Amenability of the Restricted Fourier Algebras

Abstract

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

1. Introduction and Preliminaries

An inverse semigroup 𝑆 is a discrete semigroup such that, for each 𝑠∈𝑆, there is a unique element 𝑠^∗∈𝑆 such that 𝑠𝑠^∗𝑠=𝑠,𝑠^∗𝑠𝑠^∗=𝑠^∗.(1.1) 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 ℓ¹(𝑆)=𝑓∶𝑆⟶ℂ∶_𝑠∈𝑆||𝑓(𝑠)||<∞(1.2) is a Banach algebra under convolution 𝑓∗𝑔(𝑥)=_{𝑠𝑡=𝑥}𝑓(𝑠)𝑔(𝑡)(1.3) 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)^∼=𝑔̃∗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 (𝐷_𝑘) 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 𝜆𝑥^∗𝜉,𝜂=⟨𝜉,𝜆(𝑥)𝜂⟩(1.4) 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 ℓ^1𝑟(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 𝜆_𝑟. 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 semigroup 𝑆. 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 𝑆_𝑎. If we adjoin a zero element 0 to this groupoid, and put 0^∗=0, we get an inverse semigroup 𝑆_𝑟 with the multiplication rule 𝑥•𝑦=𝑥𝑦,if𝑥^∗0,other𝑥w=ise𝑦,𝑦^∗,(1.5) for 𝑥,𝑦∈𝑆∪{0}, which is called the restricted semigroup of 𝑆. A restricted representation {𝜋,ℋ_𝜋} of 𝑆 is a pair consisting of a Hilbert space ℋ_𝜋 and a map 𝜋∶𝑆→ℬ(ℋ_𝜋) into the algebra ℬ(ℋ_𝜋) of bounded operators on ℋ_𝜋 such that 𝜋(𝑥^∗)=𝜋(𝑥)^∗ for 𝑥∈𝑆 and 𝜋(𝑥)𝜋(𝑦)=𝜋(𝑥𝑦),if𝑥^∗0,other𝑥w=ise𝑦,𝑦^∗,(1.6) for 𝑥,𝑦∈𝑆. Let Σ_𝑟=Σ_𝑟(𝑆) be the family of all restricted representations 𝜋 of 𝑆 with ‖𝜋‖≤1. It is clear that, via a canonical identification, Σ_𝑟(𝑆)=Σ₀(𝑆_𝑟) consists of all 𝜋∈Σ(𝑆_𝑟) 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 𝜆(𝑥)𝜉(𝑦)=𝜉𝑥^∗0,𝑦,ioft𝑥he𝑥r^∗w≥ise𝑦,𝑦^∗,(1.7) for 𝜉∈ℓ²(𝑆),𝑥,𝑦∈𝑆. The restricted left regular representation 𝜆_𝑟∶𝑆→ℬ(ℓ²(𝑆)) is defined in [5] by 𝜆_𝑟(𝑥)𝜉(𝑦)=𝜉𝑥^∗0,𝑦,ioft𝑥he𝑥r^∗w=ise𝑦,𝑦^∗,(1.8) 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 (𝑓•𝑔)(𝑥)=_{𝑥∗𝑥=𝑦𝑦∗}𝑓(𝑥𝑦)𝑔𝑦^∗(1.9) and 𝑓(𝑥)=𝑓(𝑥^∗), for 𝑥∈𝑆. Then, ℓ^1𝑟(𝑆)∶=(ℓ¹(𝑆),•,^∼) is a Banach ∗-algebra with an approximate identity [5]. The restricted left regular representation 𝜆_𝑟 lifts to a faithful representation 𝜆̃ of ℓ^1𝑟(𝑆). We call the completion 𝐶^∗𝜆_𝑟(𝑆) of ℓ^1𝑟(𝑆) in the 𝐶^∗-norm ‖⋅‖_𝜆𝑟∶=‖𝜆̃_𝑟(⋅)‖ the restricted reduced 𝐶^∗-algebra of 𝑆 and its completion 𝐶^∗𝑟(𝑆) in the 𝐶^∗-norm ‖⋅‖_Σ𝑟∶=sup{‖𝜋(⋅)‖,𝜋∈Σ(𝑆_𝑟)} the