# Amenability of the Restricted Fourier Algebras

Authors: Massoud Amini; Marzieh Shams Yousefi
Journal: ISRN Algebra
 (2012)

---

## Abstract

We discuss amenability of the restricted Fourier-Stieltjes algebras on inverse semigroups. We show that, for anE-unitary inverse semigroup, amenability of the restricted Fourier-Stieltjes algebra is related to the amenability of an associated Banach algebra on a Fell bundle.


---

## Body

## 1. Introduction and Preliminaries

An inverse semigroupS is a discrete semigroup such that, for each s∈S, there is a unique element s*∈S such that
(1.1)ss*s=s,s*ss*=s*.
One can show that s↦s* is an involution on S [1]. Set E consisting of idempotents of S, elements of the form ss*, s∈S, is a commutative sub-semigroup of S [1]. There is a natural order ≤ on E defined by e≤f if and only if ef=e. The semigroup algebra
(1.2)l1(S)={f:S⟶C:∑s∈S‍|f(s)|<∞}
is a Banach algebra under convolution
(1.3)f*g(x)=∑st=x‍f(s)g(t)
and norm ∥f∥1=∑s∈S‍|f(s)|. Although ℓ1(S) has some common features with the group algebra ℓ1(G), there are certain technical difficulties when one tries to do things on inverse semigroups similar to the group case, and some well-known properties of the group algebra ℓ1(G) break down for inverse semigroups. For instance, ℓ1(G) is a Banach *-algebra under the canonical involution f~(x)=f(x-1)¯. This is important when one constructs the enveloping C*-algebra of ℓ1(G) or studies the automatic continuity properties of characters and homomorphisms. For an inverse semigroup S, the natural involution on ℓ1(S) is f~(x)=f(x*)¯. This is an isometry on ℓ1(S) but does not satisfy (f*g)~=g~*f~. On the other hand, unlike the group case, ℓ1(S) does not have a bounded approximate identity (this happens when E fails to satisfy the condition (Dk) of Duncan and Namioka [2] for any positive integer k, a Brandt semigroup on an infinite index set is a concrete example). Finally, the left regular representation λ of an inverse semigroup loses its connection with positive definite functions. This is because the crucial equality
(1.4)〈λ(x*)ξ,η〉=〈ξ,λ(x)η〉
for x∈S and ξ,η∈ℓ2(S) fails in general. This makes it difficult to study positive definite functions [3] and Fourier (Fourier-Stieltjes) algebras on semigroups [4].The first author and Medghalchi introduced and studied the notion of restricted semigroup algebra in [5, 6] to overcome such difficulties. They showed that, if the convolution product on ℓ1(S) is appropriately modified, one gets a Banach *-algebra ℓr1(S), called the restricted semigroup algebra with an approximate identity (not necessarily bounded). In the new convolution product, positive definite functions fit naturally with a restricted version of the left regular representation λr. The idea is that one requires the homomorphism property of representations to hold only for those pairs of elements in the semigroup whose range and domain match. This is quite similar to what is done in the context of groupoids, but the representation theory of groupoids is much more involved [6].The basic idea of the restricted semigroup algebra is to consider the associated groupoid of an inverse semigroupS. Given x,y∈S, the restricted product of x,y is xy if x*x=yy*, and undefined, otherwise. The set S with its restricted product forms a groupoid, which is called the associated groupoid of S and is denoted by Sa. If we adjoin a zero element 0 to this groupoid, and put 0*=0, we get an inverse semigroup Sr with the multiplication rule
(1.5)x•y={xy,ifx*x=yy*,0,otherwise,
for x,y∈S∪{0}, which is called the restricted semigroup of S. A restricted representation {π,ℋπ} of S is a pair consisting of a Hilbert space ℋπ and a map π:S→ℬ(ℋπ) into the algebra ℬ(ℋπ) of bounded operators on ℋπ such that π(x*)=π(x)* for x∈S and
(1.6)π(x)π(y)={π(xy),ifx*x=yy*,0,otherwise,
