International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 531023 |

Marzieh Shams Yousefi, Massoud Amini, Fereshteh Sady, "Order Structure of the FigΓ -Talamanca-Herz Algebra", International Scholarly Research Notices, vol. 2011, Article ID 531023, 13 pages, 2011.

Order Structure of the FigΓ -Talamanca-Herz Algebra

Academic Editor: V. S. Matveev
Received30 Mar 2011
Accepted21 Apr 2011
Published22 Jun 2011


We study the interplay between the order structure and the 𝑝-operator space structure of FigΓ -Talamanca-Herz algebra 𝐴𝑝(𝐺) of a locally compact group 𝐺. We show that for amenable groups, an order and algebra isomorphism of FigΓ -Talamanca-Herz-algebras yields an isomorphism or anti-isomorphism of the underlying groups. We also give a bound for the norm of a 𝑝-completely positive linear map from FigΓ -Talamanca-Herz algebra to its dual space.

1. Introduction and Preliminaries

Throughout this paper, 𝐺 is a locally compact group, 𝑝 is a real number in (1,∞) and π‘žβˆˆ(1,∞) is the conjugate of 𝑝, that is, 1/𝑝+1/π‘ž=1. The Fourier algebra 𝐴(𝐺) consists of all coefficient functions of the left regular representation πœ† of 𝐺𝐴(𝐺)=𝑀=(πœ†πœ‰,πœ‚)βˆΆπœ‰,πœ‚βˆˆπΏ2ξ€Ύ(𝐺).(1.1) This is a Banach algebra with the norm ‖𝑀‖𝐴(𝐺)=inf{β€–πœ‰β€–2β€–πœ‚β€–2βˆΆπ‘€=(πœ†πœ‰,πœ‚)} [1]. When 𝐺 is abelian, the Fourier transform yields an isometric isomorphism from 𝐴(𝐺) onto 𝐿1(𝐺), where 𝐺 is the Pontryagin dual of 𝐺. In general, 𝐴(𝐺) is a two-sided closed ideal of the Fourier-Stieltjes algebra 𝐡(𝐺) [1]. This is the linear span of the set 𝑃(𝐺) of all positive definite continuous functions on 𝐺.

In [2], FigΓ -Talamanca introduced a natural generalization of the Fourier algebra, for a compact abelian group 𝐺, by replacing 𝐿2(𝐺) by 𝐿𝑝(𝐺). In [3], Herz extended the notion to an arbitrary group, leading to the commutative Banach algebra 𝐴𝑝(𝐺), called the FigΓ -Talamanca-Herz algebra. In many ways, this algebra behaves like the Fourier algebra. For example, Leptin's theorem remains valid, namely, 𝐺 is amenable if and only if 𝐴𝑝(𝐺) has a bounded approximate identity [4]. The 𝑝-analog, 𝐡𝑝(𝐺) of the Fourier-Stieltjes algebra is defined as the multiplier algebra of 𝐴𝑝(𝐺), by some authors in [5, 6]. In this paper, we follow [7] for the definition of 𝐡𝑝(𝐺).

This paper investigates the order structure of 𝐴𝑝(𝐺). In an earlier paper, the authors studied the order structure of the Fourier algebra 𝐴(𝐺) [8]. Here, we first introduce a positive cone on 𝐴𝑝(𝐺), then we show that for locally compact amenable groups 𝐺1 and 𝐺2, if 𝐴𝑝(𝐺1) and 𝐴𝑝(𝐺2) are order and algebra isomorphic, then 𝐺1 and 𝐺2 are isomorphic or anti-isomorphic (Theorem 3.2(ii)). This extends a result of Arendt and CanniΓ¨re on the Fourier algebra in the amenable group case [9].

