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ISRN Algebra
Volume 2012 (2012), Article ID 782953, 8 pages
http://dx.doi.org/10.5402/2012/782953
Research Article

Amenability of the Restricted Fourier Algebras

1Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran 14115-134, Iran
2Department of Pure Mathematics, Faculty of Mathematical Science, University of Guilan, Rasht 1841, Iran

Received 7 March 2012; Accepted 2 May 2012

Academic Editors: V. K. Dobrev, K. Fujii, M. Ladra, and M. Przybylska

Copyright © 2012 Massoud Amini and Marzieh Shams Yousefi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝑠𝑆, there is a unique element 𝑠𝑆 such that 𝑠𝑠𝑠=𝑠,𝑠𝑠𝑠=𝑠.(1.1) One can show that 𝑠𝑠 is an involution on 𝑆 [1]. Set 𝐸 consisting of idempotents of 𝑆, elements of the form 𝑠𝑠, 𝑠𝑆, is a commutative sub-semigroup of 𝑆 [1]. There is a natural order on 𝐸 defined by 𝑒𝑓 if and only if 𝑒𝑓=𝑒. The semigroup algebra 1(𝑆)=𝑓𝑆𝑠𝑆||||𝑓(𝑠)<(1.2) is a Banach algebra under convolution 𝑓𝑔(𝑥)=𝑠𝑡=𝑥𝑓(𝑠)𝑔(𝑡)(1.3) and norm 𝑓1=𝑠𝑆|𝑓(𝑠)|. Although 1(𝑆) has some common features with the group algebra 1(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 1(𝐺) break down for inverse semigroups. For instance, 1(𝐺) is a Banach -algebra under the canonical involution 𝑓(𝑥)=𝑓(𝑥1). This is important when one constructs the enveloping 𝐶-algebra of 1(𝐺) or studies the automatic continuity properties of characters and homomorphisms. For an inverse semigroup 𝑆, the natural involution on 1(𝑆) is 𝑓(𝑥)=𝑓(𝑥). This is an isometry on 1(𝑆) but does not satisfy (𝑓𝑔)𝑓=̃𝑔. On the other hand, unlike the group case, 1(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 𝜆𝑥𝜉,𝜂=𝜉,𝜆(𝑥)𝜂(1.4) for 𝑥𝑆 and 𝜉,𝜂2(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 1(𝑆) is appropriately modified, one gets a Banach -algebra 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 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 semigroup 𝑆. Given 𝑥,𝑦𝑆, the restricted product of 𝑥,𝑦 is 𝑥𝑦 if 𝑥𝑥=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 𝑥𝑦=𝑥𝑦,if𝑥𝑥=𝑦𝑦,0,otherwise,(1.5) for 𝑥,𝑦𝑆{0}, which is called the restricted semigroup of 𝑆. A restricted representation {𝜋,𝜋} of 𝑆 is a pair consisting of a Hilbert space 𝜋 and