for x,y∈S. Let Σr=Σr(S) be the family of all restricted representations π of S with ∥π∥≤1. It is clear that, via a canonical identification, Σr(S)=Σ0(Sr) consists of all π∈Σ(Sr) with π(0)=0 [5]. One of the central concepts in the analytic theory of inverse semigroup is the left regular representation λ:S→ℬ(ℓ2(S)) defined by
(1.7)λ(x)ξ(y)={ξ(x*y),ifxx*≥yy*,0,otherwise,
for ξ∈ℓ2(S),x,y∈S. The restricted left regular representation λr:S→ℬ(ℓ2(S)) is defined in [5] by
(1.8)λr(x)ξ(y)={ξ(x*y),ifxx*=yy*,0,otherwise,
for ξ∈ℓ2(S),x,y∈S. The main objective of [5] is to change the convolution product on the semigroup algebra to restore the relation between positive definite functions and left regular representation [6].Throughout this paper,S is an inverse semigroup. For each f,g∈ℓ1(S), define
(1.9)(f•g)(x)=∑x*x=yy*‍f(xy)g(y*)
and f~(x)=f(x*)¯, for x∈S. Then, ℓr1(S):=(ℓ1(S),•,~) is a Banach *-algebra with an approximate identity [5]. The restricted left regular representation λr lifts to a faithful representation λ~ of ℓr1(S). We call the completion Cλr*(S) of ℓr1(S) in the C*-norm ∥·∥λr:=∥λ~r(·)∥ the restricted reduced C*-algebra of S and its completion Cr*(S) in the C*-norm ∥·∥Σr:=sup{∥π~(·)∥,π∈Σ(Sr)} the restricted full C*-algebra of S. The dual space of the C*-algebra Cr*(S) is a unital Banach algebra which is called the restricted Fourier-Stieltjes algebra and is denoted by Br,e(S). The closure of the set of finitely support functions in Br,e(S) is called the restricted Fourier algebra and is denoted by Ar,e(S) [6].In this paper, we discuss the amenability of the restricted Fourier and Fourier-Stieltjes algebras on inverse semigroups. We show that, for anE-unitary inverse semigroup, the restricted Fourier algebra is amenable if and only if its maximal homomorphic group image is abelian by finite (i.e., it has an abelian subgroup of finite index). We refer the readers to [5, 6] for more details about restricted semigroup algebra, restricted semigroup C*-algebra, restricted positive definite functions, and restricted Fourier and Fourier-Stieltjes algebras.A bounded complex valued functionu:S→ℂ is called positive definite if for all positive integers n and all c1,…,cn∈ℂ, and x1,…,xn∈S we have
(1.10)∑‍∑i,jn‍ci¯cju(xi*xj)≥0,
and it is called restricted positive definite if for all positive integers n and all c1,…,cn∈ℂ and x1,…,xn∈S we have
(1.11)∑‍∑i,jn‍ci¯cj(λr(xi)u)(xj)≥0.
The sets of all positive definite and restricted positive definite functions on S are denoted by P(S) and Pr(S), respectively. Positive definite functions are usually considered on unital semigroups. Of course, one can always adjoin a unit 1 to an inverse semigroup T with 1*=1. But extending a positive definite function on T to one on T1=T∪{1} is not always possible. We denote all extendable restricted positive definite functions by Pr,e(S) which are exactly those u∈Pr(S) such that u~=u, and there exists a constant c>0 such that for all n≥1, x1,…,xn∈S, and c1,…,cn∈ℂ,
(1.12)|∑i=1n‍ciu(xi)|2≤c∑xixi*=xjxj*‍ci¯cju(xi*xj).
Then, Pr,e(S)≅ℓr1(S)+* and Br,e(S) is the linear span of Pr,e(S) [5]. Since the restricted Fourier-Stieltjes algebra is the dual space of the restricted semigroup C*-algebra, it is an ordered Banach algebra in the sense of [7], where the order structure comes from the set of extendable restricted positive definite functions as the positive cone. The same applies to the restricted Fourier algebra.For each inverse semigroupS, the states on the *-algebra ℂS (the vector space over S spanned by S with convolution and involution comes from S) are defined by Milan in [8]. A state on a *-algebra 𝒜 is a positive linear map ρ:𝒜→ℂ such that
(1.13)sup⁡{|ρ(a)|2:a∈A;ρ(a*a)≤1}=1.
If 𝒮(𝒜) is the set of states on 𝒜, we know that
(1.14)‖f‖C*(S)=sup⁡{ρ(f*f)1/2:ρ∈S(C*(S))}=sup⁡{ρ(f*f)1/2:ρ∈S(CS)}
for each f in ℂS [8]. When S is a (discrete) group, all these concepts are already discussed by Eymard in [9]. The amenability results for Fourier and Fourier-Stieltjes algebras on groups are surveyed in [10].