1.1. 𝑝-Operator Spaces

In this section, we give a brief introduction to the notion of 𝑝-operator spaces [7]. Let π‘›βˆˆβ„•,  π‘βˆˆ(1,∞), and let 𝐸 be a vector space. We denote the vector space of π‘›Γ—π‘š matrices with entries from 𝐸 by 𝕄𝑛,π‘š(𝐸). We put simply 𝕄𝑛,π‘šβˆΆ=𝕄𝑛,π‘š(β„‚). The space π•„π‘›βˆΆ=𝕄𝑛,𝑛 is equipped with the operator norm |β‹…|𝑛 from its canonical action on 𝑛-dimensional 𝐿𝑝-space, ℓ𝑛𝑝. Clearly, 𝕄𝑛 acts on 𝕄𝑛(𝐸) by matrix multiplication. For a square matrix π‘Ž=(π‘Žπ‘–π‘—)βˆˆπ•„π‘›, we have β€–π‘Žβ€–π΅(ℓ𝑛𝑝)⎧βŽͺ⎨βŽͺβŽ©βŽ›βŽœβŽœβŽ=sup𝑛𝑖=1|||||𝑛𝑗=1π‘Žπ‘–π‘—π‘₯𝑗|||||π‘βŽžβŽŸβŽŸβŽ 1/π‘βˆΆπ‘₯π‘—βˆˆβ„‚,𝑛𝑗=1||π‘₯𝑗||π‘βŽ«βŽͺ⎬βŽͺβŽ­β‰€1.(1.2)

Definition 1.1. Let 𝐸 be a vector space. A 𝑝-matricial norm on 𝐸 is a family (‖⋅‖𝑛)βˆžπ‘›=1 such that for each π‘›βˆˆβ„•,‖⋅‖𝑛 is a norm on 𝕄𝑛(𝐸) satisfying ||πœ†||β€–πœ†β‹…π‘₯β‹…πœ‡β€–β‰€β€–π‘₯‖𝑛||πœ‡||,β€–π‘₯βŠ•π‘¦β€–π‘›+π‘šξ€½=maxβ€–π‘₯‖𝑛,β€–π‘¦β€–π‘šξ€Ύ,(1.3) for each πœ†βˆˆπ•„π‘š,𝑛,πœ‡βˆˆπ•„π‘›,π‘š,π‘₯βˆˆπ•„π‘›(𝐸), and π‘¦βˆˆπ•„π‘š(𝐸). Here, πœ†β‹…π‘₯β‹…πœ‡ is the obvious matrix product, and |πœ†| and |πœ‡| are the norms of πœ† and πœ‡ as the members of ℬ(ℓ𝑛𝑝,β„“π‘šπ‘) and ℬ(β„“π‘šπ‘,ℓ𝑛𝑝), respectively.
The vector space 𝐸 equipped with a 𝑝-matricial norm (‖⋅‖𝑛)βˆžπ‘›=1 is called a 𝑝-matricial normed space. If moreover, each (𝕄𝑛(𝐸),‖⋅‖𝑛) is a Banach space, 𝐸 is called an (abstract) 𝑝-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 𝐸 and 𝐹 be 𝑝-operator spaces, and let π‘‡βˆˆβ„¬(𝐸,𝐹), then for each π‘›βˆˆβ„•, 𝑇(𝑛)βˆΆπ•„π‘›(𝐸)βŸΆπ•„π‘›(𝐹),𝑇(𝑛)π‘₯𝑖𝑗=𝑇π‘₯𝑖𝑗(1.4) is the 𝑛th amplification of 𝑇. The map 𝑇 is called 𝑝-completely bounded if ‖𝑇‖𝑝cbβ€–β€–π‘‡βˆΆ=sup(𝑛)β€–β€–<∞.(1.5) If ‖𝑇‖𝑝cb≀1, we say that 𝑇 is a 𝑝-complete contraction, and if 𝑇(𝑛) is an isometry, for each π‘›βˆˆβ„•, we call 𝑇 a 𝑝-complete isometry.

By [13, Section 4], the collection π’žβ„¬π‘(𝐸,𝐹) of all 𝑝-completely bounded maps from 𝐸 to 𝐹 is a Banach space under ‖⋅‖𝑝cb and a 𝑝-operator space through the identification π•„π‘›ξ€·π’žβ„¬π‘ξ€Έ(𝐸,𝐹)=π’žβ„¬π‘ξ€·πΈ,𝕄𝑛(𝐹)(π‘›βˆˆβ„•).(1.6)

