# An Extension of the Spectral Mapping Theorem

Authors: A. R. Medghalchi; S. M. Tabatabaie
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

---

## Abstract

We give an extension of the spectral mapping theorem
on hypergroups and prove that ifK is a commutative strong hypergroup with K^=Xb(K) and κ is a weakly continuous representation of M(K) on a W∗-algebra such that for every t∈K, κt is an ∗-automorphism, spκ is a synthesis
set for L1(K) and κ(L1(K)) is without order, then for any μ in a closed regular
subalgebra of M(K) containing L1(K), σ(κ(μ))=μ^(spκ)¯, where spκ is the
Arveson spectrum of κ.


---

## Body

## 1. Introduction and Notation

Hypergroups were introduced in a series of papers by
Jewett [1],  Dunkle [2], and
Spector [3] in 70's.
They are in fact extensions of topological groups with the difference that hypergroups
do not have necessarily an algebraic structure. Roughly speaking, the product
of two elements of a hypergroup is a probability measure. A hypergroup is a
locally compact space which has enough structure so that a convolution on the
space of finite regular Borel measures can be defined. Therefore, the extension
of Fourier analysis on hypergroups is made with more difficulties and usually
with different proofs in the group case. Classical examples of hypergroups are
locally compact groups, the space of conjugacy classes of a compact group,
spaces of orbits in the group of automorphisms, and double-cosets of certain
nonnormal closed subgroups of a compact group. We will state definition and
some basic properties of hypergroups in Section 2. Throughout this paper, K is a commutative strong hypergroup with K^=Xb(K).
In Section 4, we give some examples of this type of hypergroups. We denote by M (K) the space of all regular complex Borel
measures on K,
by M+(K) the subset of positive measures in M (K), and by δx the Dirac measure at the point x.AC*-algebra M is called W*-algebra if for a Banach algebra M*, (M*)*=M.
Any W*-algebra is unitary (with unit 1M). The famous examples of W*-algebras are von Neumann algebras. We can
consider σ(M,M*) (i.e., wk*-) topology on M [4]. In this paper, M is always a W*-algebra. We denote by Bσ(M) the set of all (σ(M,M*),σ(M,M*))-continuous operators on M.Letκ:M(K)→Bσ(M) be a norm-decreasing algebra-homomorphism. For
any t∈K, we denote κt=κ(δt). Suppose that κ has the following properties.(1)
For anyt∈K, κt:M→M is an ∗-automorphism. (Then by [5, Theorem 4.8, page 253]
any κt is an isometry.)(2)
For anyx∈M and ρ∈M*,
the function t↦〈κt(x),ρ〉 is continuous.(3)
κe=IM,
where e is the identity of K and IM is the identity mapping on M.Letx∈M and ρ∈M*.
Obviously for any measure υ∈M(K) with finite support, we have 〈κ(υ)(x),ρ〉=∫K〈κt(x),ρ〉dυ(t).
Let μ∈M(K).
Since the set E containing all measures in M(K) that have finite support is dense in M(K),
there exists a net (υβ)⊆E such that υβ→μ in M(K).
Then by [1, Lemma 2.2C], we have ∫Kfdυβ→∫Kfdμ,
where f(t)=〈κt(x),ρ〉(t∈K). On the other hand, by continuity of κ, ∫Kfdυβ=〈κ(υβ)(x),ρ〉→〈κ(μ)(x),ρ〉.
Then, for any μ∈M(K) we have(1.1)〈κ(μ)(x),ρ〉=∫K〈κt(x),ρ〉dμ(t),where x∈M and ρ∈M*.


