ISRN GeometryVolume 2011 (2011), Article ID 531023, 13 pagesdoi:10.5402/2011/531023

Research Article

Order Structure of the Figà-Talamanca-Herz Algebra

Marzieh Shams Yousefi,1,2 Massoud Amini,¹ and Fereshteh Sady¹

¹Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-134, Tehran, Iran²Department of Pure Mathematics, Faculty of Mathematical Science, University of Guilan, Rasht, Iran

Received 30 March 2011; Accepted 21 April 2011

Academic Editors: V. S. Matveev and E. Previato

Copyright © 2011 Marzieh Shams Yousefi et al. 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 study the interplay between the order structure and the -operator space structure of Figà-Talamanca-Herz algebra of a locally compact group . We show that for amenable groups, an order and algebra isomorphism of Figà-Talamanca-Herz-algebras yields an isomorphism or anti-isomorphism of the underlying groups. We also give a bound for the norm of a -completely positive linear map from Figà-Talamanca-Herz algebra to its dual space.

1. Introduction and Preliminaries

Throughout this paper, is a locally compact group, is a real number in and is the conjugate of , that is, . The Fourier algebra consists of all coefficient functions of the left regular representation of This is a Banach algebra with the norm [1]. When is abelian, the Fourier transform yields an isometric isomorphism from onto , where is the Pontryagin dual of . In general, is a two-sided closed ideal of the Fourier-Stieltjes algebra [1]. This is the linear span of the set of all positive definite continuous functions on .

In [2], Figà-Talamanca introduced a natural generalization of the Fourier algebra, for a compact abelian group , by replacing by . In [3], Herz extended the notion to an arbitrary group, leading to the commutative Banach algebra , called the Figà-Talamanca-Herz algebra. In many ways, this algebra behaves like the Fourier algebra. For example, Leptin's theorem remains valid, namely, is amenable if and only if has a bounded approximate identity [4]. The -analog, of the Fourier-Stieltjes algebra is defined as the multiplier algebra of , by some authors in [5, 6]. In this paper, we follow [7] for the definition of .

This paper investigates the order structure of . In an earlier paper, the authors studied the order structure of the Fourier algebra [8]. Here, we first introduce a positive cone on , then we show that for locally compact amenable groups and , if and are order and algebra isomorphic, then and are isomorphic or anti-isomorphic (Theorem 3.2(ii)). This extends a result of Arendt and Cannière on the Fourier algebra in the amenable group case [9].

1.1. -Operator Spaces

In this section, we give a brief introduction to the notion of -operator spaces [7]. Let ,  , and let be a vector space. We denote the vector space of matrices with entries from by . We put simply . The space is equipped with the operator norm from its canonical action on -dimensional -space, . Clearly, acts on by matrix multiplication. For a square matrix , we have

Definition 1.1. Let be a vector space. A -matricial norm on is a family such that for each is a norm on satisfying for each , and . Here, is the obvious matrix product, and and are the norms of and as the members of and , respectively.The vector space equipped with a -matricial norm is called a -matricial normed space. If moreover, each is a Banach space, is called an (abstract) -operator space.

Clearly, 2-operator spaces are the same as classical operator spaces. For more details about operator spaces see [10–12].

Definition 1.2. Let and be -operator spaces, and let , then for each , is the th amplification of . The map is called -completely bounded if If , we say that is a -complete contraction, and if is an isometry, for each , we call a -complete isometry.

By [13, Section 4], the collection of all -completely bounded maps from to is a Banach space under and a -operator space through the identification

Figà-Talamanca-Herz algebras are our main examples of -operator spaces, studied in [13, 14]. For any function we define by ,  . The Figà-Talamanca-Herz algebra consists of those functions for which there are sequences and in and , respectively, such that and

The norm of is defined as the infimum of the above sums over all possible representations of . Then is a Banach space which is embedded contractively in . It was shown by Herz that is a Banach algebra under pointwise multiplication. When , we get the Fourier algebra .

Let be the left regular representation of on , defined by . Then can be lifted to a representation of on . The algebra of pseudomeasures is defined as the -closure of in . There is a canonical duality via In particular, is the group von Neumann algebra . If the map from the projective tensor product to is defined by for , and , then is isometrically isomorphic to and

1.2. Complexification of Ordered Vector Spaces

We often consider vector spaces and algebras over the complex field. It is therefore desirable to have the notion of a complex ordered vector space. This is usually done through the complexification of real ordered spaces. We recall some basic constructions in the theory of complexification. For more details, see [15].

If is a real vector space, then the complexification of is the additive group with scalar multiplication defined by , for . Each is uniquely represented as , where . Thus, can be written as .

For real vector spaces and with complexifications and , every -linear map has a unique -linear extension given by , for , where . The map is the canonical extension of (usually again denoted by ).

A real vector space , endowed with an order relation ≤, is called a real ordered space if(i) implies , for all (ii) implies , for all and .