FigΓ -Talamanca-Herz algebras are our main examples of 𝑝-operator spaces, studied in [13, 14]. For any function π‘“βˆΆπΊβ†’β„‚ we define ξ‚π‘“βˆΆπΊβ†’β„‚ by 𝑓(π‘₯)=𝑓(π‘₯βˆ’1),  π‘₯∈𝐺. The FigΓ -Talamanca-Herz algebra 𝐴𝑝(𝐺) consists of those functions π‘“βˆΆπΊβ†’β„‚ for which there are sequences (πœ‰π‘›)βˆžπ‘›=1 and (πœ‚π‘›)βˆžπ‘›=1 in πΏπ‘ž(𝐺) and 𝐿𝑝(𝐺), respectively, such that βˆ‘π‘“=βˆžπ‘›=1πœ‰π‘›βˆ—Μƒπœ‚π‘› and βˆžξ“π‘›=1β€–β€–πœ‰π‘›β€–β€–π‘žβ€–β€–πœ‚π‘›β€–β€–π‘<∞.(1.7)

The norm ‖𝑓‖𝐴𝑝(𝐺) of π‘“βˆˆπ΄π‘(𝐺) is defined as the infimum of the above sums over all possible representations of 𝑓. Then 𝐴𝑝(𝐺) is a Banach space which is embedded contractively in 𝐢0(𝐺). It was shown by Herz that 𝐴𝑝(𝐺) is a Banach algebra under pointwise multiplication. When 𝑝=2, we get the Fourier algebra 𝐴(𝐺).

Let πœ†π‘βˆΆπΊβ†’β„¬(𝐿𝑝(𝐺)) be the left regular representation of 𝐺 on 𝐿𝑝(𝐺), defined by πœ†π‘(𝑠)(𝑓)(𝑑)=𝑓(π‘ βˆ’1𝑑). Then πœ†π‘ can be lifted to a representation of 𝐿1(𝐺) on 𝐿𝑝(𝐺). The algebra of pseudomeasures PM𝑝(𝐺) is defined as the π‘€βˆ—-closure of πœ†π‘(𝐿1(𝐺)) in ℬ(𝐿𝑝(𝐺)). There is a canonical duality PM𝑝(𝐺)≅𝐴𝑝(𝐺)βˆ— viaξ€·βŸ¨πœ‰βˆ—Μƒπœ‚,π‘‡βŸ©βˆΆ=βŸ¨πœ‰,𝑇(πœ‚)βŸ©πœ‰βˆˆπΏπ‘(𝐺),πœ‚βˆˆπΏπ‘ž(𝐺),π‘‡βˆˆPM𝑝(𝐺).(1.8) In particular, PM2(𝐺) is the group von Neumann algebra VN(𝐺). If the map Λ𝑝 from the projective tensor product πΏπ‘žξβŠ—(𝐺)𝛾𝐿𝑝(𝐺) to 𝐢0(𝐺) is defined byΛ𝑝(π‘”βŠ—π‘“)(𝑠)=𝑔,πœ†π‘ξ¬(𝑠)𝑓,(1.9) for π‘”βˆˆπΏπ‘ž(𝐺),π‘“βˆˆπΏπ‘(𝐺), and π‘ βˆˆπΊ, then 𝐴𝑝(𝐺) is isometrically isomorphic to πΏπ‘žξβŠ—(𝐺)𝛾𝐿𝑝(𝐺)/kerΛ𝑝 and 𝐴𝑝(𝐺)βˆ—=ξ‚†ξ€·πΏπ‘‡βˆˆβ„¬π‘ξ€Έ(𝐺)βˆΆπ‘‡|kerΛ𝑝=0.(1.10)

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 𝐸 is a real vector space, then the complexification 𝐸ℂ of 𝐸 is the additive group 𝐸×𝐸 with scalar multiplication defined by (𝛼,𝛽)(π‘₯,𝑦)∢=(𝛼π‘₯βˆ’π›½π‘¦,𝛽π‘₯+𝛼𝑦), for (𝛼,𝛽)=𝛼+π‘–π›½βˆˆβ„‚. Each π‘§βˆˆπΈΓ—πΈ is uniquely represented as 𝑧=π‘₯+𝑖𝑦, where π‘₯,π‘¦βˆˆπΈ. Thus, 𝐸ℂ can be written as 𝐸+𝑖𝐸.

For real vector spaces 𝐸 and 𝐹 with complexifications 𝐸ℂ and 𝐹ℂ, every ℝ-linear map π‘‡βˆΆπΈβ†’πΉ has a unique β„‚-linear extension 𝑇 given by 𝑇(𝑧)∢=𝑇π‘₯+𝑖𝑇𝑦, for 𝑧=π‘₯+π‘–π‘¦βˆˆπΈβ„‚, where π‘₯,π‘¦βˆˆπΈ. The map 𝑇 is the canonical extension of 𝑇 (usually again denoted by 𝑇).

