U.(C) For all compacta A, B c K with A n B = 0 there exists a neighborhood U ofe with (A * U) n (B * U) = 0.Proof of Theorem 1. The proof will be divided into three major steps.Step 1. Assume the hypergroup K has a countable neighborhood base of its iden-tity e. For k E N we choose inductively a neighborhood base (Uk)kcN of e consistingof open neighborhoods Uk of e with the following properties:

(a) Uk = Uk and Uk+1 * Uk+1 c Uk for all k.


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HYPERGROUPS WITH INVARIANT METRIC 2637

(b) For all I < n < k and all integers I < 1(1) < 1(2) < ... l(n) < k,Uk * U1(1) * U1(2) *.*.*. * Ul(n) C U4(l) * U1(2) * * * ** Ul(n) * Uk-l1

In fact, (a) can be achieved obviously for each sufficiently small symmetric Uk.Moreover, as cl(Ul(l) * Ui(2) * *. *i(,)) C * Ui(2) * ... * Ui(n) * Uk-I by fact(B) above, we conclude from fact (C) above that the finitly many conditions in (b)are satisfied for each sufficiently small Uk.As (Uk)kcN is a neighborhood base of e, we in particular have nk Uk = {e}.In the next step, we define V2-k := Uk for k E N. For arbitrary dyadic rationalnumbers(2) r = 2-1(1) + 2-1(2) + ... + 2-1(n) with 0 < 1(1) < 1(2) < * < I(n)we define

Vr :=V2 -(l) * V2-1(2) ...*V2-I(n) for 0 < r < 1.Moreover, for r > 1 we put Vr = K. The sets Vr are open, and we next prove thatthey have the following properties for arbitrary dyadic rational numbers r, s > 0:(1.1) Vr c V, for r < s.(1.2) Vr* Vs C Vr+s+2.min(r,s).

Our proof of (1.1) follows the exposition of the proof of the same statement in theproof of Theorem 8.2 of Hewitt and Ross [3]. In order to check (1.1), we may assume0 < r < s < 1. Let r be given as in Eq. (2) and s by 2-m(1) + 2-m(2) +... + 2-m(P)with 0 < m(l) < ... < m(p). Then there is a unique k with 1(j) = m(j) for j < kand l(k) > m(k). Letting W := V2-i(i) * V2-1(2)* . . . * V2-1(k-1), we then have

V, = W * V2-I(k) * ... * V2-I(n) C W * V2-1(k) * V2-1(k)-l * ... * V2-1(n) * V2-1(n)C ... C W * V2-1(k)+l C W * V2-m(k) C V2-m(l) * V2-m(2) * ... * V2-m(p) = V.

In order to prove (1.2), we first assume 0 < s < r with r + 3s < 1 (notice thatthe case r + 3s > 1 is obvious). It suffices here to check (1.2) for r as in (2), andfor s = 2-' with 1> 1(1). If 1 > 1(n), then we have(3) Vr* V2-1= Vr+2-1,and (1.2) holds. If 1< 1(n), then we take k E N with k > 2 and l(k - 1) < 1

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(2.2) For all a, b E K and c E {a} * {b}, one has q(c) < q(a) +(b).(2.3) The sets We(e) := {x E K q(x) < e} form a neighborhood base of e forc > 0, and, in particular, 0 is continuous at e E K.The symmetry of 0 in (2.1) is clear. Moreover, q(x) 0 means that x,x E Vrfor all dyadic r > 0, which is equivalent to the fact that x, x E Uk for all k E N,and this means that x = e. Hence, part (2.1) is clear.In order to check (2.2), take c > O. We then find dyadic numbers rl, r2 > 0 with

a E Vrl, b E Vr2, q(a)2 + e > r1, and q(b)2 + e > r2. Then by (1.2),c E {a} * {b} C Vrl * Vr2 C Vri+r2+min(ri,r2)

which implies thatq$(c)2< rl + r2 + 2 min(ri, r2) < q(a)2 + q(b)2 + 2 min(q(a)2, q(b)2) + 4e.

