View source for Hypergroups, harmonic analysis on locally compact

<!--
h1104101.png
$#A+1 = 6 n = 1
$#C+1 = 6 : ~/encyclopedia/old_files/data/H110/H.1100410 Hypergroups, harmonic analysis on locally compact
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

Locally compact hypergroups are natural generalizations of locally compact group measure algebras; the field is therefore part of the research programme designed under  "harmonic analysis without a group to do it on"  (cf. also [[Harmonic analysis|Harmonic analysis]]; [[Harmonic analysis, abstract|Harmonic analysis, abstract]]; for the notion of a hypergroup, see also [[Generalized displacement operators|Generalized displacement operators]]). The idea goes back to I.M. Gel'fand, A. Povzner and B. Levitan [[#References|[a14]]] on the one hand, and to J. Delsarte [[#References|[a4]]] on the other. Another approach has been undertaken by Yu. Berezanskii and collaborators [[#References|[a1]]].

For fundamentals and examples, as well as results oriented towards probability theory, see [[#References|[a2]]]. It contains an almost complete and up-to-date (1996) bibliography.

==Haar measure.==
Basic results from [[#References|[a5]]] prove the existence and uniqueness of the [[Haar measure|Haar measure]] for locally compact commutative hypergroups as well as for discrete hypergroups. The compact case has been settled independently and in a different way in [[#References|[a13]]], which imitates a construction by A. Weil. In the non-compact case, Spector's proof involves a deep new fixed-point theorem for not necessarily affine mappings preserving a compact convex subset in a locally convex vector space.

The problem of whether the Haar measure exists on a locally compact hypergroup is still open, but an important step towards a complete answer is given by using a generalization of the Ryll–Nardzewski fixed-point theorem [[#References|[a7]]], embodying Spector's theorem.

==Commutative locally compact hypergroups.==
The basic facts about the Plancherel measure (cf. also [[Plancherel formula|Plancherel formula]]) are presented in [[#References|[a3]]].

It is shown in [[#References|[a8]]] that the general  $  L _ {2} $-
theory appears as a consequence of the Plancherel–Godement theory for an involutive algebra.

A crucial issue is to decide which (commutative) locally compact hypergroups possess a Fourier–Eymard algebra. The question is widely open for further research; notably, an elementary finite counterexample is provided in [[#References|[a13]]].

===The compact case.===
Whilst the  $  L _ {2} $-
theory appears as generalizing known facts for compact commutative groups (at least at first glance), the  $  L _ {1} $-
theory proves to be extremely involved and deep. R. Vrem proved [[#References|[a6]]] the crucial property allowing one to work with trigonometric polynomials: this subspace is dense in  $  L _ {1} $. 
Vrem's approach makes strong use of the compact group case and does not allow further generalizations.

A radically new approach was provided in [[#References|[a8]]], which showed the existence of a universal, and explicit, bounded approximate identity comprising trigonometric polynomials, using the Dixmier symbolic calculus [[#References|[a5]]].

The latter result allows one to develop basic definitions of Sidon sets (in the classical sense) in full generality (cf. also [[Harmonic analysis, abstract|Harmonic analysis, abstract]]). Pioneering work related to that question was performed in [[#References|[a17]]]. The question here is to decide which class of commutative compact hypergroups do possess non-trivial Sidon sets. See [[#References|[a2]]], [[#References|[a9]]]. The problem seems to be related to the symmetry (or non-symmetry) of the associated measure algebra, an extremely deep problem.

===The general case.===
One basic approach is given in [[#References|[a8]]]: it relies deeply upon the exploitation of Dixmier's symbolic calculus. An important guideline is to obtain insight into the spectral picture of  $  L _ {1} $
by taking care of the growth of the hypergroup. Basic spectral synthesis results are obtained in that way, generalizing the Abelian locally compact group case as well as preliminary results [[#References|[a3]]].

Notably, amenability plays a crucial role (quite unexpectedly; cf. [[Invariant average|Invariant average]]). The theory is essentially different from the locally compact group situation (even for the commutative theory). A crucial question is to be able to formulate an appropriate Fölner property.

==Discrete hypergroups.==
The basic axioms are elementary. Aside from commutative examples associated with trees, very little has been accomplished so far (1996). Amenability is again a crucial question: some results have been obtained in [[#References|[a10]]] and [[#References|[a11]]].