restricted full 𝐶^∗-algebra of 𝑆. The dual space of the 𝐶^∗-algebra 𝐶^∗𝑟(𝑆) is a unital Banach algebra which is called the restricted Fourier-Stieltjes algebra and is denoted by 𝐵_𝑟,𝑒(𝑆). The closure of the set of finitely support functions in 𝐵_𝑟,𝑒(𝑆) is called the restricted Fourier algebra and is denoted by 𝐴_𝑟,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 an 𝐸-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 function 𝑢∶𝑆→ℂ is called positive definite if for all positive integers 𝑛 and all 𝑐₁,…,𝑐_𝑛∈ℂ, and 𝑥₁,…,𝑥_𝑛∈𝑆 we have ^𝑛_𝑖,𝑗𝑐_𝑖𝑐_𝑗𝑢𝑥^∗𝑖𝑥_𝑗≥0,(1.10) and it is called restricted positive definite if for all positive integers 𝑛 and all 𝑐₁,…,𝑐_𝑛∈ℂ and 𝑥₁,…,𝑥_𝑛∈𝑆 we have ^𝑛_𝑖,𝑗𝑐_𝑖𝑐_𝑗𝜆_𝑟𝑥_𝑖𝑢𝑥_𝑗≥0.(1.11) The sets of all positive definite and restricted positive definite functions on 𝑆 are denoted by 𝑃(𝑆) and 𝑃_𝑟(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 𝑃_𝑟,𝑒(𝑆) which are exactly those 𝑢∈𝑃_𝑟(𝑆) such that 𝑢̃=𝑢, and there exists a constant 𝑐>0 such that for all 𝑛≥1, 𝑥₁,…,𝑥_𝑛∈𝑆, and 𝑐₁,…,𝑐_𝑛∈ℂ, |||||_𝑛_𝑖=1𝑐_𝑖𝑢𝑥_𝑖|||||²≤𝑐_{𝑥𝑖𝑥∗𝑖=𝑥𝑗𝑥∗𝑗}𝑐_𝑖𝑐_𝑗𝑢𝑥^∗𝑖𝑥_𝑗.(1.12) Then, 𝑃_𝑟,𝑒(𝑆)≅ℓ^1𝑟(𝑆)^∗+ and 𝐵_𝑟,𝑒(𝑆) is the linear span of 𝑃_𝑟,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 semigroup 𝑆, 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 sup||𝜌(𝑎)||²∶𝑎∈𝒜;𝜌𝑎^∗𝑎≤1=1.(1.13) If 𝒮(𝒜) is the set of states on 𝒜, we know that ‖𝑓‖_𝐶^∗_(𝑆)=sup𝜌𝑓^∗𝑓^1/2∶𝜌∈𝒮𝐶^∗(𝑆)=sup𝜌𝑓^∗𝑓^1/2∶𝜌∈𝒮(ℂ𝑆)(1.14) 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 as 𝑆_𝑟, some modification is necessary. This is because the maximum group homomorphic image of 𝑆_𝑟 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. Let 𝑆 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 group 𝐺 is a collection of closed subspaces 𝔹={𝐵_𝑔}_𝑔∈𝐺 of a 𝐶^∗-algebra 𝐵, satisfying 𝐵_𝑔∗=𝐵_𝑔−1 and 𝐵_𝑔𝐵_ℎ=𝐵_𝑔ℎ 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 grading 𝜑. For an inverse semigroup 𝑆 with zero, the algebras 𝐶^∗0(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 𝐴_𝑔=span{𝑠∶𝜑(𝑠)=𝑔}insideℂ₀𝑆,𝐵_𝑔=𝐴_𝑔inside𝐶^∗0(𝑆).(2.1) By [8, Proposition 3.3], the collection 𝔹={𝐵_𝑔}_𝑔∈𝐺 is a Fell bundle for 𝐶^∗0(𝑆) and representations of 𝔹 are in one-one correspondence with representations of 𝐶^∗0(𝑆), and hence 𝐶^∗(𝔹) is isomorphic to 𝐶^∗0(𝑆).

