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Abstract

This is a survey paper about functional equations on hypergroups. We show how some fundamental functional equations can be treated on some types of hypergroups. We also present stability and superstability results using invariant means and other tools.
Functional equations and stability problems
on hypergroups
aszl´o Sz´ekelyhidi
Keywords. hypergroup, stability.
Abstract. This is a survey paper about functional equations on hyper-
groups. We show how some fundamental functional equations can be
treated on some types of hypergroups. We also present stability and
superstability results using invariant means and other tools.
1. Basics on hypergroups
The concept of DJS–hypergroup (according to the initials of C. F. Dunkl,
R. I. Jewett and R. Spector) depends on a set of axioms which can be for-
mulated in several different ways. The way of formulating these axioms we
follow here is due to R. Lasser (see e.g. [3], [16]). One begins with a locally
compact Hausdorff space Kand with the space CcpKqof all compactly sup-
ported complex valued functions on the space K. The space CcpKqwill be
topologized as the inductive limit of the spaces
CEpKq“tfPCcpKq: supp pFq Ď Eu,
where Eis a compact subset of Kcarrying the uniform topology. A (complex)
Radon measure µis a continuous linear functional on CcpKq. Thus, for every
compact subset Ein Kthere exists a constant αEsuch that |µpfq| ď αE}f}8
for all fin CEpKq. The set of Radon measures on Kwill be denoted by
MpKq. In the sequel by a measure we always mean a Radon measure. For
each measure µwe write
}µ} “ supt|µpfq| :fPCcpKq,}f}8ď1u.
A measure µis said to be bounded, if }µ} ă `8. In addition, µis called a
probability measure, if µis nonnegative and }µ} “ 1. The set of all bounded
measures, the set of all compactly supported measures, the set of all proba-
bility measures, and the set of all probability measures with compact support
in MpKqwill be denoted by MbpKq,McpKq,M1pKqand M1,cpKq, respec-
tively. The point mass concentrated at xis denoted by δx. Via integration
2 aszl´o Sz´ekelyhidi
theory we are able to consider measures as functions on the σ-algebra BpKq
of Borel subsets of Kand we use the notation şKf dµ rather than µpfq. We
use the notation M`pKqfor the set of positive measures on the σ-algebra
BpKqthat means, for measures which take values in r0,`8s.
Now we formulate the first part of the axioms. Suppose that we have
the following:
1. pH˚qThere is a continuous mapping px, yq ÞÑ δx˚δyfrom KˆKinto
M1,cpKq. This mapping is called convolution.
2. pH_qThere is an involutive homeomorphism xÞÑ qxfrom Kto K. This
mapping is called involution.
3. pHeqThere is a fixed element ein K. This element is called identity.
Identifying xby δxthe mapping in pH˚qhas a unique extension to a
continuous bilinear mapping from MbpKq ˆ MbpKqto MbpKq. The involu-
tion on Kextends to a continuous involution on MbpKq. Convolution maps
M1pKq ˆ M1pKqinto M1pKqand involution maps M1pKqonto M1pKq.
Then a DJS–hypergroup, or simply a hypergroup is a quadruple pK, ˚,_, eq
satisfying the following axioms: for each x, y, z in Kwe have
1. (H1) δx˚ pδy˚δzq“pδx˚δyq ˚ δz,
2. (H2) pδx˚δyqqδqy˚δqx,
3. (H3) δx˚δeδe˚δxδx,
4. (H4) eis in the support of δx˚δqyif and only if xy,
5. (H5) the mapping px, y q ÞÑ supp pδx˚δyqfrom KˆKinto the space of
nonvoid compact subsets of Kis continuous, the latter being endowed
with the Michael topology (see [3]).
For arbitrary measures µ, ν in MbpKqthe symbol µ˚νdenotes their
convolution, and qµdenotes the involution of µ. With these operations MbpKq
is an algebra with involution. If the topology of Kis discrete, then we call
the hypergroup discrete. In case of discrete hypergroups the above axioms
have a simpler form. Here we present a set of axioms for these types of
hypergroups. Clearly, in the discrete case we can simply forget about the
topological requirements in the previous axioms to get a purely algebraic
system.
Let Kbe a set and suppose that the following properties are satisfied:
1. pD˚qThere is a mapping px, yq ÞÑ δx˚δyfrom KˆKinto M1,cpKq, the
space of all finitely supported probability measures on K. This mapping
is called convolution.
2. pDqqThere is an involutive bijection xÞÑ qxfrom Kto K. This mapping
is called involution.
3. pDeqThere is a fixed element ein K. This element is called identity.
Functional equations and stability problems on hypergroups 3
Identifying xby δxas above, and extending convolution and involution,
adiscrete DJS–hypergroup is a quadruple pK, ˚,q, eqsatisfying the following
axioms : for each x, y, z in Kwe have
1. (D1) δx˚ pδy˚δzq “ pδx˚δyq ˚ δz,
2. (D2) pδx˚δyqqδq
y˚δq
x,
3. (D3) δx˚δeδe˚δxδx,
4. (D4) eis in the support of δx˚δq
yif and only if xy.
If δx˚δyδy˚δxholds for all x, y in K, then we call the hypergroup
commutative. If qxxholds for all xin K, then we call the hypergroup
Hermitian. By (H2), every Hermitian hypergroup is commutative. In any
case we have qee. For instance, if KGis a locally compact Hausdorff
group, δx˚δyδxy for all x, y in K,qxis the inverse of xand eis the identity
of G, then we obviously have a hypergroup pK, ˚,q, eq, which is commutative
if and only if the group Gis commutative. However, not every hypergroup
originates in this way.
The simplest hypergroup is obviously the trivial one, consisting of a
singleton. The next simplest hypergroup structure can be introduced on a
set consisting of two elements. As an example we describe all hypergroups of
this type.
Let K“ t0,1u. Clearly, the only Hausdorff topology on Kis the discrete
one. We specify e0 as the identity element. In this case the only involution
satisfying the above axioms is the identity, that is, q
00 and q
11. Conse-
quently, we have a Hermitian hypergroup, which is necessarily commutative.
Now we have to define the four possible products δ0˚δ0,δ0˚δ1,δ1˚δ0and
δ1˚δ1. As δ0is the identity, the first three products are uniquely determined
and the fourth one must have the form
δ1˚δ1θ¨δ0` p1´θq ¨ δ1
with some number θsatisfying 0 ďθď1. It turns out that θ0, as a
consequence of (D4). We shall denote this hypergroup by Dpθq. It is clear
that in this way we have a complete description of all possible hypergroup
structures on a set consisting of two elements. Observe that in the case θ1
we have a group isomorphic to Z2, the integers modulo 2, in any other case
the resulting structure is not a group.