A real vector space 𝐸, endowed with an order relation ≀, is called a real ordered space if(i)π‘₯≀𝑦 implies π‘₯+𝑧≀𝑦+𝑧, for all π‘₯,𝑦,π‘§βˆˆπΈ(ii)π‘₯≀𝑦 implies 𝛼π‘₯≀𝛼𝑦, for all π‘₯,π‘¦βˆˆπΈ and π›Όβˆˆβ„+.

The subset 𝐸+∢={π‘₯∈𝐸∢0≀π‘₯} is called the positive cone of the real ordered space 𝐸. In general, a cone is a subset 𝑃 of 𝐸 such that π‘₯+π‘¦βˆˆπ‘ƒ and 𝛼π‘₯βˆˆπ‘ƒ, for π‘₯,π‘¦βˆˆπ‘ƒ and π›Όβˆˆβ„+. Then 𝑃 defines an order structure on 𝐸 by π‘₯≀𝑦 if and only if π‘¦βˆ’π‘₯βˆˆπ‘ƒ. A cone 𝑃 is called proper if π‘ƒβˆ©(βˆ’π‘ƒ)={0}. If a normed space 𝐸 is an ordered space with the positive cone 𝐸+, then there is a natural order structure on its dual space πΈβˆ—. The positive cone of πΈβˆ— is defined as πΈβˆ—+=ξ€½π‘“βˆˆπΈβˆ—ξ€·βˆΆπ‘“(𝑝)β‰₯0π‘βˆˆπΈ+ξ€Έξ€Ύ.(1.11)

A complex vector space is called an ordered space, if it is the complexification of a real ordered vector space. A (complex) Banach space 𝐴 is called a Banach ordered space if it is the complexification of a real ordered space 𝐡 which is also a real Banach space, such that:(i)the inclusion map π‘–βˆΆπ΅β†’π΄ is an isometry,(ii)each element π‘Žβˆˆπ΄ can be written as π‘Ž=π‘Ž1βˆ’π‘Ž2+𝑖(π‘Ž3βˆ’π‘Ž4), where π‘Ž1,…,π‘Ž4 are positive in 𝐴 and β€–π‘Žπ‘—β€–β‰€β€–π‘Žβ€–, for 𝑗=1,…,4.

Banach lattices, πΆβˆ—-algebras, and their duals are Banach ordered spaces.

Given ordered spaces 𝐸ℂ and 𝐹ℂ, a β„‚-linear map π‘‡βˆΆπΈβ„‚β†’πΉβ„‚ is positive if 𝑇(𝐸)βŠ†πΉ and the restriction 𝑇|πΈβˆΆπΈβ†’πΉ is positive.

1.3. Order Structure of 𝑝-Operator Spaces

Each operator space 𝐸 can be embedded in ℬ(β„‹), for some Hilbert space β„‹, by Ruan's Theorem [10, Theorem 2.3.5], and the order structure of 𝐸 is induced by ℬ(β„‹) [8] (see also [16]). We have a 𝑝-analog of Ruan's theorem for 𝑝-operator spaces [17, Theorem 4.1] which asserts that each 𝑝-operator space 𝐸 is 𝑝-complete isometrically embedded in ℬ(β„°) for some QS𝐿𝑝 space β„°, which again induces an order structure on 𝐸. The main challenge is of course that the powerful methods from πΆβˆ—-algebras and von Neumann algebras are no longer at one's disposal for 𝑝≠2.