Since 𝑆_𝑟 is an inverse semigroup with zero, the grading map technique applies to 𝑆_𝑟. 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 homomorphism 𝜑_𝑟∶𝑆_𝑟→𝐺⁰, 𝜑_𝑟(𝑠)=𝜑(𝑠),𝑠∈𝑆, and 𝜑_𝑟(0)=0 that induces the natural map 𝜃∶ℂ𝑆_𝑟→ℂ𝐺⁰ defined by 𝜃⎛⎜⎜⎝_{𝑠∈𝑆𝑟}𝛼_𝑠𝑠⎞⎟⎟⎠=_{𝑠∈𝑆𝑟}𝛼_𝑠𝜑_𝑟(𝑠).(2.2)

Proposition 2.2. With the above notation, 𝜃∶ℂ𝑆_𝑟→ℂ𝐺⁰ is a positive map.

Proof. Let 𝑓=∑𝛼_𝑠𝛿_𝑠 be a typical element of ℂ𝑆_𝑟. It is enough to show that 𝜃(𝑓•𝑓^∗) is positive for each 𝑓∈ℂ𝑆_𝑟. Observe that 𝑓•𝑓^∗=𝛼_𝑠𝛿_𝑠•𝛼^𝑠𝛿_𝑠^∗=𝛼_𝑠𝛼_𝑡𝛿_𝑠•𝛿_𝑡∗=𝛼_𝑠𝛼_𝑡𝛿_{𝑠•𝑡∗}.(2.3) Hence, we have 𝜃(𝑓•𝑓^∗)=∑_𝑠,𝑡𝛼_𝑠𝛼_𝑡𝛿_𝜑(𝑠)•𝛿_𝜑(𝑡)^∗, which is a positive element of ℂ𝐺⁰.

Proposition 2.3. For each 𝑎∈ℂ𝑆_𝑟, ‖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 each 𝜌∈𝒮(ℂ𝐺⁰), 𝜌∘𝜃∈𝒮(ℂ𝑆_𝑟). Indeed, by the previous proposition, 𝜌∘𝜃 is positive, since, for each 𝑎∈ℂ𝑆_𝑟, 𝜃(𝑎)^∗•𝜃(𝑎)=𝜃(𝑎^∗•a). It follows that, for each 𝑎∈ℂ𝑆_𝑟 with 𝜌∘𝜃(𝑎^∗•𝑎)≤1, we have 𝜃(𝑎)∈ℂ𝐺⁰ with 𝜌(𝜃(𝑎)^∗•𝜃(𝑎))≤1. Therefore, ‖𝜃𝑎‖_𝐶^∗_(𝐺⁰₎==ssuupp𝜌(𝜌(∘𝜃𝜃𝑎))^∗𝑎•_∗𝜃𝑎^1/2∶𝜌∈𝒮ℂ𝐺⁰≤sup(𝜌∘𝜃)𝑎_∗•𝑎^1/2∶𝜌∈𝒮ℂ𝐺⁰≤‖𝑎‖_{𝐶∗(𝑆𝑟)}•𝑎^1/2∶𝜌∘𝜃∈𝒮ℂ𝑆_𝑟.(2.4)

Recall that a strongly 𝐸^∗-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. If 𝑆 is 𝐸-unitary, then 𝑆_𝑟 is strongly 𝐸^∗-unitary.