## 2. Basic Properties of Hypergroups

First, we
recall the definition and basic properties of a hypergroup. The main references
are [1, 6].Definition 2.1.
LetK be a locally compact Hausdorff space. The
space K is a hypergroup if there exists a binary
mapping (x,y)↦δx∗δy from K×K into M+(K) satisfying the following conditions.
(1)
The mapping(δx,δy)↦δx∗δy extends to a bilinear associative operator ∗ from M(K)×M(K) into M (K) such that(2.1)∫Kfd(μ∗ν)=∫K∫K∫Kfd(δx∗δy)dμ(x)dν(y)for all continuous functions f on K vanishing at infinity.(2)
For eachx,y∈K the measure δx∗δy is a probability measure with compact support.(3)
The mapping(μ,ν)↦μ∗ν is continuous from M+(K)×M+(K) into M+(K);
the topology on M+(K) being the cone topology.(4)
There exists ane∈K such that δe∗δx=δx=δx∗δe,
for all x∈K.(5)
There exists a homeomorphism involutionx↦x− from K onto K such that, for all x,y∈K,
we have (δx∗δy)−=δy−∗δx−,
where for μ∈M(K), μ− is defined by(2.2)∫Kf(t)dμ−(t)=∫Kf(t−)dμ(t),and also,(2.3)e∈supp(δx∗δy)iffy=x−,where supp(δx∗δy) is the support of the measure δx∗δy.(6)
The mapping(x,y)↦supp(δx∗δy) from K×K into the space C(K) of compact subsets of K is continuous, where C(K) is given the topology whose subbasis is given
by all(2.4)CU,V={A∈C(K):A∩U⁢≠∅,⁢A⊆V},where U, V are open subsets of K.Note thatδx∗δy is not necessarily a Dirac measure. A
hypergroup K is commutative if δx∗δy=δy∗δx,
for all x,y in K.
Each commutative hypergroup K carries a Haar measure m such that δx∗m=m, for all x∈K,
as shown by Spector [7]. In any commutative hypergroup K, we have m=m− (see [1, Section 5.3]). Let f,g be Borel functions on K and μ∈M(K).
For any x,y∈K we denote fx(y)=f(x∗y):=∫Kfd(δx∗δy). Also we define(2.5)(μ∗f)(x):=∫Kf(y−∗x)dμ(y),(f∗g)(x):=∫kf(x∗y)g(y−)dm(y),where x∈K.
If x,y∈K and A,B⊆K, we denote {x}∗{y}=supp(δx∗δy),
and A∗B=⋃x∈A,y∈B{x}∗{y}.A complex-valued continuous functionξ on K is said to be multiplicative if ξ(x∗y)=ξ(x)ξ(y) holds for all x,y∈K.
The space of all multiplicative functions on K is denoted by Xb(K).
A nonzero multiplicative function ξ on K is called a character if ξ(x−)=ξ(x),¯ for all x in K.
The dual K^ of K is the locally compact Hausdorff space of all
characters with the topology of uniform convergence on compacta. In general, K^ is not necessarily a hypergroup. A hypergroup K is called strong if its dual K^ is also a hypergroup with complex conjugation as
involution, pointwise product as convolution, that is(2.6)η(x)χ(x)=∫K^ξ(x)dδη∗δχ(ξ),for all η,χ∈K^ and x∈K,
and has the constant function 1 as the identity
element.For anyp>0,
we denote Lp(K)=Lp(K,m) and Lp(K^)=Lp(K^,π), where π is the Plancherel measure on K^ associated with m.
The structure space Δ(L1(K)) of Banach algebra L1(K) doses not necessarily equal with K^ and we only have Δ(L1(K))=Xb(K),
while K^⊆Xb(K).For anyf∈L1(K) and μ∈M(K),
the Fourier-Stieltjes transform μ^ of μ and the Fourier transform f^ of f are defined by(2.7)μ^(ξ)=∫Kξ(t)¯dμ(t),f^(ξ)=∫Kξ(t)¯f(t)dm(t),where ξ∈K^.
For any f,g∈L2(K),
we have f^,g^∈L2(K^) and f^∗g^=fg^ [6].