Very important is the fact that discrete hypergroups need not be unimodular [[#References|[a18]]]. P.G.A. Floris and T.H. Koornwinder have found connections between quantum groups and discrete hypergroups (cf. also [[Quantum groups|Quantum groups]]). A remarkable example is provided in [[#References|[a6]]]: this example (based on  $  q $-
Jacobi polynomials) is non-unimodular, GCR, hence type-I; one may compare this example with the relevant result for discrete groups: E. Thoma proved that a discrete group is type-I if and only if it is a finite extension of a commutative discrete group.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Yu. Berezanskii,   "Hypercomplex systems with a discrete basis"  ''Dokl. Akad. Nauk SSSR (N.S.)'' , '''81'''  (1954)  pp. 825–828  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.R. Bloom,   H. Heyer,   "Harmonic analysis of probability measures on hypergroups" , ''Studies in Mathematics'' , '''20''' , W. de Gruyter  (1995)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Chilana,   K. Ross,   "Spectral synthesis in hypergroups"  ''Pacific J. Math.'' , '''76'''  (1978)  pp. 313–328</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Delsarte,   "Hypergroupes et opérateurs de permutation et de transmutation" , ''La théorie des équations aux dérivées partielles (Nancy, 9–15 avril 1956)'' , ''Colloq. CNRS'' , '''71''' , CNRS  (1956)  pp. 29–45</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Dixmier,   "Opérateurs de rang fini dans les représentation unitaires"  ''IHES Publ. Math.''  (1960)  pp. 305–317</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P. Floris,   "On quantum groups, hypergroups and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110410/h1104107.png" />-special functions"  ''Ph.D. Thesis RU Leiden''  (1995)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M.O. Gebuhrer,   "The Haar measure on a locally compact hypergroup"  Pr. Komrakov (ed.)  Pr. Litvinov (ed.) , ''Proc. Conf. Differential Geometry on Homogeneous Spaces and Harmonic Analysis on Lie Groups (Moscow, 1994)'' , Kluwer Acad. Publ.  (1996)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M.O. Gebuhrer,   "Analyse harmonique sur les espaces de Gel'fand–Levitan et applications à la théorie des semigroupes de convolution"  ''Thæse de doctorat d'etat Univ. Louis Pasteur, Strasbourg''  (1989)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M.O. Gebuhrer,   "About the fine structure of compact commutative hypergroups" , ''Preprint'' , IRMA , Strasbourg  (1996)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  M.O. Gebuhrer,   "Remarks on amenability of discrete hypergroups"  M.A. Picardello (ed.) , ''Harmonic Analysis and Discrete Potential Theory'' , Plenum  (1992)  pp. 479–482</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  M.O. Gebuhrer,   A. Kumar,   "The Wiener property for a class of discrete hypergroups"  ''Math. Z.'' , '''202'''  (1989)  pp. 271–274</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  M.O. Gebuhrer,   A.L. Schwartz,   "Sidon sets and Riesz sets on the disk algebra"  ''Colloq. Math.''  (1996)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  R.I. Jewett,   "Spaces with an abstract convolution of measures"  ''Adv. in Math.'' , '''18'''  (1975)  pp. 1–101</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  B. Levitan,   A. Povzner,   "Differential equations of the Sturm–Liouville type on the semi-axis and Plancherel's theorem"  ''Dokl. Akad. Nauk SSSR (N.S.)'' , '''52'''  (1946)  pp. 479–482  (In Russian)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  R. Spector,   "Mesures invariantes sur les hypergroupes"  ''Trans. Amer. Math. Soc.'' , '''239'''  (1978)  pp. 147–165</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  R. Vrem,   "Harmonic analysis on compact hypergroups"  ''Pacific J. Math.'' , '''85'''  (1979)  pp. 239–251</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  R. Vrem,   "Lacunarity on compact hypergroups"  ''Math. Z.'' , '''164'''  (1968)  pp. 13–104</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  W. Woess,   V. Kaimanovitch,   "Construction of discrete non-unimodular hypergroups" , ''Preprint Quaderno'' :  9 , Univ. Milano  (1995)</TD></TR></table>