Proof. Let 𝜑∶𝑆→𝐺 be the natural epimorphism of 𝑆 onto its maximal group homomorphic image. Then, 𝜑_𝑟∶𝑆_𝑟→𝐺∪{0} with 𝜑_𝑟(0)=0 is a grading map and 𝜑^𝑟−1(𝑒)=𝐸.

Proposition 2.5. Let 𝑆 be an 𝐸-unitary inverse semigroup with the maximal group homomorphic image 𝐺. Then, the natural map 𝜃∶ℂ𝑆_𝑟→ℂ𝐺⁰ is an isometry.

Proof. Let 𝜋∶𝑆_𝑟→ℬ(ℋ) be a representation of the inverse semigroup 𝑆_𝑟 such that 𝜋(0)=0. Clearly, 𝜋 maps the idempotents of 𝑆 to projections on ℋ. Since 𝑆_𝑟 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 that 𝑎∈ℂ𝑆_𝑟 is a hermitian element. Then, ‖𝑎‖_𝐶^∗_(𝑆r)=sup_𝜋‖𝜋(𝑎)‖. By the definition of the quotient norm for 𝜋(𝜃(𝑎))=[𝜋(𝑎)], for each 𝜀>0, there is a projection 𝑖∈ℬ(ℋ) such that ‖𝜋(𝑎)+𝑖‖≤‖[𝜋(𝑎)]‖+𝜀. 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(𝜀(𝑎))‖=‖𝜋(𝑎)‖, and the result follows from Proposition 2.3 and the definition of the 𝐶^∗-norm.

Proposition 2.6. Let 𝜑∶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)→𝐵_𝑟,𝑒(𝑆) such that, for each 𝑓 in 𝐵(𝔹), 𝜙(𝑓)=𝑓∘𝜑, where 𝔹 is the Fell bundle for 𝐶^∗0(𝐺⁰).

Proof. Let 𝜑_𝑟∶𝑆_𝑟→𝐺⁰ be the induced homomorphism of 𝜑 on 𝑆_𝑟 and 𝜃 the natural map defined in Proposition 2.3. Then, Proposition 2.5 says that the natural map 𝜃∶ℂ𝑆_𝑟→ℂ𝐺⁰ extends to an isometric surjection 𝜃∶𝐶^∗𝑆_𝑟⟶𝐶^∗𝐺⁰(2.5) which maps zero to zero. Hence, we have the following map, again denoted by 𝜃∶𝜃∶𝐶^∗𝑆_𝑟ℂ𝛿₀⟶𝐶^∗𝐺⁰ℂ𝛿₀(2.6) which induces an ∗-homomorphism 𝜃̃∶𝐶^∗𝑟(𝑆)⟶𝐶^∗0𝐺⁰(2.7) and gives the isometric linear isomorphism (𝐶^∗(𝐺⁰)/ℂ𝛿₀)^∗≅𝐵_𝑟,𝑒(𝑆). By the paragraph before Proposition 2.2, there is an isometric isomorphism of Banach algebras 𝜃̃^∗∶𝐵(𝔹)⟶𝐵_𝑟,𝑒(𝑆),(2.8) where 𝔹 is the Fell bundle for 𝐶^∗0(𝐺⁰). Note that, for each 𝑢∈𝐶^∗(𝔹)^∗ and 𝛿_𝑠∈ℓ¹(𝑆_𝑟), 𝜃̃^∗(𝑢)(𝛿_𝑠)=∑𝑢(𝑥)𝜃(𝛿_𝑠)(𝑥)=𝑢∘𝜑(𝑠). This means that 𝜃̃^∗(𝑢)=𝑢∘𝜑. Therefore, 𝜙=𝜃̃^∗ 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 semigroup 𝑆 has restricted weak containment property if 𝐶^∗𝑟(𝑆)≅𝐶^∗𝜆_𝑟(𝑆).

Proposition 2.8. For an inverse semigroup 𝑆, 𝑆_𝑟 has weak containment property if and only if 𝑆 has restricted weak containment property.