The subset is called the positive cone of the real ordered space . In general, a cone is a subset of such that and , for and . Then defines an order structure on by if and only if . A cone is called proper if . If a normed space is an ordered space with the positive cone , then there is a natural order structure on its dual space . The positive cone of is defined as

A complex vector space is called an ordered space, if it is the complexification of a real ordered vector space. A (complex) Banach space is called a Banach ordered space if it is the complexification of a real ordered space which is also a real Banach space, such that:(i)the inclusion map is an isometry,(ii)each element can be written as , where are positive in and , for .

Banach lattices, -algebras, and their duals are Banach ordered spaces.

Given ordered spaces and , a -linear map is positive if and the restriction is positive.

1.3. Order Structure of -Operator Spaces

Each operator space can be embedded in , for some Hilbert space , by Ruan's Theorem [10, Theorem ], and the order structure of is induced by [8] (see also [16]). We have a -analog of Ruan's theorem for -operator spaces [17, Theorem 4.1] which asserts that each -operator space is -complete isometrically embedded in for some space , which again induces an order structure on . The main challenge is of course that the powerful methods from -algebras and von Neumann algebras are no longer at one's disposal for .

In this paper, we confine ourselves to the special case of , where is a measure space and . We say that is positive if for each , where the pairing is the canonical dual action of on . For , this order is the natural order on the -algebra . We also put an order on subspaces of for . We say is positive if , for each . It is easy to see that these orders define proper cones on and , respectively. Also for , we say is positive, if for each and , the natural matrix action is positive.

2. Order Structure of Some -Operator Spaces

2.1. The -Fourier Stieltjes Algebra

The Fourier-Stieltjes algebra was defined by Eymard as the algebra of coefficient functions of unitary representations of on a Hilbert space , where [1]. In this section, we consider the -analog of the Fourier-Stieltjes algebra, introduced by Runde [7] and study its order structure given by the -analog of positive definite continuous functions.

Definition 2.1. A representation of on a Banach space is a pair , where is a group homomorphism from into the group of invertible isometries on which is continuous with respect to the given topology on and the strong operator topology on .

Definition 2.2. A Banach space is called (i)an -space if it is of the form , for some measure space ,(ii)a -space if it is isometrically isomorphic to a quotient of a subspace of an -space (or equivalently, a subspace of a quotient of an -space [7, Section 1, Remark 1]).

We denote by , the collection of all (equivalence classes of) representations of on -spaces.

Definition 2.3. Let be a locally compact group and let be a representation of . A coefficient function of is a function of the form where and . Define For , put .

Using a suitable definition of tensor product of -spaces, it is shown in [7] that is a commutative unital Banach algebra with the pointwise multiplication which contains contractively as a closed ideal. Also we know that can be embedded in isometrically if is amenable [7, Corollary 5.3].

A compatible couple of Banach spaces in the sense of interpolation theory (see [18]) is a pair of Banach spaces such that both and are embedded continuously in some (Hausdorff) topological vector space. In this case, the intersection is again a Banach space under the norm . For example, the pairs and are compatible couples.

Definition 2.4. Let be a representation of such that is a compatible couple. Then a -positive definite function on is a function which has a representation as , where . We call each element in the closure of the set of all -positive definite functions on in , where is a representation of on an -space, a -positive definite function on , and the set of all -positive definite functions on will be denoted by .

It follows from [7, Lemma 4.3] and the definition of that for each , associated to a representation , there exist a sequence of cyclic representations of on closed subspaces of , and in , such that

The closed subspace of is the closure of the set of all functions of the form , for some representation , where is an -space, , and .

Proposition 2.5. The linear span of is dense in .

Proof. Let have a representation as , where is a representation on some -space and . It is clear that is dense in both and . Hence, there exist sequences and in converging to and in and , respectively. Put , then By the definition of the norm in , we have which tends to zero, as .Now, it is enough to consider the following decomposition for , where , and note that each , for , belongs to . So belongs to the closed linear span of .

Each representation of induces a representation of the group algebra on via Let be the -universal representation of . The Banach space of all pseudofunctions on is the closure of in . By [7, Theorem 6.6], we know that is the dual space of via the pairing

We say that is positive if , for all , where is the corresponding linear functional of . Let be the full group -algebra of , which is the enveloping -algebra of [1].

Proposition 2.6. There exists a positive contraction from to .

Proof. Since each Hilbert space is a -space [3] and for each , where , it follows that the identity map is continuous. Thus, it induces a continuous map Consider the conjugate map Since for each and , is the inclusion map. By the definition of positivity in and is a positive map. Therefore is positive, and so is .

2.2. Figà-Talamanca-Herz Algebra

In this section, we study the order structure of the Figà-Talamanca-Herz algebra . Since is the set of coefficient functions of the left regular representation of on , we have . We define the positive cone of as the closure in , of the set of all function of the form , for a sequence in , and denote it by . It is clear that is contained in . Since , this order structure, in the case where , is the same as the order structure of , induced by the set as a positive cone.

Clearly, is positive as an element of if and only if it is positive as an element of , where is the dual cone induced by the positive cone . Also, since is closed, if and only if for each ,  [19].

Theorem 2.7. Let be a locally compact amenable group. Then , where is the identity of . In particular, is a proper cone.