If Kis any hypergroup and His an arbitrary set, then for the function
f:KÑHwe define q
fby the formula
q
fpxq “ fpqxq
for each xin K. Obviously, q
q
ff. Each measure µin MbpKqsatisfies
q
µpfq “ µpq
fq
whenever f:KÑCis a bounded Borel function.
4 aszl´o Sz´ekelyhidi
Let Kbe an arbitrary hypergroup. Then, for each x, y in Kthe measure
δx˚δyis a compactly supported probability measure on Kwhich makes the
measurable space pK, BpKq, δx˚δyqaprobability space. An arbitrary function
f:KÞÑ Cwhich is δx˚δy-measurable, can be considered as a random
variable on this probability space. In particular, each continuous complex
valued function on Kis a random variable with respect to every measure of
the form δx˚δy. Clearly, each fis integrable with respect to every δx, and
its expectation is
Expfq “ żf dδxfpxq,
hence it seems to be reasonable to define the ”value” of fat δxas fpxq. This
can be extended to an arbitrary probability measure µon Kby defining
fpµq “ Eµpfq “ żf dµ ,
whenever fis integrable with respect to µ. In particular,
fpδx˚δyq “ żf dpδx˚δyq,
whenever fis integrable with respect to δx˚δy. In this case we shall use the
suggestive notation fpx˚yqfor fpδx˚δyq. In fact, in every hypergroup Kwe
identify xby δx.
Here we call the attention to the fact that fpx˚yqhas no meaning on its
own, because x˚yis in general not an element of K, hence fis not defined
at x˚y. The expression x˚ydenotes a kind of ”indistinct” product. If Bis
a Borel subset of K, then δx˚δypBqexpresses the probability of the event
that this ”blurred product” of xand ybelongs to the set B. In the special
case of groups this probability is 1, if Bcontains xy and is 0 otherwise, that
is, exactly δxypBq.
We define the right translation operator τyby the element yin Kac-
cording to the formula
τyfpxq “ żK
f dpδx˚δyq
for each fintegrable with respect to δx˚δy. In particular, τyis defined for
every continuous complex valued function on K. Similarly, we can define left
translation operators, denoted by yτ. In general, one uses the above notation
fpx˚yq “ żK
f dpδx˚δyq,
for each x, y in K. Obviously, in case of commutative hypergroups the simple
term translation operator is used. The function τyfis the translate of fby y.
Functional equations and stability problems on hypergroups 5
As an example, we consider a function f:Dpθq Ñ Con the Dpθq
hypergroup. Then we have
fp1˚1q “ żfptqdpδ1˚δ1q “
żfptqdpθ¨δ0` p1´θq ¨ δ1q “ θf p0q`p1´θqfp1q.
Convolution of functions and measures is defined in the following obvi-
ous way: for each measure µin MbpKqand for every continuous bounded
function f:KÑCwe let
f˚µpxq “ żK
fpx˚qyqpyq
whenever xis in K. Then f˚µis a continuous bounded function on K. For
more details see [3].
In what follows we simply refer to the hypergroup pK, ˚,q, eqas a hyper-
group K.
2. Functional equations
The presence of translation operators makes it possible to introduce and
to study some basic functional equations on hypergroups.
Let Kbe a hypergroup. The non-identically zero continuous function
m:KÑCis called an exponential, if it satisfies
(1) mpx˚yq “ mpxqmpyq
for each x, y in K. Exponentials play a basic role in harmonic analysis, spec-
tral synthesis and functional equations. More explicitly, the above functional
equation can be written in the form of the integral equation
żK
mptqdpδx˚δyqptq “ mpxqmpyq.
For each exponential mwe have mpeq “ 1. An exponential mwith mpqxq “
mpxqfor all xin Kis called a semi-character, and bounded semi-characters
are called characters. We note the inconvenient facts that in contrast with
the case of groups exponentials can take the value zero, and the product of
two exponentials is not necessary an exponential.
Here we give a simple illustration by describing all exponential functions
on the hypergroup Dpθqwith 0 ăθď1.
Suppose that m:Dpθq Ñ Cis an exponential that is
żDpθq
mptqdpδx˚δyqptq “ mpxqmpyq
6 aszl´o Sz´ekelyhidi
holds for each x, y in Dpθq. According to the definition of convolution in
Dpθqthe only nontrivial consequence of this equation we obtain in the case
xy1, by our above remark:
θmp0q`p1´θqmp1q “ mp1qmp1q.
Using mp0q “ 1 and solving the quadratic for mp1qwe have the two possi-
bilities: mp1q “ 1 or mp1q“´θ. The first case gives the trivial exponential
which is identically 1, and the second case is the nontrivial one: mp0q “ 1
and mp1q“´θ. Obviously, both are characters.
Another important function class is the following. Given the commuta-
tive hypergroup Kthe continuous function a:KÑCis called additive, if it
satisfies
(2) apx˚yq “ apxq ` apyq
for each x, y in K. This equation can be written in the integral form
żK
aptqdpδx˚δyqptq “ apxq ` apyq.
Every additive function asatisfies apeq “ 0, and all additive functions on K
form a complex linear space.
Considering the hypergroup Dpθqagain, let a:Dpθq Ñ Cbe an additive
function. Then we have
θap0q`p1´θqap1q “ ap0q ` ap1q,
and ap0q “ 0 implies ap1q “ 0, as θ0. Hence every additive function is
zero on Dpθq. We note that, more generally, every additive function is zero
on compact hypergroups.
Exponential and additive functions are fundamental – they are the so-
lutions of the basic Cauchy functional equations on hypergroups. We remark
that, obviously, these functions can be defined on non-commutative hyper-
groups, too. We can also consider their pexiderized versions.
Moment functions and moment function sequences play an important
role in probability theory. These functions can be defined in the following
manner. Let Kbe a hypergroup. The sequence of continuous complex valued
functions pϕnqnPNon Kis called a generalized moment function sequence, if
ϕ0is not identically zero and for each natural number nwe have
(3) ϕnpx˚yq “
n
ÿ
k0ˆn
k˙ϕkpxqϕn´kpyq
for each x, y in Kand for every natural number n. Obviously, ϕ0is an ex-
ponential. If ϕ01, then this sequence is called a moment function se-
quence, and, in general, we say that the generalized moment function sequence
pϕnqnPNis associated with the exponential ϕ0. Given a natural number Nwe
Functional equations and stability problems on hypergroups 7
say that the functions tϕk:k0,1, . . . , Nufor a generalized moment func-
tion sequence of order N, if the above equations hold for n0,1, . . . , N.