In this paper, we confine ourselves to the special case of β„°=𝐿𝑝(𝑋), where 𝑋 is a measure space and 1<𝑝<∞. We say that π‘‡βˆˆβ„¬(𝐿𝑝(𝑋)) is positive if βŸ¨π‘‡π‘“,π‘“βŸ©β‰₯0 for each π‘“βˆˆπΏπ‘(𝑋)βˆ©πΏπ‘ž(𝑋), where the pairing is the canonical dual action of πΏπ‘ž(𝑋) on 𝐿𝑝(𝑋). For 𝑝=2, this order is the natural order on the πΆβˆ—-algebra ℬ(𝐿2(𝑋)). We also put an order on subspaces of 𝕄𝑛(ℬ(𝐿𝑝(𝑋))) for 𝑛β‰₯1. We say π‘‡βˆˆπ•„π‘›(ℬ(𝐿𝑝(𝑋)))≅ℬ(𝐿𝑛𝑝(𝑋)) is positive if βˆ‘π‘›π‘–,𝑗=1βŸ¨π‘‡π‘“π‘–,π‘“π‘—βŸ©β‰₯0, for each 𝑓1,…,π‘“π‘›βˆˆπΏπ‘(𝑋)βˆ©πΏπ‘ž(𝑋). It is easy to see that these orders define proper cones on ℬ(𝐿𝑝(𝑋)) and 𝕄𝑛(ℬ(𝐿𝑝(𝑋))), respectively. Also for 𝑇=[𝑇𝑖𝑗]βˆˆπ•„π‘›(ℬ(𝐿𝑝(𝑋))βˆ—), we say 𝑇 is positive, if for each π‘šβˆˆβ„• and πœ™=[πœ™π‘–π‘—]βˆˆπ•„π‘š(ℬ(𝐿𝑝(𝑋)))+, the natural matrix action βŸ¨πœ™,π‘‡βŸ© is positive.

2. Order Structure of Some 𝑝-Operator Spaces

2.1. The 𝑝-Fourier Stieltjes Algebra

The Fourier-Stieltjes algebra 𝐡(𝐺) was defined by Eymard as the algebra of coefficient functions π‘₯β†¦βŸ¨πœ‹(π‘₯)πœ‰,πœ‚βŸ© of unitary representations πœ‹ of 𝐺 on a Hilbert space β„‹, where πœ‰,πœ‚βˆˆβ„‹ [1]. In this section, we consider the 𝑝-analog 𝐡𝑝(𝐺) of the Fourier-Stieltjes algebra, introduced by Runde [7] and study its order structure given by the 𝑝-analog 𝑃𝑝(𝐺) of positive definite continuous functions.

Definition 2.1. A representation of 𝐺 on a Banach space β„° is a pair (πœ‹,β„°), where πœ‹ is a group homomorphism from 𝐺 into the group of invertible isometries on β„° which is continuous with respect to the given topology on 𝐺 and the strong operator topology on ℬ(β„°).

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

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

Definition 2.3. Let 𝐺 be a locally compact group and let (πœ‹,β„°) be a representation of 𝐺. A coefficient function of (πœ‹,β„°) is a function π‘“βˆΆπΊβ†’β„‚ of the form 𝑓(π‘₯)=βŸ¨πœ‹(π‘₯)πœ‰,πœ™βŸ©(π‘₯∈𝐺),(2.1) where πœ‰βˆˆβ„° and πœ™βˆˆβ„°βˆ—. Define 𝐡𝑝(𝐺)∢=π‘“βˆΆπΊβŸΆβ„‚βˆΆπ‘“isacoefficientfunctionofsome(πœ‹,𝐸)∈Rep𝑝(𝐺).(2.2) For π‘“βˆˆπ΅π‘(𝐺), put β€–π‘“β€–βˆΆ=inf{β€–πœ‰β€–β€–πœ™β€–βˆΆπ‘“(β‹…)=βŸ¨πœ‹(β‹…)πœ‰,πœ™βŸ©}.

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

A compatible couple of Banach spaces in the sense of interpolation theory (see [18]) is a pair (β„°0,β„°1) of Banach spaces such that both β„°0 and β„°1 are embedded continuously in some (Hausdorff) topological vector space. In this case, the intersection β„°0βˆ©β„°1 is again a Banach space under the norm β€–β‹…β€–(β„°0,β„°1)=max{β€–β‹…β€–β„°0,β€–β‹…β€–β„°1}. For example, the pairs (𝐴𝑝(𝐺),π΄π‘ž(𝐺)) and (𝐿𝑝(𝐺),πΏπ‘ž(𝐺)) are compatible couples.

Definition 2.4. Let (πœ‹,β„°) be a representation of 𝐺 such that (β„°,β„°βˆ—) is a compatible couple. Then a πœ‹-positive definite function on 𝐺 is a function which has a representation as 𝑓(π‘₯)=βŸ¨πœ‹(π‘₯)πœ‰,πœ‰βŸ©(π‘₯∈𝐺), where πœ‰βˆˆβ„°βˆ©β„°βˆ—. We call each element in the closure of the set of all πœ‹-positive definite functions on 𝐺 in 𝐡𝑝(𝐺), where πœ‹ is a representation of 𝐺 on an 𝐿𝑝-space, a 𝑝-positive definite function on 𝐺, and the set of all 𝑝-positive definite functions on 𝐺 will be denoted by 𝑃𝑝(𝐺).