Proof. Since , it follows that . But when is amenable, the identity map is an embedding with [3, Theorem C]. For each , we have where is the supremum norm on , that is, . In particular, is closed in and consequently . Now, let be an arbitrary element of . Then there exists a sequence in of the form converging to in . Since for each , it follows that converges to in . Clearly, is contained in and since is closed, .The second equality follows immediately from the above-mentioned fact that for each ,  .

3. Order Maps between -Operator Spaces

In this section, we study the positive maps between various -operator spaces. Also, we give the general form of the algebra and order isomorphisms between Figà-Talamanca-Herz algebras.

Proposition 3.1. Let be a Banach ordered space with a closed positive cone , and, be a normed space and an ordered space such that for , implies . Then every positive linear map from into is continuous.

Proof. Assume towards a contradiction that is a positive linear map that is not continuous. Then is unbounded on the unit ball of , and hence, on since . This implies that there exists a sequence in such that , for each . Since is closed, is in . Hence, , for each . Therefore, , for each , which is impossible.

We note that the above proposition remains true for a Banach space with a closed positive cone satisfying the following property: for each , there is a sequence converging to , such that with and

A linear map between two ordered spaces is called an order isomorphism if is one-to-one and onto, and moreover and are both positive maps.

Theorem 3.2. Let and be amenable locally compact groups with identities and , respectively, and let be an order and algebra isomorphism. Then (i), where is the evaluation homomorphism at ,  ,(ii)there exists an isomorphism or anti-isomorphism from onto such that for all .

Proof. (i) Consider the adjoint map , which is clearly an order isomorphism. Since is an algebra isomorphism, is a multiplicative linear functional on , and since for each locally compact group , any (non zero) multiplicative linear functional on is an evaluation homomorphism at some point of , it follows that, , for some . We note that is positive because is a positive element of , hence, by [20, Proposition 4.3]. (ii) We first note that ,  , by Theorem 2.7. Hence maps onto , and since for ,   spans , it follows that the restriction map is an order and algebra isomorphism from onto . Now, [9, Theorem 3.2] implies that there exists an isomorphism or anti-isomorphism such that for all . Now, the result follows from the density of in and continuity of .

Proposition 3.3. Let and be locally compact groups, and let be a completely positive linear map. Then there are infinitely many such that .

Proof. We first note that is continuous by Proposition 3.1. Now, assume on the contrary that there exists such that for each . Fix . Then there is some , the unit ball of , such that . Since , where is the algebra of all -continuous, completely bounded linear maps from to , and since is an injective von Neumann algebra, by [7, Corollary 2.6] or [4, Theorem 2.1], it follows that can be decomposed into positive elements such that the norm of each element in this decomposition is less than or equal to the norm of . Considering instead of , without loss of generality, we may assume that is positive, , and . Let be defined by . Then clearly, .Now, is positive in , hence, , and for each ,  is self-adjoint. Hence, , where for each and ,   is a unitary operator in . For each unit vector in , set ,  . Then Hence, we have On the other hand, By the Hahn-Banach theorem, for each , there is some such that . Now, put , then . Hence, and . Therefore, This means that the norm of as an element of is less than or equal to . Therefore, by inequality (3.2), Hence, is unbounded, which is a contradiction.

Let , and let be an arbitrary measure space. For subspaces and , we say that a linear map is -completely positive, if for all , the th amplification ,   of is a positive map. For simplicity in the case where , we call such maps -completely positive.

Recall that for a locally compact group , has the natural -operator space structure as the predual of [13]. Now, we define an order structure on as follows. Let , we say that is positive if for every -completely positive and -completely bounded linear map , the natural action is a positive scalar matrix.

For and or , we say that a linear map is -completely positive if for each positive integer , the th amplification of is a positive map. If this is the case for , we say that is - completely positive.

Let . Then since and are equivalent norms on , for each , there exists a positive scalar such that on . For the case where , for each , we choose such that on .

Proposition 3.4. Let be an amenable locally compact group, let and let be a linear map. Let be the restriction of to . (i)If is -completely positive, then is a completely positive linear map from to . (ii)If and is a bounded linear map, then for each , we have .(iii)If and is bounded, then for each , we have .(iv)If and is a -completely positive map, then there are infinitely many such that .

Proof. (i) Let be -completely positive. Let and . Since , where is the set of all -continuous completely positive linear maps from to , it follows that if and only if for each , we have . Since is amenable, the embedding is norm decreasing [3]. Therefore, each element in can be considered as an element of . Moreover, if and only if , for all with . It is clear that if , then it also belongs to . Therefore, is a -completely positive map. This implies that is a completely positive linear map. (ii) As we noted before, the embedding is a norm decreasing map with dense range. So, can be considered as a subspace of . We first show that for each , is continuous and has dense range. Let , then for each and which shows that is continuous. Consider now an element . Then for each , there exists a sequence in converging to in . Hence, which clearly converges to zero as . This implies that has dense range. Hence, for each , since we have that is, .(iii)Using the weak-star density of in , the same proof as in (ii) can be applied.(iv)Let be a -completely positive linear map. By part (i), the restriction map is a completely positive map. Hence, by Proposition 3.3, there are infinitely many such that . Now, since by part (iii), for each ,   the statement is clear.

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