## 3. The Main Result

Recall that for any complex commutative algebraA, Δ(A),
the structure space of A, is a locally compact Hausdorff space with Gelfand
topology and for each a∈A,
the Gelfand transform a^ is in C0(Δ(A)).
For an ideal I of A,
the hull of I is defined by h(I)=ξ∈Δ(A):a^(ξ)=0 for every a∈I}.
For each closed F⊆Δ(A),
we denote k(F)={a∈A:a^(ξ)=0 for every ξ in F} and J0(F)={a∈A:a^=0 on some neighborhood of F,
and supp(a^) is compact},
where supp(a^) is the support of the function a^.
We denote J(F)=J0(F)¯,
the closure of J(F).
For any Banach algebra A,
the spectrum of any a∈A is denoted by σA(a).Definition 3.1.
The Arveson spectrum ofκ is defined by(3.1)spκ:=h({f∈L1(K):κ(f)=0})={ξ∈K^:foranyf∈L1(K),κ(f)=0⁢impliesf^(ξ)=0}.The Arveson spectrum and
spectral subspaces on hypergroups have been studied in [8]. In this paper, we
concentrate on an extension of the spectral mapping theorem to hypergroups. The
spectral mapping theorem gives the relation σ(κ(μ))=μ^(spκ)¯ for some measures μ∈M(K).
In the case that K is a locally compact Abelian group,  Connes
proved the spectral mapping theorem for every Dirac measure μ [9]. Then,   D'Antoni et al.
proved that
the spectral mapping theorem holds for those measures whose continuous part
belongs to L1(G) [10]. Furthermore,  Eschmeier proved the spectral mapping
theorem for κ as the translation group representation, in
the case that M is Banach algebra L1(G) or M(G) and the convolution operator induced by μ has the weak 2-SDP [11].  Takahasi and  Inoue
proved the spectral mapping theorem for any regular subalgebra of M(G) in the case that G is compact [12]. Also,  Seferoğlu proved that σ(κ(μ))=μ^(spec(κ))¯ holds when κ(L1(G)) is without order and spκ is a synthesis set for L1(G) [13]. (A subset C of Banach algebra A is called without order if for all a∈A, a.C={0} implies a=0 [14].)A Banach algebraA is called regular if for any closed subset E of Δ(A) and any φ∈Δ(A)∖E,
there is an a∈A such that a^(φ)=1 and a^≡0 on E.
For any commutative and strong hypergroup K with K^=Xb(K), L1(K) is a regular Banach algebra [15]. Also, since the Fourier transform ^:L1(K)→C0(K^) is a norm-decreasing ∗-homomorphism, L1(K) is semisimple. If A is a regular commutative Banach algebra, then
for any closed F⊆Δ(A), h(J(F))=F.
If A is also semisimple, then k(F) and J(F) are the largest closed ideal and the smallest
ideal with hull equal to F,
respectively [16]. F is called a set of spectral
synthesis if and only if J(F)=k(F).
Then, for such sets, k(F) is the only closed ideal in A such that its hull is F.It is well known that the correspondenceF↦h(k(F)) is a proper closure operation and that the
topology on Δ(A) determined by this operation, called
hull-kernel topology, is weaker than the Gelfand topology [17]. A commutative Banach