It follows from [7, Lemma 4.3] and the definition of 𝑃𝑝(𝐺) that for each π‘“βˆˆπ‘ƒπ‘(𝐺), associated to a representation (πœ‹,β„°), there exist a sequence (πœ‹π‘›,ℰ𝑛)βˆžπ‘›=1 of cyclic representations of 𝐺 on closed subspaces ℰ𝑛 of β„°βˆ©β„°βˆ—, and {πœ‰π‘›} in ℰ𝑛, such that 𝑓(π‘₯)=βˆžξ“π‘›=1βŸ¨πœ‹π‘›(π‘₯)πœ‰π‘›,πœ‰π‘›βŸ©(π‘₯∈𝐺).(2.3)

The closed subspace 𝐡𝑝𝑝(𝐺) of 𝐡𝑝(𝐺) is the closure of the set of all functions π‘“βˆˆπ΅π‘(𝐺) of the form 𝑓(π‘₯)=βŸ¨πœ‹(π‘₯)πœ‰,πœ‚βŸ©,π‘₯∈𝐺, for some representation (πœ‹,β„°), where β„° is an 𝐿𝑝-space, πœ‰βˆˆβ„°, and πœ‚βˆˆβ„°βˆ—.

Proposition 2.5. The linear span of 𝑃𝑝(𝐺) is dense in 𝐡𝑝𝑝(𝐺).

Proof. Let π‘’βˆˆπ΅π‘π‘(𝐺) have a representation as 𝑒(π‘₯)=βŸ¨πœ‹(π‘₯)πœ‰,πœ‚βŸ©,π‘₯∈𝐺, where πœ‹ is a representation on some 𝐿𝑝-space β„°,πœ‰βˆˆβ„° and πœ‚βˆˆβ„°βˆ—. It is clear that β„°βˆ©β„°βˆ— is dense in both β„° and β„°βˆ—. Hence, there exist sequences {πœ‰π‘›} and {πœ‚π‘›} in β„°βˆ©β„°βˆ— converging to πœ‰ and πœ‚ in β„° and β„°βˆ—, respectively. Put 𝑒𝑛(π‘₯)∢=βŸ¨πœ‹(π‘₯)πœ‰π‘›,πœ‚π‘›βŸ©, then ξ€·π‘’βˆ’π‘’π‘›ξ€Έξ«πœ‹ξ€·(π‘₯)=(π‘₯)πœ‰,πœ‚βˆ’πœ‚π‘›+ξ«πœ‹ξ€·ξ€Έξ¬(π‘₯)πœ‰βˆ’πœ‰π‘›ξ€Έ,πœ‚π‘›ξ¬.(2.4) By the definition of the norm in 𝐡𝑝(𝐺), we have β€–β€–π‘’βˆ’π‘’π‘›β€–β€–β‰€β€–πœ‰β€–π‘β€–β€–πœ‚βˆ’πœ‚π‘›β€–β€–π‘ž+β€–β€–πœ‰βˆ’πœ‰π‘›β€–β€–π‘β€–β€–πœ‚π‘›β€–β€–π‘ž,(2.5) which tends to zero, as π‘›β†’βˆž.
Now, it is enough to consider the following decomposition for 𝑒𝑛, βŸ¨πœ‹(π‘₯)πœ‰π‘›,πœ‚π‘›1⟩=43𝑗=0ξ«ξ€·πœ‰πœ‹(π‘₯)𝑛+π‘–π‘—πœ‚π‘›ξ€Έ,πœ‰π‘›+π‘–π‘—πœ‚π‘›ξ¬,(2.6) where βˆšπ‘–=βˆ’1, and note that each πœ‰π‘›+π‘–π‘—πœ‚π‘›, for 𝑗=0,1,2,3, belongs to β„°βˆ©β„°βˆ—. So 𝑒 belongs to the closed linear span of 𝑃𝑝(𝐺).