a map 𝜋𝑆(𝜋) into the algebra (𝜋) of bounded operators on 𝜋 such that 𝜋(𝑥)=𝜋(𝑥) for 𝑥𝑆 and 𝜋(𝑥)𝜋(𝑦)=𝜋(𝑥𝑦),if𝑥𝑥=𝑦𝑦,0,otherwise,(1.6) for 𝑥,𝑦𝑆. Let Σ𝑟=Σ𝑟(𝑆) be the family of all restricted representations 𝜋 of 𝑆 with 𝜋1. It is clear that, via a canonical identification, Σ𝑟(𝑆)=Σ0(𝑆𝑟) consists of all 𝜋Σ(𝑆𝑟) with 𝜋(0)=0 [5]. One of the central concepts in the analytic theory of inverse semigroup is the left regular representation 𝜆𝑆(2(𝑆)) defined by 𝜉𝑥𝜆(𝑥)𝜉(𝑦)=𝑦,if𝑥𝑥𝑦𝑦,0,otherwise,(1.7) for 𝜉2(𝑆),𝑥,𝑦𝑆. The restricted left regular representation 𝜆𝑟𝑆(2(𝑆)) is defined in [5] by 𝜆𝑟𝜉𝑥(𝑥)𝜉(𝑦)=𝑦,if𝑥𝑥=𝑦𝑦,0,otherwise,(1.8) for 𝜉2(𝑆),𝑥,𝑦𝑆. 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,𝑔1(𝑆), define (𝑓𝑔)(𝑥)=𝑥𝑥=𝑦𝑦𝑦𝑓(𝑥𝑦)𝑔(1.9) and 𝑓(𝑥)=𝑓(𝑥), for 𝑥𝑆. Then, 1𝑟(𝑆)=(1(𝑆),,) is a Banach -algebra with an approximate identity [5]. The restricted left regular representation 𝜆𝑟 lifts to a faithful representation ̃𝜆 of 1𝑟(𝑆). We call the completion 𝐶𝜆𝑟(𝑆) of 1𝑟(𝑆) in the 𝐶-norm 𝜆𝑟̃𝜆=𝑟() the restricted reduced 𝐶-algebra of 𝑆 and its completion 𝐶𝑟(𝑆) in the 𝐶-norm Σ𝑟=sup{𝜋(),𝜋Σ(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 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 𝑢𝑆 is called positive definite if for all positive integers 𝑛 and all 𝑐1,,𝑐𝑛, and 𝑥1,,𝑥𝑛𝑆 we have 𝑛𝑖,𝑗𝑐𝑖𝑐𝑗𝑢𝑥𝑖𝑥𝑗0,(1.10) and it is called restricted positive definite if for all positive integers 𝑛 and all 𝑐1,,𝑐𝑛 and 𝑥1,,𝑥𝑛𝑆 we have 𝑛𝑖,𝑗𝑐𝑖𝑐𝑗𝜆𝑟𝑥𝑖𝑢𝑥𝑗0.(1.11) 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 𝑢𝑃𝑟(𝑆) such that ̃𝑢=𝑢, and there exists a constant 𝑐>0 such that for all 𝑛1, 𝑥1,,𝑥𝑛𝑆, and 𝑐1,,𝑐𝑛, |||||𝑛𝑖=1𝑐𝑖𝑢𝑥𝑖|||||2𝑐𝑥𝑖𝑥𝑖=𝑥𝑗𝑥𝑗𝑐𝑖𝑐𝑗𝑢𝑥𝑖𝑥𝑗.(1.12) Then, 𝑃𝑟,𝑒(𝑆)1𝑟(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 semigroup 𝑆, the states on the -algebra 𝑆 (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 𝜌𝒜 such that ||||sup𝜌(𝑎)2𝑎𝑎𝒜;𝜌𝑎1=1.(1.13) If 𝒮(𝒜) is the set of states on 𝒜, we know that 𝑓𝐶(𝑆)𝜌𝑓=sup𝑓1/2𝐶𝜌𝒮𝜌𝑓(𝑆)=sup𝑓1/2𝜌𝒮(𝑆)(1.14) for each 𝑓 in 𝑆 [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 𝑆𝑟, 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. Let 𝑆 be an inverse semigroup with zero, a grading of 𝑆 by the group 𝐺 is a map 𝜑𝑆𝐺{0}such that 𝜑1(0)={0} and 𝜑(𝑎𝑏)=𝜑(𝑎)𝜑(𝑏) provided that 𝑎𝑏0.