algebra A is regular if and only if the Gelfand and the
hull-kernel topologies coincide on Δ(A) (see [18, Theorem 3.2.10]). Another property of commutative
regular semisimple Banach algebras appears in the following lemma. It is given
here for the convenience of the reader.Lemma 3.2.
LetA be a commutative regular semisimple Banach
algebra, let B be a unitary Banach algebra (with unit 1B), and let κ:A→B be a continuous one-to-one homomorphism.
(i)
IfA is unitary with unit 1A and κ(1A)=1B, then for any a∈A,(3.2)σB(κ(a))=a^(Δ(A)).(ii)
IfA is without unit, then for any a∈A,(3.3)σB(κ(a))=a^(Δ(A))¯.Proof.
See [13, Lemmas 1 and 2]
for the proof.Lemma 3.3.
Suppose thatκ:M(K)→B(M) is the function introduced in Section 1. Let κ(L1(K)) be without order and let spκ be a spectral synthesis set for L1(K).
Then, κ(μ)=0 if and only if μ^|spκ=0.Proof.
Letμ∈M(K) and κ(μ)=0.
For any f∈L1(K), μ∗f∈L1(K) and κ(μ∗f)=κ(μ)κ(f)=0.
Then(3.4)μ∗f∈kh({μ∗f})⊆kh({f∈L1(K):κ(f)=0})=k(spκ).So μ^(ξ)f^(ξ)=μ∗f^(ξ)=0,
where ξ∈spκ.
For each ξ∈K^, there exists a g∈Cc+(K^) such that ξ∈supp(g).
We can consider h∈Cc(K) such that h^≥0 and h^>g on supp(g) (see [6, Proposition 2.2.5]). In other words, the set {f^:f∈Cc(K)} separates the points of K^.
Then, for any ξ∈spκ we have μ^(ξ)=0.
Conversely, letμ∈M(K) and μ^|spκ≡0.
Then, for each ξ∈K^ and f∈L1(K) we have μ∗f^(ξ)=μ^(ξ)f^(ξ)=0.
Now, since k(spκ) is the only closed ideal in L1(K) whose hull is spκ,(3.5)μ∗f∈k(spκ)=kh({f∈L1(K):κ(f)=0})={f∈L1(K):κ(f)=0}.Therefore, κ(μ)⋅κ(L1(K))=0.
Since κ(L1(K)) is without order, κ(μ)=0.
LetM0(K) be a closed subalgebra of M(K) which contains L1(K).
Then, K^ can be considered as a subset of Δ(M0(K)) and for every μ in K,
the restriction of the Gelfand transform Tμ of μ to K^ coincides with the Fourier-Stieltjes transform μ^ of μ.
This implies that M0(K) is a semisimple algebra.Theorem 3.4.
Suppose thatκ:M(K)→Bσ(M) is the function introduced in Section 1. Let κ(L1(K)) be without order and let spκ be a spectral synthesis set. Then, for any μ∈M0(K),(3.6)σB(M)(κ(μ))=μ^(spκ)¯,where M0(K) is a closed regular subalgebra of M(K) containing L1(K).Proof.
SinceL1(K)⊆M0(K)⊆M(K), K^≅Δ(L1(K)) is a subset of Δ(M0(K)).
For each μ∈M0(K), we denote by Tμ the Gelfand transform of μ.
We have Tμ∈C0(Δ(M0(K))) and μ^=Tμ|K^.
The function κ:M0(K)/Iκ→B(M) naturally defined by κ(μ+Iκ):=κ(μ) is a continuous one to one homomorphism, where Iκ={μ∈M0(K):κ(μ)=0}.
(Iκ clearly is a closed ideal of M0(K).)
Since0→Iκ→M0(K)→M0(K)/Iκ→0
is an exact sequence, by
[18, page 324] we have Δ(M0(K)/Iκ)≅h(Iκ).
Consider spκ as a subset of Δ(M0(K)) and denote by spκ¯g the closure of spκ in Gelfand topology. By regularity of M0(K),
Gelfand and hull-kernel topologies coincide on Δ(M0(K)).
So spκ¯g is closed in hull-kernel topology. But the