Each representation (πœ‹,β„°) of 𝐺 induces a representation of the group algebra 𝐿1(𝐺) on β„° via ξ€œπœ‹(𝑓)∢=𝐺𝑓(π‘₯)πœ‹(π‘₯)π‘“βˆˆπΏ1(𝐺).(2.7) Let (πœ‹π‘,β„°) be the 𝑝-universal representation of 𝐺. The Banach space UPF𝑝(𝐺) of all pseudofunctions on 𝐺 is the closure of πœ‹π‘(𝐿1(𝐺)) in 𝐡(β„°). By [7, Theorem 6.6], we know that 𝐡𝑝(𝐺) is the dual space of UPF𝑝(𝐺) via the pairing ξ«πœ‹π‘(=ξ€œπ‘“),𝑔𝐺𝑓(π‘₯)𝑔(π‘₯)π‘“βˆˆπΏ1(𝐺),π‘”βˆˆπ΅π‘(𝐺).(2.8)

We say that π‘“βˆˆUPF𝑝(𝐺) is positive if πœ™π‘”(𝑓)β‰₯0, for all π‘”βˆˆπ΅π‘(𝐺)+=𝑃𝑝(𝐺), where πœ™π‘” is the corresponding linear functional of 𝑔. Let πΆβˆ—(𝐺) be the full group πΆβˆ—-algebra of 𝐺, which is the enveloping πΆβˆ—-algebra of 𝐿1(𝐺) [1].

Proposition 2.6. There exists a positive contraction from UPF𝑝(𝐺) to πΆβˆ—(𝐺).

Proof. Since each Hilbert space is a QS𝐿𝑝-space [3] and for each π‘“βˆˆπΏ1(𝐺),‖𝑓‖Rep2≀‖𝑓‖Rep𝑝, where ‖𝑓‖Rep𝑝=supπœ‹βˆˆRep𝑝(𝐺),β€–πœ‹β€–β‰€1β€–πœ‹(𝑓)β€–, it follows that the identity map ξ‚€πΏπ‘–βˆΆ1β€–(𝐺),β‹…β€–Repπ‘ξ‚βŸΆξ€·πΏ1β€–(𝐺),β‹…β€–Rep2ξ€Έ(2.9) is continuous. Thus, it induces a continuous map π‘–π‘βˆΆUPF𝑝(𝐺)βŸΆπΆβˆ—(𝐺).(2.10) Consider the conjugate map ξ€·π‘–π‘ξ€Έβˆ—βˆΆπ΅(𝐺)βŸΆπ΅π‘(𝐺).(2.11) Since for each π‘’βˆˆπ΅(𝐺) and π‘“βˆˆπΏ1(𝐺), π‘–ξ«ξ€·π‘ξ€Έβˆ—ξ¬(𝑒),𝑓=βŸ¨π‘’βˆ˜π‘–,π‘“βŸ©=βŸ¨π‘’,π‘“βŸ©,(2.12)(𝑖𝑐)βˆ— is the inclusion map. By the definition of positivity in 𝐡𝑝(𝐺) and 𝐡(𝐺),(𝑖𝑐)βˆ— is a positive map. Therefore (𝑖𝑐)βˆ—βˆ— is positive, and so is 𝑖𝑐=(𝑖𝑐)βˆ—βˆ—βˆ£UPF𝑝(𝐺).

2.2. FigΓ -Talamanca-Herz Algebra

In this section, we study the order structure of the FigΓ -Talamanca-Herz algebra 𝐴𝑝(𝐺). Since 𝐴𝑝(𝐺) is the set of coefficient functions of the left regular representation of 𝐺 on 𝐿𝑝(𝐺), we have 𝐴𝑝(𝐺)βŠ†π΅π‘π‘(𝐺). We define the positive cone of 𝐴𝑝(𝐺) as the closure in 𝐴𝑝(𝐺), of the set of all function of the form βˆ‘π‘“=𝑛𝑖=1πœ‰π‘–βˆ—Μƒπœ‰π‘–, for a sequence (πœ‰π‘–) in 𝐿𝑝(𝐺)βˆ©πΏπ‘ž(𝐺), and denote it by 𝐴𝑝(𝐺)+. It is clear that 𝐴𝑝(𝐺)+ is contained in 𝐡𝑝(𝐺)+. Since 𝐢𝑐(𝐺)βˆ©π‘ƒ(𝐺)=𝐴(𝐺)βˆ©π‘ƒ(𝐺), this order structure, in the case where 𝑝=2, is the same as the order structure of 𝐴(𝐺), induced by the set 𝑃(𝐺)∩𝐴(𝐺) as a positive cone.