The second equation of the above system is
(4) ϕ1px˚yq “ ϕ0pxqϕ1pyq ` ϕ1pxqϕ0pyq,
which is called sine equation, for obvious reasons.
An important functional equation related to involution is the d’Alembert
functional equation, or the square norm functional equation which has the
form
(5) fpx˚yq ` fpx˚qyq “ 2fpxq ` 2fpyq
for each x, y in K, where f:KÑCis a continuous function on the hyper-
group K. Clearly, on Hermitian hypergroups this equation is identical with
the additive Cauchy equation.
On Abelian groups an important function class is formed by the so-
called polynomial functions. There are several different ways to introduce
these functions, and in some cases they result in different function classes. In
the subsequent sections we follow the way used in [24, 26].
3. The measure algebra
Exponential monomials are the basic building bricks of spectral analysis
and spectral synthesis on Abelian groups. Recently there have been some
attempts to extend the most important spectral analysis and spectral syn-
thesis results from groups to hypergroups. For this purpose it is necessary
to introduce a reasonable concept of exponential monomials. In the group
case this concept arises from additive and exponential functions. Roughly
speaking, in that case exponential monomials are the functions which can be
represented as the product of an exponential and an ordinary polynomial of
additive functions. It turns out that this definition will not work in the case of
hypergroups. In fact, even in the simplest case, when we consider the square
of an exponential, it will not be an exponential. In [26] we reconsidered this
problem, and using a ring-theoretical approach we proved characterization
theorems for particular function classes, which can be considered as ”expo-
nential monomials” on commutative hypergroups. Some of those ideas cen be
extended to the non-commutative case, too. Nevertheless, here we consider
the commutative setting only.
The basic structure is the measure algebra. If KK,q, eqis a commu-
tative hypergroup, then CpKqdenotes the locally convex topological vector
space of all continuous complex valued functions defined on K, equipped with
the pointwise linear operations and the topology of compact convergence.
8 aszl´o Sz´ekelyhidi
It is well-known (see e.g. [7], p. 551) that the dual of CpKqcan be iden-
tified with McpKq, the space of all compactly supported complex measures
on K. If Kis discrete, then this space is also identified with the set of all
finitely supported complex valued functions on K. The pairing between CpKq
and McpKqis given by the formula
xµ, f y “ żf dµ .
Convolution on McpKqis defined by
µ˚νpxq “ żµpx˚qyqpyq
for each µ, ν in McpKqand xin K. Convolution converts the space McpKq
into a commutative algebra with unit δe. We call this algebra the measure
algebra of K. If Kis discrete, then we call it the hypergroup algebra of K, as
in the case when Kis a group it is identical with the group algebra of this
group.
We also define convolution of measures in McpKqwith arbitrary func-
tions in CpKqby the same formula
µ˚fpxq “ żfpx˚qyqpyq
for each µin McpKq,fin CpKqand xin K. It is easy to see that equipped
with this action CpKqturns into a module over the measure algebra.
Translation operators are closely related to convolution. In fact, τyis a
convolution operator, namely, it is the convolution with the measure δqy. A
subset of CpKqis called translation invariant, if it contains all translates of
its elements. A closed linear subspace of CpKqis called a variety on K, if it
is translation invariant. For each function fthe smallest variety containing f
is called the variety generated by f, or simply the variety of fand is denoted
by τpfq. It is the intersection of all varieties containing f.
We recall the concept of the annihilator. Given a subset Hin CpKqits
annihilator in McpKqis the set
Ann H“ tµ:µPMcpKq, µ ˚f0 for each fPHu.
It is easy to see that this is an ideal in McpKq. Analogously, for each subset
Lin McpKqits annihilator in CpKqis defined by
Ann L“ tf:fPCpKq, µ ˚f0 for each µPLu.
It follows that Ann Lis a variety in CpKq.
The concept of the annihilator is closely related to the notion of the
orthogonal. Given a subset Hin CpKqits orthogonal in McpKqis the set
HK“ tµ:µPMcpKq, µpfq “ 0 for each fPHu.
Functional equations and stability problems on hypergroups 9
It is easy to see again that this is an ideal in McpKq. Analogously, for each
subset Lin McpKqits orthogonal in CpKqis defined by
LK“ tf:fPCpKq, µpfq “ 0 for each µPLu.
It follows that Ann Lis a variety in CpKq. The relation between annihilators
and orthogonals of varieties and ideals is easy to describe. Indeed, we have
for each variety Vin CpKqand for each ideal Iin McpKqthe identities
Ann q
V“ pAnn Vqq,Ann q
I“ pAnn Iqq,
further
VKAnn q
V , IKAnn q
I .
Hence, in the case of varieties and ideals the use of annihilators or orthogonals
is more or less a question of taste.
It is also obvious that VĎVKK and IĎIKK holds for each variety V
on Kand for each ideal Iin McpKqand the similar relations hold for Ann V
and Ann I. Moreover, using the Hahn–Banach Theorem, it is easy to show
that VVKK and VAnn pAnn Vqholds for each variety. Unfortunately,
we don’t have the corresponding equality for ideals, as it is shown by an
example in [9] in the case, when Kis a group. However, if Kis a discrete
hypergroup, then IAnn pAnn Iqholds for each ideal in McpKq. This is
also shown in [9] in the group case, and one can see immediately that the
proof given there works on hypergroups, too.
4. Exponential polynomials
Exponential polynomials play a fundamental role in the theory of func-
tional equations. In fact, all the functional equations mentioned above char-
acterize functions on Abelian groups which belong to the class of exponential
polynomials. Hence in order to build up a satisfactory theory of functional
equations on hypergroups it is necessary to find a reasonable definition of
exponential polynomials which, in the group case, coincides with the usual
concept. As we mentioned above, an obvious copy of the definitions, by re-
placing the group operation with convolution does not work, at all. Now we
shortly summarize the way we have offered in the papers [24, 26]. For the
sake of simplicity here we consider the commutative case only. Nevertheless,
it will be clear for the reader that some of the methods can be extended to
non-commutative hypergroups as well.