A Fell bundle over a discrete group 𝐺 is a collection of closed subspaces 𝔹={𝐵𝑔}𝑔𝐺 of a 𝐶-algebra 𝐵, satisfying 𝐵𝑔=𝐵𝑔1 and 𝐵𝑔𝐵=𝐵𝑔 for all 𝑔 and in 𝐺. The 1 cross-sectional algebra 1(𝔹) of 𝔹 is the Banach -algebra consisting of the 1 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 1(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 𝐶0(𝑆) and 0𝑆 are the quotients of the algebras 𝐶(𝑆) and 𝑆 by the (closed) ideal generated by the zero of 𝑆. For each 𝑔𝐺, let 𝐴𝑔=span{𝑠𝜑(𝑠)=𝑔}inside0𝑆,𝐵𝑔=𝐴𝑔inside𝐶0(𝑆).(2.1) By [8, Proposition 3.3], the collection 𝔹={𝐵𝑔}𝑔𝐺 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).

Since 𝑆𝑟 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 𝜑𝑆𝐺, we define the following new product on 𝐺{0}. Put 𝜑(𝑥)𝜑(𝑦)=𝜑(𝑥𝑦) 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 𝜑𝑟𝑆𝑟𝐺0, 𝜑𝑟(𝑠)=𝜑(𝑠),𝑠𝑆, and 𝜑𝑟(0)=0 that induces the natural map 𝜃𝑆𝑟𝐺0 defined by 𝜃𝑠𝑆𝑟𝛼𝑠𝑠=𝑠𝑆𝑟𝛼𝑠𝜑𝑟(𝑠).(2.2)

Proposition 2.2. With the above notation, 𝜃𝑆𝑟𝐺0 is a positive map.

Proof. Let 𝛼𝑓=𝑠𝛿𝑠 be a typical element of 𝑆𝑟. It is enough to show that 𝜃(𝑓𝑓) is positive for each 𝑓𝑆𝑟. Observe that 𝑓𝑓=𝛼𝑠𝛿𝑠𝛼𝑠𝛿𝑠=𝛼𝑠𝛼𝑡𝛿𝑠𝛿𝑡=𝛼𝑠𝛼𝑡𝛿𝑠𝑡.(2.3) Hence, we have 𝜃(𝑓𝑓)=𝑠,𝑡𝛼𝑠𝛼𝑡𝛿𝜑(𝑠)𝛿𝜑(𝑡), which is a positive element of 𝐺0.

Proposition 2.3. For each 𝑎𝑆𝑟, 𝜃𝑎𝐶(𝐺0)𝑎𝐶(𝑆𝑟).

Proof. It is enough to check the relation between states on these spaces. It is easy to check that, for each 𝜌𝒮(𝐺0), 𝜌𝜃𝒮(𝑆𝑟). Indeed, by the previous proposition, 𝜌𝜃 is positive, since, for each 𝑎𝑆𝑟, 𝜃(𝑎)𝜃(𝑎)=𝜃(𝑎a). It follows that, for each 𝑎𝑆𝑟 with 𝜌𝜃(𝑎𝑎)1, we have 𝜃(𝑎)𝐺0 with 𝜌(𝜃(𝑎)𝜃(𝑎))1. Therefore, 𝜃𝑎𝐶(𝐺0)𝜌=sup(𝜃𝑎)𝜃𝑎1/2𝜌𝒮𝐺0𝑎=sup(𝜌𝜃)𝑎1/2𝜌𝒮𝐺0𝑎sup(𝜌𝜃)𝑎1/2𝜌𝜃𝒮𝑆𝑟𝑎𝐶(𝑆𝑟).(2.4)

Recall that a strongly 𝐸-unitary inverse semigroup 𝑆 is an inverse semigroup that admits a grading 𝜑𝑆𝐺{0} such that 𝜑1(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 𝑆𝑟 is strongly 𝐸-unitary.

Proof. Let 𝜑𝑆𝐺 be the natural epimorphism of 𝑆 onto its maximal group homomorphic image. Then, 𝜑𝑟𝑆𝑟𝐺{0} with 𝜑𝑟(0)=0 is a grading map and 𝜑𝑟1(𝑒)=𝐸.

Proposition 2.5. Let 𝑆 be an 𝐸-unitary inverse semigroup with the maximal group homomorphic image 𝐺. Then, the natural map 𝜃𝑆𝑟𝐺0 is an isometry.

Proof. Let 𝜋𝑆𝑟() be a representation of the inverse semigroup 𝑆𝑟 such that 𝜋(0)=0. Clearly, 𝜋 maps the idempotents of 𝑆 to projections on . Since 𝑆𝑟 is strongly 𝐸-unitary, 𝜋 induces a representation on 𝐺0 defined by 𝜋(𝜑(𝑥))=[𝜋(𝑥)], for each 𝑥𝑆{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 𝑎𝑆𝑟 is a hermitian element. Then, 𝑎𝐶(𝑆r)=sup𝜋𝜋(𝑎). By the definition of the quotient norm for 𝜋(𝜃(𝑎))=[𝜋(𝑎)], for each 𝜀>0, there is a projection 𝑖() such that 𝜋(𝑎)+𝑖[𝜋(𝑎)]+𝜀. But 𝑖 is positive as an element in the 𝐶-algebra () 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 𝜑𝑆𝐺 be the quotient map of an 𝐸-unitary inverse semigroup 𝑆 onto its maximal group homomorphic image. Then, there exists an isometric isomorphism 𝜙𝐵(𝔹)𝐵𝑟,𝑒(𝑆) such that, for each 𝑓 in 𝐵(𝔹), 𝜙(𝑓)=𝑓𝜑, where 𝔹 is the Fell bundle for 𝐶0(𝐺0).

Proof. Let 𝜑𝑟𝑆𝑟𝐺0 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 𝜃𝑆𝑟𝐺0 extends to an isometric surjection 𝜃𝐶𝑆𝑟𝐶𝐺0(2.5) which maps zero to zero. Hence, we have the following map, again denoted by 𝜃𝐶𝜃𝑆𝑟𝛿0𝐶𝐺0𝛿0(2.6) which induces an -homomorphism ̃𝜃𝐶𝑟(𝑆)𝐶0𝐺0(2.7) and gives the isometric linear isomorphism (𝐶(𝐺0)/𝛿0)𝐵𝑟,𝑒(𝑆). By the paragraph before Proposition 2.2, there is an isometric isomorphism of Banach algebras ̃𝜃𝐵(𝔹)𝐵𝑟,𝑒(𝑆),(2.8) where 𝔹 is the Fell bundle for 𝐶0(𝐺0). Note that, for each 𝑢𝐶(𝔹) and 𝛿𝑠1(𝑆𝑟), ̃𝜃(𝑢)(𝛿𝑠)=𝑢(𝑥)𝜃(𝛿𝑠)(𝑥)=𝑢𝜑(𝑠). This means that ̃𝜃(𝑢)=𝑢𝜑. Therefore, ̃𝜃𝜙= 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 𝐶𝑟(𝑆)𝐶𝜆𝑟(𝑆).

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

Proof. The result follows from the following isomorphisms [5]: 𝐶𝑟𝐶(𝑆)𝑆𝑟𝛿0,𝐶𝜆𝑟𝐶(𝑆)Λ𝑆𝑟𝛿0,(2.9) where Λ is the left regular representation of 𝑆𝑟 and 𝐶Λ(𝑆𝑟) is the completion of 1(𝑆𝑟) in the norm 𝑓Λ=supΛ(𝑓).

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

Proof. If 𝐸𝑆 is finite then so is 𝐸𝑆𝑟=𝐸𝑆{0}, hence 1(𝑆𝑟) is amenable [2], and so is 1𝑟(𝑆)1(𝑆𝑟)/𝛿0.
Conversely, if 1𝑟(𝑆) is amenable, then 1(𝑆𝑟)/𝛿0 and 𝛿0 are both amenable. Hence, 1(𝑆𝑟) is amenable, therefore 𝐸𝑆𝑟 and so 𝐸𝑆 are finite [2].

Let 𝜑𝑆𝐺 be the quotient map of the inverse semigroup 𝑆 onto its maximal group homomorphic image and 𝜑𝑟𝑆𝑟𝐺0 a grading of 𝑆𝑟 by the group 𝐺, let 𝐻𝑟=𝜑𝑟1(𝑒){0} and 𝐻=𝜑1(𝑒). There is a conditional expectation 𝜀𝐶0𝑆𝑟𝐶𝑆𝑟𝛿0𝐶𝑟(𝑆)𝐶0𝐻𝑟𝐶𝐻𝑟𝛿0𝐶𝑟𝜀(𝐻),𝑟𝐶𝜆,0𝑆𝑟𝐶𝜆𝑆𝑟𝛿0𝐶𝜆𝑟(𝑆)0𝐻𝑟=1𝐻𝑟𝛿0=1𝑟(𝐻)𝜆𝐶𝜆𝑟(𝐻).(2.10) These are extensions of the restriction map 1𝑟(𝑆)1𝑟(𝐻), where 𝐻𝑆. By [8, Theorem 4.2], 𝑆𝑟 has weak containment property if and only if 𝜀 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 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=𝜑1(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 𝜀𝐶𝑟(𝑆)𝐶𝑟(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)𝜀𝐶𝑟(𝑆)𝐶𝑟(𝐸) is faithful,(iii)the Fell bundle of 𝐶𝑟(𝑆) is amenable.

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