As this holds for all c > 0, we conclude thatq$(c)2 ? 0 and dyadic numbers r with0 < r < c/2, the set Vr n Vr is an open neighborhood of e with Vr n vr c we4/(e))which shows that all We(e) are neighborhoods of e. Conversely, for each openneighborhood U of e we find Uk with Uk C U. Therefore, W2-k C Uk C U, and(2.3) becomes clear.Step 3. We define the metric d on K by

d(x, y) :- inf{q$(z): z E {x} * {Y}} for x,y E K.We still have to prove that d has the required properties:(3.1) It is clear that for all x, y E K, d(x, y) > 0 and d(x, x) = 0. Moreover,if x 7& ,then e is not contained in the compactum {x} * {y}. By (2.3) this yields thatthere exists c > 0 with We(e) n ({x} * {y}) = 0, and hence d(x, y) > e > 0.(3.2) d is symmetric, as {x} * {y} = ({y} * {x}) and as 0 is symmetric by (2.1).(3.3) To check the triangle inequality, take x, y, z E K and c > O. We then finda E {x} *{y} and b E {y} *{z} with q(a) < d(x, y) + c and q(b) < d(y, z) + C.Thus, x E {a} * {y} and y E {b} * {z}, and x E {a} * {b} * {z}. This impliesthat ({a} *{b}) n ({x} * }) =A . Now choose some c E ({al * {b}) n ({x} *{The definition of d together with (2.2) ensures that

d(x, z) < 0(c) < +(a) + +(b) < d(x, y) + d(y, z) + 2e.The triangle inequality now follows for c -> 0, and we now know that d is ametric.(3.4) In order to prove the right-invariance of d, choosee > 0 and x E K and define

We(x) := {z E K: d(x, z) < }.Then this notion is consistent with that of (2.3) for x = e. Moreover, for eachy E We(e)* {x} we find z E K with d(z, e) < c and y E {z} * {x}, which yieldsz E {y} * {x}, and hence, by the definition of d, d(x, y) < d(z, e) < e. Thisimplies that

We(e) * {X} C We(x)


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HYPERGROUPS WITH INVARIANT METRIC 2639Conversely, if y E We(x), then d(x, y) 0 there is no with ({x} * {Zn})nfW (e) $& for n > nO(c) For c > 0 there is no with Zn E We(x) for n > nOIn fact, the equivalence (a)>??(b) follows from the continuity axiom (2) ofhypergroups above while (b)>===(c)is a consequence of (3.4).The proof of Theorem 1 is now complete.

Proof of Theorem 2. The proof will again be divided into three steps, which aresimilar to, but sometimes easier than, the steps in the proof of Theorem 1.Step 4. Choose a neighborhood base (Uk)k>l of e such that for all k E N,

(a) Uk = Uk and Uk+1* Uk+1 C Uk, and(b) Uk is conjugation-invariant.The symmetry and the conjugation-invariance of the Uk ensures that the sets Vras defined in step 1 of the proof of Theorem 1 have the following stronger propertiesfor all dyadic rational numbers r, s > 0:

(4.1) Vr c V, for r < s,(4.2) Vr * Vs C Vr+s,(4.3) Vr is conjugation-invariant and symmetric.Step 5. The function

q(x):=inf{r: xEVr} forxEKhas the following properties:(5.1) 0q(x)= O if and only if x = e,(5.2) +(x) = q(x) for all x E K,(5.3) The sets We(e) := {x E K : $(x) < e} form a neighborhood base of e for

e > 0.(5.4) All We(e) are conjugation invariant.In fact, it suffices to check (5.4): Take y E {x}*W,(e). Then there is z E K withy E {x} * {z} and q(z) < c. We find a dyadic rational r < c with z E Vr. Hence, by

the conjugation invariance of Vr, y E {x} * {z} c {x} * Vr = Vr * {x} c We * {x},which proves {x} * We,(e)C We* {x}. The converse inclusion can be checked in thesame way.Step 6. This step can be carried out in the same way as in the previous theoremwhere the additional left-invariance of d follows from (5.4). This completes theproof of Theorem 2. D

Examples. Hypergroup structures on intervals I c IRwere studied and partiallyclassified by Zeuner [6]. He proved that on IReach hypergroup structure is iso-morphic with the group (IR,+). He also showed that each hypergroup structure on


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2640 MICHAEL VOIT

[0, oo[ or [0, 1] is commutative and can be normalized (after a suitable isomorphism)as follows:(1) If K = [0, oo[, then Ix-yl, x+y c {x}*{y} c [lx-yl, x+y] for all x, y c [0,oo[.(2) If K = [0,1], then Ix-yl,x+y c {x}*{y} c [lx-yl,x+y] for all x,y c [0,1]with x+y

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