The basic idea is the use of modified difference operators. Given a hy-
pergroup K, a continuous function f:KÑC, and an element yin Kwe
define the modified difference f;yas the element
f;yδqy´fpyqδe,
where eis the identity of K. For the products of such elements we use the
following notation: given a natural number nand elements y1, y2, . . . , yn`1
10 aszl´o Sz´ekelyhidi
in Kwe write
f;y1,y2,...,yn`1Πn`1
k0f;yk,
where the product means convolution. In the case of the exponential m1 we
use the simplified notation ∆yfor ∆1;yand ∆y1,y2,...,yn`1for ∆1;y1,y2,...,yn`1.
A fundamental role is played by the ideals generated by modified differ-
ences. Given the continuous function f:KÑCthe closure of the ideal in
the measure algebra generated by all modified differences of the form ∆f;y
with yis in Kwill be denoted by Mf. The following theorem shows that this
ideal is relevant if and only if fis an exponential.
Theorem 4.1. Let Kbe a commutative hypergroup and f:KÑCa function
with fpeq “ 1. Then the following statements are equivalent:
1. fis an exponential.
2. The ideal Mfis proper.
3. The ideal Mfis maximal.
4. MfAnn τpfq.
This theorem can be proved following the lines of [27] (see also [26]). We
use this result to define generalized exponential monomials on commutative
hypergroups as follows. Let Kbe a commutative hypergroup. The continuous
function f:KÑCis called a generalized exponential monomial, if there
exists an exponential m, and a natural number nsuch that the relation
(6) Mn`1
mĎAnn τpfq
holds. It can be shown that if fis nonzero, then mis uniquely determined and
we say that fis associated with m. In other words, the continuous function
f:KÑCis a generalized exponential monomial associated with mif and
only if there exists a natural number nsuch that
(7) ∆m;y1,y2,...,yn`1˚f0
holds for each y1, y2, . . . , yn`1in K. If fis nonzero, then the smallest n
with this property is defined as the degree of f. Obviously, every exponen-
tial is a generalized exponential monomial of degree zero, associated with
itself. Indeed, the exponential mclearly satisfies ∆m;y˚m0 for each y.
To understand the difference between the group case and the case of general
hypergroups we note that, for instance, in the group case every generalized
exponential monomial ϕof degree at most 1 associated with the given expo-
nential mhas the form
ϕpxq “ `apxq ` c˘mpxq
with some complex number c, where ais additive, while son hypergroups
there are other functions of this type. This can be verified using the general
description of exponentials and additive functions on some special hyper-
groups, like polynomial hypergroups, Sturm–Liouville hypergroups, etc. The
interested reader will find further details in [25].
Functional equations and stability problems on hypergroups 11
Generalized exponential monomials associated with the exponential iden-
tically 1 are called generalized polynomials. It is known that on Abelian groups
generalized polynomials can be represented in a unique manner as the sum of
the diagonalizations of symmetric multi-additive functions. According to our
knowledge a similar result on hypergroups has not been published yet. Lin-
ear combinations of generalized exponential monomials are called generalized
exponential polynomials.
An important subclass of generalized exponential monomials is formed
by the ones whose variety is finite dimensional. In fact, the generalized ex-
ponential monomial fis called simply an exponential monomial, if τpfqis
a finite dimensional vector space. Similarly, a generalized polynomial fis
called a polynomial, if τpfqis a finite dimensional vector space. Accordingly,
linear combinations of exponential monomials are called exponential poly-
nomials. As we noted above in the group case an exponential monomial is
always the product of an exponential and an ordinary polynomial of addi-
tive functions. However, in the hypergroup case we have different situation.
In fact, a complete description of all exponential monomials on commuta-
tive hypergroups is still missing. In other words, the solution space of the
functional equation (7) on arbitrary commutative hypergroups has not been
characterized yet. Still there are some types of hypergroups on which a com-
plete description of some of the above function classes is available (see e.g.
[28, 25, 11, 13, 12, 21, 22, 14, 17]). In the subsequent sections we present
some examples where such a description has been obtained.
5. Polynomial hypergroups
An important special class of Hermitian hypergroups is closely related
to orthogonal polynomials.
Let panqnPN,pbnqnPNand pcnqnPNbe real sequences with the following
properties: cną0, bně0, an`1ą0 for each nin N, moreover a0b00
and an`bn`cn1 for each nin N. We define the sequence of polynomials
pPnqnPNby P0pλq “ 1, P1pλq “ λand by the recursive formula
λPnpλq “ anPn´1pλq ` bnPnpλq ` cnPn`1pλq
for each ně1 and λin R. The following theorem holds (see [3]).
Theorem 5.1. If the sequence of polynomials pPnqnPNsatisfies the above con-
ditions, then there exist constants cpn, l, kqfor each n, l, k in Nsuch that
Pn¨Pl
n`l
ÿ
k“|n´l|
cpn, l, kqPk
holds for each n, l in N.
12 aszl´o Sz´ekelyhidi
The formula in the theorem is called linearization formula and the co-
efficients cpn, l, kqare called linearization coefficients. The recursive formula
for the sequence pPnqnPNimplies Pnp1q “ 1 for each nin N, hence we have
n`l
ÿ
k“|n´l|
cpn, l, kq “ 1
for each nin N. If the linearization is nonnegative, that is, the linearization
coefficients are nonnegative: cpn, l, kq ě 0 for each n, l, k in N, then we can
define a hypergroup structure on Nby the following rule:
δn˚δl
n`l
ÿ
k“|n´l|
cpn, l, kqδk
for each n, l in N, with involution as the identity mapping and with eas 0.
The resulting discrete Hermitian (hence commutative) hypergroup is called
the polynomial hypergroup associated with the sequence pPnqnPN. We shall
denote it by pN,pPnqnPN.
As an example we consider the hypergroup associated with the Legendre
polynomials. The corresponding recurrence relation is
λPnpλq “ n`1
2n`1Pn`1pλq ` n
2n`1Pn´1pλq
for each ně1 and λin R. It can easily be seen that the linearization
coefficients are nonnegative and the resulting hypergroup associated with
the Legendre polynomials is the Legendre hypergroup.
Another interesting example for polynomial hypergroups is presented
by the Chebyshev polynomials. The corresponding recurrence relation in the
case of Chebyshev polynomials of the first kind is
λTnpλq “ 1
2Tn`1pλq ` 1
2Tn´1pλq
for each ně1 and λin R. Again, it is easy to see that the linearization
coefficients are nonnegative and the resulting hypergroup associated with
the Chebyshev polynomials of the first kind is the Chebyshev hypergroup.