## 2. Amenability and Restricted Weak Containment Property

Working with an inverse semigroup, it is quite natural to go to the maximal group homomorphic image. However, when one deals with an inverse semigroup with zero such asSr, some modification is necessary. This is because the maximum group homomorphic image of Sr is trivial. To remedy this, we work with maps which are not quite homomorphism. This is the idea of Milan in [8, Section 4].Definition 2.1.
LetS be an inverse semigroup with zero, a grading of S by the group G is a map φ:S→G∪{0}such that φ-1(0)={0} and φ(ab)=φ(a)φ(b) provided that ab≠0.A Fell bundle over a discrete groupG is a collection of closed subspaces 𝔹={Bg}g∈G of a C*-algebra B, satisfying Bg*=Bg-1 and BgBh=Bgh for all g and h in G. The ℓ1 cross-sectional algebra ℓ1(𝔹) of 𝔹 is the Banach *-algebra consisting of the ℓ1 cross-sections of 𝔹 under the canonical multiplication, involution, and norm, and the cross-sectional C*-algebra C*(𝔹) of 𝔹 is the enveloping C*-algebra of ℓ1(𝔹) [11]. We also denote the dual space of C*(𝔹) by B(𝔹).Next, let us define the Fell bundle arising from a gradingφ. For an inverse semigroup S with zero, the algebras C0*(S) and ℂ0S are the quotients of the algebras C*(S) and ℂS by the (closed) ideal generated by the zero of S. For each g∈G, let
(2.1)Ag=span⁡{s:φ(s)=g}insideC0S,Bg=Ag¯insideC0*(S).
By [8, Proposition 3.3], the collection 𝔹={Bg}g∈G is a Fell bundle for C0*(S) and representations of 𝔹 are in one-one correspondence with representations of C0*(S), and hence C*(𝔹) is isomorphic to C0*(S).SinceSr is an inverse semigroup with zero, the grading map technique applies to Sr. Let S be an inverse semigroup and G its maximal group homomorphic image with φ:S→G, we define the following new product on G∪{0}. Put φ(x)•φ(y)=φ(x•y) and φ(s)*:=φ(s*). It is easy to see that, with this new multiplication, G∪{0} is an inverse semigroup, which is denoted by G0.Now, there is a homomorphismφr:Sr→G0, φr(s)=φ(s),s∈S, and φr(0)=0 that induces the natural map θ:ℂSr→ℂG0 defined by
(2.2)θ(∑s∈Sr‍αss)=∑s∈Sr‍αsφr(s).Proposition 2.2.
With the above notation,θ:ℂSr→ℂG0 is a positive map.Proof.
Letf=∑‍αsδs be a typical element of ℂSr. It is enough to show that θ(f•f*) is positive for each f∈ℂSr. Observe that
(2.3)f•f*=(∑‍αsδs)•(∑‍αsδs)*=∑‍αsαt¯δs•δt*=∑‍αsαt¯δs•t*.
Hence, we have θ(f•f*)=∑s,t‍αsαt¯δφ(s)•δφ(t)*, which is a positive element of ℂG0.Proposition 2.3.
For eacha∈ℂSr, ∥θa∥C*(G0)≤∥a∥C*(Sr).Proof.
It is enough to check the relation between states on these spaces. It is easy to check that, for eachρ∈𝒮(ℂG0), ρ∘θ∈𝒮(ℂSr). Indeed, by the previous proposition, ρ∘θ is positive, since, for each a∈ℂSr, θ(a)*•θ(a)=θ(a*•a). It follows that, for each a∈ℂSr with ρ∘θ(a*•a)≤1, we have θ(a)∈ℂG0 with ρ(θ(a)*•θ(a))≤1. Therefore,
(2.4)‖θa‖C*(G0)=sup⁡{ρ((θa)*•θa)1/2:ρ∈S(CG0)}=sup⁡{(ρ∘θ)(a*•a)1/2:ρ∈S(CG0)}≤sup⁡{(ρ∘θ)(a*•a)1/2:ρ∘θ∈S(CSr)}≤‖a‖C*(Sr).Recall that a stronglyE*-unitary inverse semigroup S is an inverse semigroup that admits a grading φ:S→G∪{0} such that φ-1(e) is equal to the set of nonzero idempotent of S, where e is the identity of G.Lemma 2.4.
IfS is E-unitary, then Sr is strongly E*-unitary.Proof.
Letφ:S→G be the natural epimorphism of S onto its maximal group homomorphic image. Then, φr:Sr→G∪{0} with φr(0)=0 is a grading map and φr-1(e)=E.Proposition 2.5.
LetS be an E-unitary inverse semigroup with the maximal group homomorphic image G. Then, the natural map θ:ℂSr→ℂG0 is an isometry.Proof.
Letπ:Sr→ℬ(ℋ) be a representation of the inverse semigroup Sr such that π(0)=0. Clearly, π maps the idempotents of S to projections on ℋ. Since Sr is strongly E*-unitary, π induces a representation on G0 defined by π~(φ(x))=[π(x)], for each x∈S∖{0} and π~(0)=0, where [π(x)] is the equivalence class of π(x). Therefore, representations of S lift to representations on the corresponding inverse semigroup G0.