only hull-kernel-closed subsets of Δ(M0(K)) are hull-sets. Then, there exists a subset I⊆M0(K) such that spκ¯g=h(I).
Therefore,(3.7)hk(spκ¯g)=hkh(I)=h(I)=spκ¯g.On the other hand, for every μ∈M0(K),μ∈k(spκ¯g)  if and only if Tμ=0 on spκ¯g.
This is also equivalent to vanishing μ^ on spκ.
By Lemma 3.3, this holds if and only if κ(μ)=0.
Therefore, h(Iκ)=spκ¯g.
By the regularity ofM0(K),
the Banach algebra M0(K)/Iκ is also regular. For any μ∈M0(K), μ+Iκ∈rad(M0(K)/Iκ) if and only if μ+Iκ^=0 on Δ(M0(K)/Iκ).
This is also equivalent to vanishing Tμ on h(Iκ).
By the previous paragraph, this holds if and only if κ(μ)=0,
that is, μ∈Iκ.
Therefore, M0(K)/Iκ is a semisimple algebra.
In the sequel, we consider two cases.
Case 1.
Letspκ¯g be compact. There exists υ∈M0(K) such that Tυ=1 on a neighborhood spκ¯g.
Then, for any μ∈M0(K) we have(3.8)T(μ∗υ−μ)=Tμ⁢Tυ−Tμ=0onspκ¯g.So μ∗υ−μ^=0 on spκ and by Lemma 3.3μ∗υ−μ∈Iκ(⋆).
Then, for any μ∈M0(K), (μ+Iκ)(υ+Iκ)=μ+Iκ.
By (⋆),
for any f∈L1(K),(3.9)0=κ(υ∗f−f)=κ(υ)κ(f)−κ(f)=(κ(υ)−IB(M))⋅κ(f).That is (κ(υ)−IB(M))⋅κ(L1(K))=0.
Now, since κ(L1(K)) is without order, κ(υ)=IB(M).
Therefore, κ(υ+Iκ)=IB(M).
Now, we are in a position using Lemma 3.2(i)  for κ,M0(K)/Iκ and B(M).
Then, for every μ∈M0(K),(3.10)σB(M)(κ(μ+Iκ))=μ+Iκ^(Δ(M0(K)Iκ)),and so
(⋆ ⋆)σB(M)(κ(μ))=Tμ(spκ¯g).Now compactness of spκ¯g in Δ(M0(K)) and continuity of Tμ on Δ(M0(K)) imply that Tμ(spκ¯g) is compact and so is closed. This fact by
using relation μ^(spκ)=Tμ(spκ)⊆Tμ(spκ¯g) shows that(3.11)μ^(spκ)¯⊆Tμ(spκ¯g).On the other hand, since Tμ is continuous, we have(3.12)Tμ(spκ¯g)⊆Tμ(spκ)¯=μ^(spκ)¯.Thus,(3.13)μ^(spκ)¯=Tμ(spκ¯g).Now by
(⋆⋆), σB(M)(κ(μ))=μ^(spκ)¯.
Case 2.
Letspκ¯g be noncompact. Then, by Δ(M0(K)/Iκ)≅spκ¯g,M0(K)/Iκ   is not unitary.
So by Lemma 3.2(ii),
for any μ∈M0(K),(3.14)σB(M)(κ(μ+Iκ))=μ+Iκ^(Δ(M0(K)Iκ))¯.Now the inclusion μ^(spκ)⊆Tμ(spκ¯g) implies μ^(spκ)¯⊆Tμ(spκ¯g)¯.
Also since Tμ is continuous, we have Tμ(spκ¯g)⊆μ^(spκ)¯,
and so that Tμ(spκ¯g)¯⊆μ^(spκ)¯.
Therefore,(3.15)σB(M)(κ(μ))=Tμ(spκ¯g)¯=μ^(spκ)¯.


## 4. Examples

In this
section, we give some examples of hypergroupsK that are commutative strong hypergroups and K^=Xb(K) (we refer to these conditions by notation (℘)). Observe that any locally compact Abelian
group has these properties. Also, if G is a locally compact Abelian group and H is a compact subgroup of Aut(G),
then the space GH containing all H-orbits is a commutative hypergroup satisfying
in (℘). In fact, GH^≅(G^)H.
We refer to [19] for
more details. As another example, let G be a group such that G/Z is compact, where Z={x∈G:foranyy∈G,xy=yx}.
If K is the hypergroup containing all conjugacy
classes of G,
then K and its dual K^ satisfy in (℘) [6]. On the other hand, an interesting example of Naimark
given in [1, Section 9.5] does not satisfy in conditions (℘).



---

*Source: 531424-2008-05-07.xml*