The previous examples about the exponential and additive functions
on the hypergroup Dpθqsuggest that there is some hope to describe all ex-
ponential and additive functions on different polynomial hypergroups, too.
We start with the Chebyshev hypergroup. We recall that m:NÑCis an
exponential on the Chebyshev hypergroup if and only if it satisfies
mpk˚lq “ mpkqmplq
for each k, l in N. From the linearization formula it follows easily by induction
that
TkpλqTlpλq “ 1
2`Tk`lpλq ` T|k´l|pλq˘
Functional equations and stability problems on hypergroups 13
holds for each k, l in Nand λin C. This means that for each function f:
NÑCwe have
fpk˚lq “ 1
2ˆfpk`lq ` fp|k´l|q˙
for each k, l in N. Consequently, exponentials of the Chebyshev hypergroup
are exactly the nonzero solutions of the functional equation
mpk`lq ` mp|k´l|q “ 2mpkqmplq
for each k, l in N. This functional equation is closely related to d’Alembert’s
functional equation and has been treated – among others – in [4] indepen-
dently of hypergroups and in [17] on hypergroups. From our consideration it
is clear that the functions kÞÑ Tkpλqsatisfy this functional equation. In other
words, the Chebyshev polynomials evaluated at any complex λas functions
of the subscript present exponential functions on the Chebyshev hypergroup.
It turns out that this is true for every polynomial hypergroup. It turns out
that the converse is also true: every exponential on a polynomial hypergroup
is generated in this way. As different complex values of λproduce differ-
ent exponentials, this means that the set of all exponentials of a polynomial
hypergroup can be identified with the set of all complex numbers.
The following theorem presents a complete description of the exponen-
tials on arbitrary polynomial hypergroups (see [3], [21]).
Theorem 5.2. Let Kbe the polynomial hypergroup associated with the se-
quence of polynomials pPnqnPN. The function m:NÑCis an exponential
on Kif and only if there exists a complex number λsuch that
mpkq “ Pkpλq
holds for each kin N.
Applying this result for the Legendre hypergroup we have that the ex-
ponential functions in that case are exactly the functions nÞÑ Pnpλqon N,
where λis any complex number and Pnis the n-th Legendre polynomial.
For the description of the additive functions on the Chebyshev hyper-
group we know that a:NÑCis additive on the Chebyshev hypergroup if
and only if it satisfies the functional equation
apk`lq ` ap|k´l|q “ 2apkq ` 2aplq
for each k, l in N. Surprisingly, this functional equation is closely related to
the square-norm functional equation. In fact, any solution of this functional
equation has the form apkq “ c¨k2with some complex number c. This means
that additive functions on the Chebyshev hypergroup are exactly the qua-
dratic functions on N. We can interpret this result in a somewhat surprising
manner by observing that T1
np1q “ n2holds for each nin N, where T1
nis
the derivative of the n-th Chebyshev polynomial of the first kind. Conse-
quently, additive functions of the Chebyshev hypergroup have the general
14 aszl´o Sz´ekelyhidi
form: nÞÑ c¨T1
np1qwith some complex number c. This is a special case of
the following remarkable result (see [21]).
Theorem 5.3. Let Kbe the polynomial hypergroup associated with the se-
quence of polynomials pPnqnPN. The function a:NÑCis an additive func-
tion on Kif and only if there exists a complex number csuch that
apnq “ c P 1
np1q
holds for each nin N.
Finally we note that the study of generalized moment functions on hy-
pergroups leads to the study of the system of functional equations (3). We
remark that a similar system of functional equation on groupoids has been in-
vestigated and solved in [1]. The following theorem describes the generalized
moment function sequences of order Nin the case of polynomial hypergroups
(see [13]).
Theorem 5.4. Let Kbe the polynomial hypergroup associated with the se-
quence of polynomials pPnqnPN. The functions ϕ0, ϕ1, ..., ϕN:KÑCform
a generalized moment function sequence of order Non Kif and only if
ϕkpnq“pPn˝fqpkqp0q
holds for each nin Nand for k0,1, . . . , N, where
(8) fptq “
N
ÿ
j0
cj
j!tj
for each tin R, where cjis a complex number pj0,1, . . . , Nq.
6. Stability of additive functions
The study of stability problems concerning functional equations started
with S. Ulam’s question at the Mathematics Club of the University of Wis-
consin: Suppose that a group Gand a metric group Hare given. For any
εą0, does there exist a δą0 such that if a function f:GÑHsatisfies
dpfpxyq, f pxqfpyqq ă δ
for all x, y in G, then a homomorphism a:GÑHexists with
dpfpxq, apxqq ă ε
for all xin G? These kind of questions form the material of the stability
theory and D. H. Hyers obtained the first important result on this field (see
[8]). Later several mathematicians joined these investigations but the work
of D. H. Hyers is still decisive. In fact, he proved the following theorem.
Theorem 6.1. Let X, Y be Banach spaces and let f:XÑYbe a mapping
satisfying
}fpx`yq ´ fpxq ´ fpyq} ď ε
Functional equations and stability problems on hypergroups 15
for all x, y in X. Then the limit
apxq “ lim
nÑ8
fp2nxq
2n
exists for all xin Xand a:XÑYis the unique additive function satisfying
}fpxq ´ apxq} ď ε
for all xin X.
We note that the uniqueness follows immediately from the obvious fact
that the difference of two additive functions is additive, too, and the only
bounded additive function is 0.
This pioneer result of Hyers can be expressed in the following way:
Cauchy’s functional equation is stable for any pair of Banach spaces. The
function px, yq ÞÑ fpx`yq ´ fpxq ´ fpyqis called the Cauchy difference of
the function f. Functions with bounded Cauchy difference are called approx-
imately additive mappings. The sequence ´fp2nxq
2n¯nPNis called the Hyers–
Ulam sequence.
There are several possible ways to generalize the result of Hyers. A
natural way is to generalize the domain Xdepending on a more general
result of J. R¨atz [15]. Here we give a corresponding result on hypergroups.
Our proof has the novelty of using Banach limits.
Theorem 6.2. Let Kbe a hypergroup with the property that for each x, y in
Kthere exists an integer Ně2such that for něNwe have
(9) px˚yqnxn˚yn.
Then the functional equation (2) is stable for the pair pK, Cq.
We note that here powers are meant in the sense of convolution which
is associative, by virtue of the hypergroup axiom (H1) above.
Proof. By assumption, the function f:KÑCsatisfies
(10) |fpx˚yq ´ fpxq ´ fpyq| ď L
for each x, y in Kwith some positive number L. Putting xywe have
(11) |fpx2q ´ 2fpxq| ď L
for each xin K. For x2in place of xthis yields
|fpx4q ´ 2fpx2q| ď L ,
hence, by (11), it follows
|fpx4q ´ 4fpxq| ď 3L .
Repeating this argument we get by induction
|fpx2nq ´ 2nfpxq| ď p2n´1qL
16 aszl´o Sz´ekelyhidi
for each xin K. Division by 2ngives
(12) ˇˇˇfpx2nq
2n´fpxqˇˇˇď`1´1
2n˘L ,
which shows that the Hyers–Ulam sequence ´fpx2nq
2n¯nPNis bounded. Let
LIM denote any Banach limit on N, then, by (12), we have that the function
a:KÑCdefined by
(13) apxq “ LIM fpx2nq
2n
is well-defined for xin K, and it satisfies
|apxq ´ fpxq| ď L
for each xin K. On the other hand, for each x, y in Kwe have
(14) apx˚yq ´ apxq ´ apyq “ LIM´f`px˚yq2n˘´fpx2nq ´ fpy2nq
2n¯.
By assumption, if nis large enough then we have
|f`px˚yq2n˘´fpx2nq ´ fpy2nq| “ |fpx2n˚y2nq ´ fpx2nq ´ fpy2nq| ď L ,
which implies, by (11), that ais additive.
This theorem gives the stability of Cauchy’s functional equation in the
group case, too, moreover, in contrast with Hyers’ Theorem, we don’t need
the commutativity of the domain. Nevertheless, the condition of Theorem 6.2
is quite sophisticated and artificial. On non-commutative groups and semi-
groups the present author proposed another approach based on the concept
of invariant means (see [20]). Now we show the application of this method on
hypergroups.
Let Kbe a hypergroup and let BpKqdenote the Banach space of all
bounded complex valued functions on Kequipped with the sup norm },}.
A linear functional Mof the space BpKqis called a right invariant mean, if
Mp1q “ 1 and Mpτyfq “ Mpfqholds for each yin Kand fin BpKq. We
call Kleft amenable, if there exists a left invariant mean on BpKq. Right
invariant means and right amenability are defined in a similar way. In case
of commutative hypergroups we simply use the terms invariant mean and
amenable hypergroup. For more about invariant means see e.g. [6]. It turns out
that wide classes of groups and even semigroups are amenable. Amenability
of commutative hypergroups has been proved in [27] (see also [10]). Now we
prove the stability of additive functions on right amenable hypergroups.
Theorem 6.3. Let Kbe a right amenable hypergroup. Then the functional
equation (2) is stable for the pair pK, Cq.
Proof. Let Mbe a left invariant mean on Kand f:KÑCa function
satisfying
(15) |fpx˚yq ´ fpxq ´ fpyq| ď L
Functional equations and stability problems on hypergroups 17
for each x, y in Kwith some positive number L. For each yin Kthe function
xÞÑ fpx˚yq ´ fpxqis bounded, and we define
(16) apyq “ Mx`fpx˚yq ´ fpxq˘.
Here Mydenotes that Mis applied to the argument as a function of x. Now
we have
apy˚zq ´ apyq ´ apzq “ Mx`fpx˚y˚zq ´ fpx˚yq ´ fpx˚zq ` fpxq˘
Mx`fpx˚y˚zq ´ fpx˚yq˘´Mx`fpx˚zq ´ fpxq˘0,
as the argument of Min the first term is the right translate of the second
term by y. It follows that ais additive. On the other hand, for each yin K
we have
|fpyq ´ apyq| “ |fpyq ´ Mx`fpx˚yq ´ fpxq˘|“|Mx`fpxq ` fpyq ´ fpx˚yq˘| ď
Mx`|fpxq ` fpyq ´ fpx˚yq|˘ďL .
The theorem is proved.
This result extends easily to the pexiderized equation of additive func-
tions as it is shown in the following theorem.
Theorem 6.4. Let Kbe a right amenable hypergroup and let f, g, h :KÑC
be functions such that the function px, yq ÞÑ fpx˚yq ´ gpxq ´ hpyqis bounded.
Then there exists an additive function a:KÑCsuch that f´a, g ´aand
h´aare bounded.
7. Stability of exponential functions
The stability of exponential functions was proved firstly in [2] for real
valued functions defined on linear spaces. By the results in [19] we have the
following result.
Theorem 7.1. Let Sbe a commutative semigroup with identity, and suppose
that for the functions f, m :SÑCthe function xÞÑ fpx`yq ´ fpxqmpyqis
bounded for each yin S. Then either fis bounded, or mis an exponential.
This result shows the so-called superstability property of the exponential
functional equation: the difference fpx`yq´ fpxqfpyqcan be bounded if and
only if it is either zero, or fitself is bounded. Now we study this problem on
hypergroups.
Theorem 7.2. Let Kbe a hypergroup and let f, g, h :KÑCbe continuous
functions. If the function
xÞÑ fpx˚yq ´ gpxqhpyq
is bounded for each yin K, then either fis bounded, or hpeq ‰ 0and h{hpeq
is an exponential.
18 aszl´o Sz´ekelyhidi
Proof. Suppose that fis unbounded. Then, putting yeinto the above
condition, hpeq ‰ 0 follows, and we have that f´hpeqgis bounded. Moreover,
by assumption, we have
|hpeqgpx˚yq ´ gpxqhpyq| ď lpyq
for each x, y in Kwith some function l:KÑC. Dividing by hpeq2it follows
that the function
xÞÑ 1
hpeqgpx˚yq ´ 1
hpeqgpxq ¨ 1
hpeqhpyq
is bounded for each yin K. Theorem 11.1 in [25] implies that either gis
bounded, or hpeq ‰ 0 and h{hpeqis an exponential. However, gcannot be
bounded, otherwise fis bounded, too. This implies that hpeq ‰ 0 and h{hpeq
is an exponential.
Obviously, this result implies the following.
Theorem 7.3. Let Kbe a hypergroup and f:KÑCa continuous function
such that the function
px, yq ÞÑ fpx˚yq ´ fpxqfpyq
is bounded. Then either fis bounded or it is an exponential.
In other words, the exponential functional equation is superstable on
any hypergroup. In the following section we will have an application of this
result for spherical functions.
8. Double coset hypergroups
In this section we exhibit another important type of hypergroups: the
double coset hypergroups. The idea is that if Gis a locally compact group
and Kis a subgroup then, in general, the left, or right, or double coset
spaces with respect to Kdo not bear any reasonable structure. Nevertheless,
if Kis a compact subgroup then a quite useful hypergroup structure can be
introduced on the double coset space with respect to K. In particular, this
hypergroup structure reduces to the usual group structure if Kis a normal
subgroup. In the subsequent paragraphs we present the details (see also [3]).
Let Gbe a locally compact group with identity eand Ka compact
subgroup with normed Haar measure ω:şKpkq “ 1. As Kis unimodular
ωis left and right invariant, and also inversion invariant. For each xin Gwe
define the double coset of xas the set
KxK “ tkxl :k, l PKu.
We introduce a hypergroup structure on the set LG{{Kof all double
cosets: the topology of Lis the factor topology, which is locally compact.
The identity ois the coset KKeK itself and the involution is defined by
pKxK q_K x´1K .
Functional equations and stability problems on hypergroups 19
Finally, the convolution of δKxK and δK yK is defined by
δKxK ˚δK yK żK
δKxky K pkq.
It is known that this gives a hypergroup structure on L(see [3], p. 12.),
which is non-commutative, in general. If Kis a normal subgroup, then Lis
isomorphic to the hypergroup arising from the factor group G{K.
We note that continuous functions on Lcan be identified with those
continuous functions on Gwhich are K-invariant: fpxq “ fpkxlqfor each x
in Gand k, l in K. Hence, for a continuous function f:LÑCthe simplified
– and somewhat loose – notation fpxqcan be used for the function value
fpKxK q. Using this convention we can write for each continuous function
f:LÑCand for each x, y in G:
fpx˚yq “ żK
fpxkyqpkq.
The following theorem exhibits a close connection between exponen-
tials on double coset hypergroups and spherical functions on locally compact
groups. Following the terminology of [3] (see also [5]) we recall the concept of
spherical functions. Let Gbe a locally compact group and Ka compact sub-
group with Haar measure ω. The continuous bounded K-invariant function
f:GÑCis called a K-spherical function if fpeq “ 1 and
(17) żK
fpxkyqpkq “ fpxqfpyq
holds for each x, y in G. A generalized Kspherical function on Gis the same as
above without the boundedness hypothesis. For the sake of simplicity in this
paper we use the term spherical function for continuous functions satisfying
(17) without the boundedness assumption. The following theorem, which is an
immediate consequence of the previous considerations gives the link between
spherical functions and exponentials of double coset hypergroups.
Theorem 8.1. Let Gbe a locally compact group, and KĎGa compact
subgroup. Then the nonzero continuous complex valued function mis a K-
spherical function on Gif and only if it is an exponential on the double coset
hypergroup G{{K. In particular, K-spherical functions on Gcan be identified
with the characters of G{{K.
In virtue of this theorem, as an application of Theorem 7.2, we obtain
the following result on the superstability of functional equations related to
spherical functions (see [18]).
Theorem 8.2. Let Gbe a locally compact group and Ka compact subgroup
with normed Haar measure ω. Let f, g, h :GÑCbe continuous K-invariant
functions such that the function
xÞÑ żK
fpxkyqpkq ´ gpxqhpyq
20 aszl´o Sz´ekelyhidi
is bounded for each yin G. Then either fis bounded, or hpeq ‰ 0and h{hpeq
is a K-spherical function.
9. Superstability of generalized moment functions
In this section we prove that generalized moment functions on hyper-
groups also have the remarkable superstability property (see also [23, 25]).
Theorem 9.1. Let Kbe hypergroup, na nonnegative integer, and suppose that
for the unbounded functions fk:KÑC(k0,1, . . . , nqthe functions
px, yq ÞÑ fkpx˚yq ´
k
ÿ
j0ˆk
j˙fjpxqfk´jpyq
are bounded on KˆK. Then the sequence pfkqkďnforms a moment function
sequence of order non K.
Proof. We prove the theorem for a fixed nby induction on k. For k0, by
our assumption, the function
px, yq ÞÑ f0px˚yq ´ f0pxqf0pyq
is bounded on KˆK. By Theorem 7.2, this implies that f0is an exponential
on K.
Suppose now that kě1 and we have proved that the functions fjfor
j0,1, . . . , k ´1 form a moment function sequence of order k´1 on K. By
assumption, we have that the function
px, y, zq ÞÑ Fpx, y, zq “ fkpx˚y˚zq ´
k
ÿ
j0ˆk
j˙fjpx˚yqfk´jpzq,
and also the function
px, y, zq ÞÑ Gpx, y, zq fkpx˚y˚zq ´
k
ÿ
j0ˆk
j˙fjpxqfk´jpy˚zq
is bounded on KˆKˆK. Then their difference
px, y, zq ÞÑ Fpx, y, zq ´ Gpx, y , zq “
k
ÿ
j0ˆk
j˙fjpx˚yqfk´jpzq ´
k
ÿ
j0ˆk
j˙fjpxqfk´jpy˚zq
is also bounded. By our induction hypothesis, this means that the function
px, y, zq ÞÑ Fpx, y, zq ´ Gpx, y, z q “ Hpx, y, zq “
k´1
ÿ
j1ˆk
j˙fjpxq
k´j
ÿ
i0ˆk´j
i˙fipyqfk´j´ipz
k´1
ÿ
j1ˆk
j˙j
ÿ
i0ˆj
i˙fipxqfj´ipyqfk´jpzq`
Functional equations and stability problems on hypergroups 21
`f0pxqfkpy˚zq ´ fkpx˚yqf0pzq ` fkpxqf0pyqf0pzq ´ f0pxqf0pyqfkpzq
is bounded, too. By reordering the terms in this sum we obtain
Hpx, y, zq “ f0pxqfkpy˚zq ´
k
ÿ
j0ˆk
j˙fjpyqfk´jpzq
´f0pzqfkpx˚yq ´
k
ÿ
j0ˆk
j˙fjpxqfk´jpyq`
k´1
ÿ
j1
k´j´1
ÿ
i0ˆk
j˙ˆk´j
i˙fjpxqfipyqfk´j´ipz
k´1
ÿ
j1
j
ÿ
i1ˆk
j˙ˆj
i˙fipxqfj´ipyqfk´jpzq
for all x, y, z in K. We show that the two terms on the right hand side of the
last equality cancel. In the first term replacing iby t´j, and in the second
term interchanging the sums we have
Hpx, y, zq “ f0pxqfkpy˚zq ´
k
ÿ
j0ˆk
j˙fjpyqfk´jpzq´
f0pzqfkpx˚yq ´
k
ÿ
j0ˆk
j˙fjpxqfk´jpyq`
k´1
ÿ
j1
k´1
ÿ
tjˆk
j˙ˆk´j
k´t˙fjpxqft´jpyqfk´tpz
k´1
ÿ
i1
k´1
ÿ
jiˆk
j˙ˆj
i˙fipxqfj´ipyqfk´jpzq
for all x, y, z in K. In the second term we write jfor iand tfor jto get
Hpx, y, zq “ f0pxqfkpy˚zq ´
k
ÿ
j0ˆk
j˙fjpyqfk´jpzq´
f0pzqfkpx˚yq ´
k
ÿ
j0ˆk
j˙fjpxqfk´jpyq`
k´1
ÿ
j1
k´1
ÿ
tjˆk
j˙ˆk´j
k´t˙fjpxqft´jpyqfk´tpz
k´1
ÿ
j1
k´1
ÿ
tjˆk
t˙ˆt
j˙fjpxqft´jpyqfk´tpzq
for all x, y, z in K. On the other hand, we have
ˆk
j˙ˆk´j
k´t˙k!
j!pk´jq!
pk´jq!
pk´tq!pt´jq!k!
t!pk´tq!
t!
j!pt´jq!ˆk
t˙ˆt
j˙,
22 aszl´o Sz´ekelyhidi
hence the function
px, y, zq ÞÑ Lpx, y, zq f0pxqfkpy˚zq ´
k
ÿ
j0ˆk
j˙fjpyqfk´jpzq´
f0pzqfkpx˚yq ´
k
ÿ
j0ˆk
j˙fjpxqfk´jpyq
is bounded. If there are y, z in Ksuch that
fkpy˚zq ´
k
ÿ
j0ˆk
j˙fjpyqfk´jpzq ‰ 0,
then f0is bounded, which is impossible. Thus we have
fkpy˚zq ´
k
ÿ
j0ˆk
j˙fjpyqfk´jpzq “ 0
for all y, z in K, and the proof is complete.
10. Stability problems of other functional equations
In the previous sections we discussed two different types of stability: one
of them is for additive-type equations, where the method is based on invariant
means, and the other is for exponential-type equations, where superstability
appears, and the method is direct. The question arises about stability results
of mixed type, where both the additive and exponential equations come into
the picture and we can combine the two methods.
We mentioned above that the second equation of the system defining
generalized moment function sequences is interesting on its own: we can con-
sider it independently from the other equations. It has the form
(18) fpx˚yq “ fpxqgpyq ` gpxqfpyq.
It turns out that the pexiderized version
(19) fpx˚yq “ gpxqhpyq ` lpxqkpyq,
can be treated in the case l1. We have also considered another similar
equation in [25]. In the special case of this equation, when k0 we have the
pexiderized exponential equation, and in the case hl1 we obtain the
pexiderized additive equation. We recall the corresponding results of [25].
Theorem 10.1. Let Kbe an amenable discrete hypergroup and suppose that
the functions f, g, h, k :KÑCare given, where fis unbounded. Then the
function
xÞÑ fpx˚yq ´ gpxqhpyq ´ kpyq
Functional equations and stability problems on hypergroups 23
is bounded for each yin Kif and only if we have
fpxq “ ϕpxq ` b1pxq
gpxq “ ϕpxq ` b2pxq
hpxq “ mpxq
kpxq “ ϕpxq ` b3pxq
where m:KÑCis an exponential, b1, b2, b3:KÑCare bounded functions,
and ϕ:KÑCsatisfies the functional equation
(20) ϕpx˚yq “ ϕpxqmpyq ` ϕpyq
for each x, y in K, further, if b2is nonzero, then mis bounded.
We can see here that we have a superstability with respect to hand
stability with respect to the other three functions. Another specialty is that
we have the stability result without knowing the general solution of the cor-
responding functional equation
(21) fpx˚yq “ gpxqhpyq ` kpyq.
At this point we mention the open problem about the stability of the sine
equation (18). Another related stability result in [25] is the following.
Theorem 10.2. Let Kbe a discrete commutative hypergroup and suppose that
f, g, h, k, l :KÑCare unbounded functions. Then the function
(22) xÞÑ fpx˚yq ´ gpxqhpyq ´ kpxq ´ lpyq
is bounded for each yin Kif and only if either
(23) fpxq “ λ
2apxq2`d0apxq ` a0pxq ` b1pxq
gpxq “ apxq ` c0, hpxq “ λapxq ` d0
kpxq “ λ
2apxq2`a0pxq ` b2pxq
lpxq “ λ
2apxq2` pd0´λc0qapxq ` a0pxq ` b3pxq,
or
(24) fpxq “ cdrmpxq ´ 1s ` apxq ` b1pxq
gpxq “ crmpxq ´ 1s ` c0, hpxq “ drmpxq ´ 1s ` d0
kpxq “ cpd´d0qrmpxq ´ 1s ` apxq ` b2pxq
lpxq “ dpc´c0qrmpxq ´ 1s ` apxq ` b3pxq,
where m:KÑCis an exponential, a, a0:KÑCare additive functions,
b1, b2, b3:KÑCare bounded functions, and λ, c, d, c0, d0are complex num-
bers.
The proof is again a combination of direct methods and of the invariant
mean technique. We note that – in contrast with the previous theorem –
commutativity is used instead of amenability.
24 aszl´o Sz´ekelyhidi
Acknowledgment
Research was supported by OTKA Grant No. K111651.
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aszl´o Sz´ekelyhidi
Department of Mathematics, University of Debrecen
Institute of Mathematics, H-4010 Debrecen, Hungary
E-mail: lszekelyhidi@gmail.com
... As M is maximal, it is prime and we infer that Δ m i ;u is in M for each u in H. It follows that the ideal generated by all Δ m i ;u 's with u in H, is in M , too, but this ideal is exactly the maximal ideal Ann τ (m i ), the annihilator of τ (m i ) (see [12]). By maximality, Ann τ (m i ) = M , and also Ann τ (m j ) = M , a contradiction as m i = m j . ...
... whence, by taking annihilators (see [12]) ...
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