Proof. The result follows from the following isomorphisms [5]: 𝐶^∗𝑟(𝑆)≅𝐶^∗𝑆_𝑟ℂ𝛿₀,𝐶^∗𝜆_𝑟(𝑆)≅𝐶^∗Λ𝑆_𝑟ℂ𝛿₀,(2.9) where Λ is the left regular representation of 𝑆_𝑟 and 𝐶^∗Λ(𝑆_𝑟) is the completion of ℓ¹(𝑆_𝑟) in the norm ‖𝑓‖_Λ∶=sup‖Λ(𝑓)‖.

Proposition 2.9. For an inverse semigroup 𝑆, the following three conditions are equivalent:(i)ℓ^1𝑟(𝑆) is amenable,(ii)𝐸_𝑆 is finite,(iii)ℓ^1𝑟(𝑆) has a bounded approximate identity.

Proof. If 𝐸_𝑆 is finite then so is 𝐸_𝑆𝑟=𝐸_𝑆∪{0}, hence ℓ¹(𝑆_𝑟) is amenable [2], and so is ℓ^1𝑟(𝑆)≅ℓ¹(𝑆_𝑟)/ℂ𝛿₀.
Conversely, if ℓ^1𝑟(𝑆) is amenable, then ℓ¹(𝑆_𝑟)/ℂ𝛿₀ and ℂ𝛿₀ are both amenable. Hence, ℓ¹(𝑆_𝑟) is amenable, therefore 𝐸_𝑆𝑟 and so 𝐸_𝑆 are finite [2].

Let 𝜑∶𝑆→𝐺 be the quotient map of the inverse semigroup 𝑆 onto its maximal group homomorphic image and 𝜑_𝑟∶𝑆_𝑟→𝐺∪0 a grading of 𝑆_𝑟 by the group 𝐺, let 𝐻_𝑟=𝜑^𝑟−1(𝑒)∪{0} and 𝐻=𝜑⁻¹(𝑒). There is a conditional expectation 𝜀∶𝐶^∗0𝑆_𝑟≅𝐶^∗𝑆_𝑟ℂ𝛿₀≅𝐶^∗𝑟(𝑆)⟶𝐶^∗0𝐻_𝑟≅𝐶^∗𝐻_𝑟ℂ𝛿₀𝜀≅𝐶^∗𝑟(𝐻),_𝑟∶𝐶^∗𝜆,0𝑆_𝑟≅𝐶^∗𝜆𝑆_𝑟ℂ𝛿₀≅𝐶^∗𝜆_𝑟(𝑆)⟶ℂ₀𝐻_𝑟=ℓ¹𝐻_𝑟ℂ𝛿₀=ℓ^1𝑟(𝐻)^𝜆⊆𝐶^∗𝜆_𝑟(𝐻).(2.10) These are extensions of the restriction map ℓ^1𝑟(𝑆)→ℓ^1𝑟(𝐻), where 𝐻≤𝑆. By [8, Theorem 4.2], 𝑆_𝑟 has weak containment property if and only if 𝜀 is faithful and 𝐻_𝑟 has weak containment property, that is 𝐶^∗(𝐻_𝑟)≅𝐶^∗𝜆(𝐻_𝑟). It follows that 𝑆 has restricted weak containment property if and only if 𝜀 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. Let 𝐻=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 𝜀∶𝐶^∗𝑟(𝑆)→𝐶^∗𝑟(𝐻) is faithful and 𝐻 has restricted weak containment property.

Corollary 2.11. For an 𝐸-unitary inverse semigroup 𝑆, the following three conditions are equivalent:(i)𝑆 has restricted weak containment property,(ii)𝜀∶𝐶^∗𝑟(𝑆)→𝐶^∗𝑟(𝐸) is faithful,(iii)the Fell bundle of 𝐶^∗𝑟(𝑆) is amenable.