Clearly, π‘‡βˆˆPM𝑝(𝐺) is positive as an element of 𝐡(𝐿𝑝(𝐺)) if and only if it is positive as an element of 𝐴𝑝(𝐺)βˆ—+, where 𝐴𝑝(𝐺)βˆ—+ is the dual cone induced by the positive cone 𝐴𝑝(𝐺)+. Also, since 𝐴𝑝(𝐺)+ is closed, π‘’βˆˆπ΄π‘(𝐺)+ if and only if for each π‘‡βˆˆPM𝑝(𝐺)+,  𝑇(𝑒)β‰₯0 [19].

Theorem 2.7. Let 𝐺 be a locally compact amenable group. Then 𝐴𝑝(𝐺)+=𝑃(𝐺)∩𝐴(𝐺)={π‘’βˆˆπ΄(𝐺)βˆΆβ€–π‘’β€–π΄π‘(𝐺)=𝑒(𝑒)}, where 𝑒 is the identity of 𝐺. In particular, 𝐴𝑝(𝐺)+ is a proper cone.

Proof. Since 𝐿𝑝(𝐺)βˆ©πΏπ‘ž(𝐺)βŠ†πΏ2(𝐺), it follows that 𝐴𝑝(𝐺)+βŠ†π‘ƒ(𝐺)∩𝐴(𝐺)‖⋅‖𝐴𝑝(𝐺). But when 𝐺 is amenable, the identity map π‘–βˆΆπ΄(𝐺)→𝐴𝑝(𝐺) is an embedding with ‖⋅‖𝐴𝑝(𝐺)≀‖⋅‖𝐴(𝐺) [3, Theorem C]. For each π‘’βˆˆπ‘ƒ(𝐺)∩𝐴(𝐺), we have ‖𝑒‖𝐴(𝐺)=𝑒(𝑒)β‰€β€–π‘’β€–βˆžβ‰€β€–π‘’β€–π΄π‘(𝐺)≀‖𝑒‖𝐴(𝐺),(2.13) where β€–β‹…β€–βˆž is the supremum norm on 𝐴(𝐺), that is, ‖𝑒‖𝐴(𝐺)=‖𝑒‖𝐴𝑝(𝐺). In particular, 𝑃(𝐺)∩𝐴(𝐺) is closed in 𝐴𝑝(𝐺) and consequently 𝐴𝑝(𝐺)+βŠ†π‘ƒ(𝐺)∩𝐴(𝐺). Now, let 𝑣 be an arbitrary element of 𝐴(𝐺)βˆ©π‘ƒ(𝐺). Then there exists a sequence in 𝑃(𝐺)βˆ©πΆπ‘(𝐺) of the form {π‘“π‘–βˆ—ξ‚π‘“π‘–},π‘“π‘–βˆˆπΆπ‘(𝐺),π‘–βˆˆβ„• converging to 𝑣 in 𝐴(𝐺). Since for each π‘’βˆˆπ΄(𝐺),‖𝑒‖𝐴𝑝(𝐺)≀‖𝑒‖𝐴(𝐺), it follows that {π‘“π‘–βˆ—ξ‚π‘“π‘–} converges to 𝑣 in 𝐴𝑝(𝐺). Clearly, {π‘“π‘–βˆ—ξ‚π‘“π‘–} is contained in 𝐴𝑝(𝐺)+ and since 𝐴𝑝(𝐺)+ is closed, π‘£βˆˆπ΄π‘(𝐺)+.
The second equality follows immediately from the above-mentioned fact that for each π‘’βˆˆπ‘ƒ(𝐺)∩𝐴(𝐺),  ‖𝑒‖𝐴(𝐺)=‖𝑒‖𝐴𝑝(𝐺).

3. Order Maps between 𝑝-Operator Spaces

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

Proposition 3.1. Let 𝐸 be a Banach ordered space with a closed positive cone 𝐸+, and, 𝐹 be a normed space and an ordered space such that for π‘₯,π‘¦βˆˆπΉ, 0≀π‘₯≀𝑦 implies β€–π‘₯‖≀‖𝑦‖. Then every positive linear map from 𝐸 into 𝐹 is continuous.

Proof. