Research Article | Open Access
Restricted Algebras on Inverse Semigroups—Part II: Positive Definite Functions
The relation between representations and positive definite functions is a key concept in harmonic analysis on topological groups. Recently this relation has been studied on topological groupoids. In this paper, we investigate the concept of restricted positive definite functions and their relation with restricted representations of an inverse semigroup. We also introduce the restricted Fourier and Fourier-Stieltjes algebras of an inverse semigroup and study their relation with the corresponding algebras on the associated groupoid.
In  we introduced the concept of restricted representations for an inverse semigroup and studied the restricted forms of some important Banach algebras on . In this paper we continue our study by considering the relation between the restricted positive definite functions and restricted representations. In particular, in Section 2 we prove restricted version of the Godement's characterization of positive definite functions on groups (Theorem 2.9). In Section 3 we study the restricted forms of the Fourier and Fourier-Stieltjes algebras on . The last section is devoted to the study of the Fourier and Fourier-Stieltjes algebras on the associated groupoid of , as well as the -algebra of certain related graph groupoids.
An inverse semigroup is a discrete semigroup such that for each there is a unique element such that The set of idempotents of consists of elements of the form . Then is a commutative subsemigroup of . There is a natural order ≤ on defined by if and only if . A - representation of is a pair consisting of a (possibly infinite dimensional) Hilbert space and a map satisfying that is, a -semigroup homomorphism from into the -semigroup of partial isometries on . Let be the family of all -representations of with For , is the Banach space of all complex valued functions on satisfying For , consists of those with . Recall that is a Banach algebra with respect to the product and is a Hilbert space with inner product
Let also put for each .
Given , the restricted product of is if and undefined, otherwise. The set with its restricted product forms a groupoid , called the associated groupoid of . If we adjoin a zero element 0 to this groupoid and put , we get an inverse semigroup with the multiplication rule which is called the restricted semigroup of . A restricted representation of is a map such that and
Let be the family of all restricted representations of with . It is not hard to guess that should be related to . Let be the set of all with . Note that contains all cyclic representations of . Now it is clear that, via a canonical identification, . Two basic examples of restricted representations are the restricted left and right regular representations and of . For each put then is a semisimple Banach -algebra  which is denoted by and is called the restricted semigroup algebra of .
All over this paper, denotes a unital inverse semigroup with identity 1. denotes the set of idempotents of which consists of elements of the form . is the family of all -representations of with
2. Restricted Positive Definite Functions
A bounded complex valued function is called positive definite if for all positive integers and all , and , we have and it is called restricted positive definite if for all positive integers and all , and , we have We denote the set of all positive definite and restricted positive definite functions on by and , respectively. The two concepts coincide for (discrete) groups.
It is natural to expect a relation between and . Before checking this, note that is hardly ever unital. This is important, as the positive definite functions in nonunital case should be treated with extra care . Let us take any inverse semigroup with possibly no unit. Of course, one can always adjoin a unit 1 to with to get a unital inverse semigroup (if happened to have a unit we put ). However, positive definite functions on do not necessarily extend to positive definite functions on . Following , we consider the subset of extendible positive definite functions on which are those such that , and there exists a constant such that for all , , and , If is the canonical isomorphism, then maps onto the set of extendible positive bounded linear functionals on (those which are extendible to a positive bounded linear functional on ), and the restriction of to is an isometric affine isomorphism of convex cones [3, 1.1]. Also the linear span of is an algebra [3, 3.4] which coincides with the set of coefficient functions of -representations of [3, 3.2]. If has a zero element, then so is . In this case, we put and . To each , there corresponds a cyclic - representation of which restricts to a cyclic representation of (see the proof of [3, 3.2]). Let be the direct sum of all cyclic representations of obtained in this way, then the set of all coefficient functions of is the linear span of [3, 3.2]. We call the universal representation of .
All these arrangements are for , as it is an inverse 0-semigroup which is not unital unless is a group. We remind the reader that our blanket assumption is that is a unital inverse semigroup. From now on, we also assume that has no zero element (see Example 2.2).
Lemma 2.1. The restriction map is an affine isomorphism of convex cones.
Proof. Let . For each , and , we have
in particular if , then
so maps into .
is clearly an injective affine map. Also if and is extension by zero of on , then from the above calculation applied to , and , so is surjective.
It is important to note that the restriction map may fail to be surjective when already has a zero element.
Example 2.2. If with discrete topology and operations then is a zero inverse semigroup with identity. Here , as sets, , but the constant function is in . This in particular shows that the map is not necessarily surjective, if happens to have a zero element. To show that note that for each , each and each , if are distinct elements in , then for , we have , whenever , for each . Hence
Notation 1. Let be the set of all extendible elements of . This is a subcone which is mapped isomorphically onto a subcone by . The elements of are called extendible restricted positive definite functions on . These are exactly those such that , and there exists a constant such that for all , and ,
Proposition 2.3. There is an affine isomorphism of convex cones from onto
Proof. The affine isomorphism is just the restriction of the linear isomorphism of [1, Theorem 4.1] to the corresponding positive cones. Let us denote this by . In Notation 1 we presented an affine isomorphism from onto . Finally [3, 1.1], applied to , gives an affine isomorphism from onto , whose restriction is an affine isomorphism from onto . Now the obvious map , which makes the diagram (2.10) commutative, is the desired affine isomorphism.
In  the authors defined the Fourier algebra of a topological foundation -semigroups (which include all inverse semigroups) and in particular studied positive definite functions on these semigroups. Our aim in this section is to develop a parallel theory for the restricted case and among other results prove the generalization of the Godement's characterization of positive definite functions on groups  in our restricted context (Theorem 2.9).
For , put This is clearly a finite set, when and are finite.
Lemma 2.4. If is an inverse semigroup and , then . In particular, when and are of finite supports, then so is .
Proof. if and only if , for some with . This is clearly the case if and only if , for some and with . Hence .
The following lemma follows from the fact that the product is linear in each variable.
Lemma 2.5 (polarization identity). For each where .
Lemma 2.6. For each , one has .
Proof. For each Now if then for we have , and conversely and imply that , and then and , so , that is, . Hence the last sum is if , and it is zero, otherwise. Summing up, Now for , we get
Lemma 2.7. With the above notation, for each .
Proof. Given and , put and , then and, for each , which are obviously the same.
Lemma 2.8. For each and each , the coefficient function is in .
Proof. For each , and , noting that is a restricted representation,we have and, regarding as an element of and using the fact that , we have so .
Theorem 2.9. Let be a unital inverse semigroup. Given , the following statements are equivalent:(i),(ii)there is an such that .Moreover if is of finite support, then so is .
Proof. By the above lemma applied to , (ii) implies (i). Also if , then by Lemma 2.4, is of finite support.
Conversely assume that . Choose an approximate identity for consisting of positive, symmetric functions of finite support, as constructed in [1, Proposition 3.2]. Let be the restricted right regular representation of , then by the above lemma . Take , then if is the identity element, then for each we have as , where the last equality follows from [1, Lemma 3.2(ii)]. Hence, there is such that in . Now for each The last equality follows from the remark after Proposition 3.2 in  and the fact that . Hence , as required.
3. Restricted Fourier and Fourier-Stieltjes Algebras
Let be a unital inverse semigroup and, let be the set of all bounded positive definite functions on (see  for the group case and  for inverse semigroups). We use the notation with indices , , , and 0 to denote the positive definite functions which are restricted, extendible, of finite support, or vanishing at zero, respectively. Let be the linear span of . Then is a commutative Banach algebra with respect to the pointwise multiplication and the following norm : Also coincides with the set of the coefficient functions of elements of . If one wants to get a similar result for the set of coefficient functions of elements of , one has to apply the above facts to . But is not unital in general, so one is led to consider a smaller class of bounded positive definite functions on . The results of  suggests that these should be the class of extendible positive definite functions on . Among these, those which vanish at 0 correspond to elements of .
The structure of algebras and is far from being well understood, even in special cases. From the results of [4, 10], it is known that for a commutative unital discrete -semigroup , via Bochner theorem . Even in this case, the structure of seems to be much more complicated than the group case. This is mainly because of the lack of an appropriate analog of the group algebra. If is a discrete idempotent semigroup with identical involution. Then is a compact topological semigroup with pointwise multiplication. We believe that in this case where is the Baker algebra on (see e.g., ) however we are not able to prove it at this stage. In this section we show that the linear span of is a commutative Banach algebra with respect to the pointwise multiplication and an appropriate modification of the above norm. We call this the restricted Fourier-Stieltjes algebra of and show that it coincides with the set of all coefficient functions of elements of .
As before, the indices , 0, and are used to distinguish extendible elements, elements vanishing at 0, and elements of finite support, respectively. We freely use any combination of these indices. Consider the linear span of which is clearly a two-sided ideal of , whose closure is called the restricted Fourier algebra of . We show that it is a commutative Banach algebra under pointwise multiplication and norm of .
In order to study properties of , we are led by Proposition 2.3 to consider . More generally we calculate this algebra for any inverse 0-semigroup . Let be the linear span of with pointwise multiplication and the norm and let be the closed ideal of consisting of elements vanishing at 0. First let us show that is complete in this norm. The next lemma is quite well known and follows directly from the definition of the functional norm.
Lemma 3.1. If is a Banach space, is dense, and , then
Lemma 3.2. If is an inverse 0-semigroup (not necessarily unital), then we have the following isometric isomorphism of Banach spaces: (i),(ii).In particular and are Banach spaces.
Proof. (ii) clearly follows from (i). To prove (i), first recall that is affinely isomorphic to [3, 1.1] via
This defines an isometric isomorphism from into (with the dual norm). By the brevious lemma, one can lift to an isometric isomorphism from into . We only need to check that is surjective. Take any , and let be the restriction of to . Since , for each , The norm of as a linear functional on is not bigger than the norm of as a functional on . In particular, and so there is a with . Then , as required.
We know that the restriction map is a surjective linear isomorphism. Also is clearly an algebra homomorphism ( is an algebra under pointwise multiplication [3, 3.4], and the surjectivity of implies that the same fact holds for ). Now we put the following norm on then using the fact that is a Banach algebra (it is a closed subalgebra of which is a Banach algebra [5, Theorem 3.4]) we have the following.
Lemma 3.3. The restriction map is an isometric isomorphism of normed algebras. In particular, is a commutative Banach algebra under pointwise multiplication and above norm. Proof. The second assertion follows from the first and Lemma 2.4 applied to . For the first assertion, we only need to check that is an isometry. But this follows directly from [1, Theorem 3.2] and the fact that .
Corollary 3.4. is the set of coefficient functions of elements of .
Proof. Given , let be the extension by zero of to a function on , then , so there is a cyclic representation , say with cyclic vector , such that (see the proof of [3, 3.2]). But that is, . But is the cyclic vector of , which means that for each , there is a net of elements of the form , converging to in the norm topology of , and so , and so . This means that . Now a standard argument, based on the fact that is closed under direct sum, shows that each is a coefficient function of some element of . The converse follows from Lemma 2.8.
Corollary 3.5. One has the isometric isomorphism of Banach spaces .
Next, as in , we give an alternative description of the norm of the Banach algebra . For this we need to know more about the universal representation of . The universal representation of is the direct sum of all cyclic representations corresponding to elements of . To be more precise, this means that given any we consider as a positive linear functional on , then by [7, 21.24], there is a cyclic representation of , with , such that Therefore is a cyclic representation of and on . Now is the direct sum of all 's, where ranges over . There is an alternative construction in which one can take the direct sum of 's with ranging over to get a subrepresentation of . Clearly and . It follows from [3, 3.2] that the set of coefficient functions of and are and , respectively. As far as the original semigroup is concerned, we prefer to work with , since it could be considered as an element of . Now is a nondegenerate - representation of which uniquely extends to a nondegenerate representation of the restricted full -algebra , which we still denote by . We gather some of the elementary facts about in the next lemma.
Lemma 3.6. With the above notation, we have the following. (i) is the direct sum of all nondegenerate representations of associated with elements via the GNS, construction, namely, is the universal representation of . In particular, is faithfully represented in .(ii)The von Numann algebras and the double commutant of in are isomorphic. They are generated by elements , with , as well as by elements , with .(iii)Each representation of uniquely decomposes as .(iv)For each and , let , then and
Proof. Statement (i) follows by a standard argument. Statement (iii) and the first part of (ii) follow from (i), and the second part of (ii) follows from the fact that both sets of elements described in (ii) have clearly the same commutant in as the set of elements , with which generate . The first statement of (iv) follows from Lemma 2.8 and Corollary 3.5. As for the second statement, first note that for each , is the image of under the canonical embedding of in . Therefore, by (iii), Taking limit in we get the same relation for any , and then, using , by taking limit in the ultraweak topology of , we get the desired relation.
Lemma 3.7. Let 1 be the identity of , then for each one has .
Proof. As and , we have . Conversely, by the proof of Corollary 3.4, there is and such that . Hence .
Lemma 3.8. For each and , consider , then . Conversely each is of this form and one may always choose so that .
Proof. The first assertion follows directly from the definition of (see the paragraph after Lemma 3.6). The first part of the second assertion is the content of Corollary 3.4. As for the second part, basically the proof goes as in . Consider as an element of and let be the polar decomposition of , with and , and the dot product is the module action of on . Again, by the proof of Corollary 3.4, there is a cyclic representation , say with cyclic vector , such that . Put , then and, by Lemma 3.6 applied to , and, by Corollary 3.5 and Lemma 3.7,
Note that the above lemma provides an alternative (direct) way of proving the second statement of Lemma 3.3 (just take any two elements in and represent them as coefficient functions of two representations such that the equality holds for the norms of both and , then use the tensor product of those representations to represent and apply the first part of the lemma to ). Also it gives the alternative description of the norm on as follows.
Corollary 3.9. For each ,
Corollary 3.10. For each ,
Proof. Just apply Kaplansky's density theorem to the unit ball of .
Corollary 3.11. The unit ball of is closed in the topology of pointwise convergence.
Proof. If with , then for each , each and each , If , pointwise on with , , for each , then all 's satisfy above inequality, and so does . Hence, by above corollary, and .
Lemma 3.12. For each , and if is the norm of , .
The next theorem extends Eymard's theorem [9, 3.4] to inverse semigroups.
Theorem 3.13. Consider the following sets: Then , and the closures of all of these sets in are equal to .
Proof. The inclusion follows from Lemma 2.5, and the inclusions and follow from Theorem 2.9. The inclusions and are trivial. Now is dense in by Lemma 3.12, and the fact that is dense in . Finally , by definition, and by Theorem 2.9; hence , for each .
Lemma 3.14. separates the points of .
Proof. We know that has a faithful representation (namely the left regular representation ), so separates the points of [3, 3.3]. Hence separates the points of .
Proposition 3.15. For each there is with . Also separates the points of .
Proof. Given , let , then . Also given and as above, if , then separates and . If , then use above lemma to get some which separates and . Then , so ; that is, separates and .
Proposition 3.16. For each finite subset , there is such that .
Proof. For , let and note that (since , for each ). Now given a finite set , put ; then since we have , and since and we have . Hence . Now is a finite set, and if , then and .
Corollary 3.17. and .
Proof. Clearly . Now if , then , for some and . Let and be as in the above proposition, then so is in the linear span of .
4. Fourier and Fourier-Stieltjes Algebras of Associated Groupoids
We observed in Section 1 that one can naturally associate a (discrete) groupoid to any inverse semigroup . The Fourier and Fourier-Stieltjes algebras of (topological and measured) groupoids are studied in [11–14]. It is natural to ask if the results of these papers, applied to the associated groupoid of , could give us some information about the associated algebras on . In this section we explore the relation between and its associated groupoid and resolve some technical difficulties which could arise when one tries to relate the corresponding function algebras. We also investigate the possibility of assigning graph groupoids to and find relations between the corresponding -algebras.
Let us recall some general terminology and facts about groupoids. There are two parallel approaches to the theory of groupoids, theory of measured groupoids, and theory of locally compact groupoids (compare  with ). Here we deal with discrete groupoids (like ), and so basically it doesn't matter which approach we take, but the topological approach is more suitable here. Even if one wants to look at the topological approach, there are two different interpretations about what we mean by a “representation” (compare  with ). The basic difference is that whether we want representations to preserve multiplications everywhere or just almost everywhere (with respect to a Borel measure on the unit space of our groupoid which changes with each representation). Again the “everywhere approach” is more suitable for our setting. This approach, mainly taken by [11, 12], is the best fit for the representation theory of inverse semigroups (when one wants to compare representation theories of and ). Even then, there are some basic differences which one needs to deal with them carefully.
We mainly follow the approach and terminology of . As we only deal with discrete groupoids we drop the topological considerations of . This would simplify our short introduction and facilitate our comparison. A (discrete) groupoid is a set with a distinguished subset of pairs of multiplicable elements, a multiplication map, and an inverse map , such that for each (i),(ii)if are in , then so are , and ,(iii) is in and if is in then ,(iv)if is in , then .
For , and are called the source and range of , respectively. is called the unit space of . For each we put , , and . Note that for each , is a (discrete) group, called the isotropy group at . Any (discrete) groupoid is endowed with left and right Haar systems and , where and are simply counting measures on and , respectively. Consider the algebra of finitely supported functions on . We usually make this into a normed algebra using the so-called -norm where the above supremums are denoted, respectively, by and . Note that in general is not complete in this norm. We show the completion of in by . There are also natural -norms in which one can complete and get a -algebra. Two-well known groupoid -algebras obtained in this way are the full and reduced groupoid -algebras and . Here we briefly discuss their construction and refer the reader to  for more details.
A Hilbert bundle over is just a field of Hilbert spaces indexed by . A representation of is a pair consisting of a map and a Hilbert bundle over such that, for each ,(i) is a surjective linear isometry,(ii),(iii)if is in , then .
We usually just refer to as the representation, and it is always understood that there is a Hilbert bundle involved. We denote the set of all representations of by . Note that here a representation corresponds to a (continuous) Hilbert bundle, where as in the usual approach to (locally compact or measured) categories representations are given by measurable Hilbert bundles (see  for more details).
A natural example of such a representation is the left regular representation of . The Hilbert bundle of this representation is whose fiber at is . In our case that is discrete, this is simply . Each could be regarded as a section of this bundle (which sends to the restriction of to ). Also acts on bounded sections of via Let be the set of sections of vanishing at infinity. This is a Banach space under the supnorm and contains . Furthermore, it is a canonical -module via