Now, assume thata∈ℂSr is a hermitian element. Then, ∥a∥C*(Sr)=supπ∥π(a)∥. By the definition of the quotient norm for π~(θ(a))=[π(a)], for each ɛ>0, there is a projection i∈ℬ(ℋ) such that ∥π(a)+i∥≤∥[π(a)]∥+ɛ. But i is positive as an element in the C*-algebra ℬ(ℋ) thus π(a)≤π(a)+i, and therefore ∥π(a)∥≤∥π(a)+i∥. Hence, ∥π~(ɛ(a))∥=∥π(a)∥, and the result follows from Proposition 2.3 and the definition of the C*-norm.Proposition 2.6.
Letφ:S→G be the quotient map of an E-unitary inverse semigroup S onto its maximal group homomorphic image. Then, there exists an isometric isomorphism ϕ:B(𝔹)→Br,e(S) such that, for each f in B(𝔹), ϕ(f)=f∘φ, where 𝔹 is the Fell bundle for C0*(G0).Proof.
Letφr:Sr→G0 be the induced homomorphism of φ on Sr and θ the natural map defined in Proposition 2.3. Then, Proposition 2.5 says that the natural map θ:ℂSr→ℂG0 extends to an isometric surjection
(2.5)θ:C*(Sr)⟶C*(G0)
which maps zero to zero. Hence, we have the following map, again denoted by θ:(2.6)θ:C*(Sr)Cδ0⟶C*(G0)Cδ0
which induces an *-homomorphism
(2.7)θ~:Cr*(S)⟶C0*(G0)
and gives the isometric linear isomorphism (C*(G0)/ℂδ0)*≅Br,e(S). By the paragraph before Proposition 2.2, there is an isometric isomorphism of Banach algebras
(2.8)θ~*:B(B)⟶Br,e(S),
where 𝔹 is the Fell bundle for C0*(G0). Note that, for each u∈C*(𝔹)* and δs∈ℓ1(Sr), θ~*(u)(δs)=∑‍u(x)θ(δs)(x)=u∘φ(s). This means that θ~*(u)=u∘φ. Therefore, ϕ=θ~* is the required map.Next, we adapt the notion of weak containment property of inverse semigroups [8] to the restricted case.Definition 2.7.
The inverse semigroupS has restricted weak containment property if Cr*(S)≅Cλr*(S).Proposition 2.8.
For an inverse semigroupS, Sr has weak containment property if and only if S has restricted weak containment property.Proof.
The result follows from the following isomorphisms [5]:
(2.9)Cr*(S)≅C*(Sr)Cδ0,Cλr*(S)≅CΛ*(Sr)Cδ0,
where Λ is the left regular representation of Sr and CΛ*(Sr) is the completion of ℓ1(Sr) in the norm ∥f∥Λ:=sup∥Λ(f)∥.Proposition 2.9.
For an inverse semigroupS, the following three conditions are equivalent:(i)
ℓr1(S) is amenable,(ii)
ES is finite,(iii)
ℓr1(S) has a bounded approximate identity.Proof.
IfES is finite then so is ESr=ES∪{0}, hence ℓ1(Sr) is amenable [2], and so is ℓr1(S)≅ℓ1(Sr)/ℂδ0.
Conversely, ifℓr1(S) is amenable, then ℓ1(Sr)/ℂδ0 and ℂδ0 are both amenable. Hence, ℓ1(Sr) is amenable, therefore ESr and so ES are finite [2].Letφ:S→G be the quotient map of the inverse semigroup S onto its maximal group homomorphic image and φr:Sr→G∪0 a grading of Sr by the group G, let Hr=φr-1(e)∪{0} and H=φ-1(e). There is a conditional expectation
(2.10)ɛ:C0*(Sr)≅C*(Sr)ℂδ0≅Cr*(S)⟶C0*(Hr)≅C*(Hr)ℂδ0≅Cr*(H),ɛr:Cλ,0*(Sr)≅Cλ*(Sr)Cδ0≅Cλr*(S)⟶C0Hr¯=l1(Hr)Cδ0¯=lr1(H)¯λ⊆Cλr*(H).
These are extensions of the restriction map ℓr1(S)→ℓr1(H), where H≤S. By [8, Theorem 4.2], Sr has weak containment property if and only if ɛ is faithful and Hr has weak containment property, that is C*(Hr)≅Cλ*(Hr). It follows that S has restricted weak containment property if and only if ɛ is faithful and H has restricted weak containment property. Now, most of the results in [8] extend to the restricted version. In particular, we have the following two results, where, in the latter, the amenability of the Fell bundle is in the sense of Exel [11].Theorem 2.10.
LetH=φ-1(e)≤S where φ is the quotient map of S onto its maximal group homomorphic image. Then, S has restricted weak containment property if and only if ɛ:Cr*(S)→Cr*(H) is faithful and H has restricted weak containment property.Corollary 2.11.
For anE-unitary inverse semigroup S, the following three conditions are equivalent:(i)
S has restricted weak containment property,(ii)
ɛ:Cr*(S)→Cr*(E) is faithful,(iii)
the Fell bundle ofCr*(S) is amenable.



---

*Source: 782953-2012-06-27.xml*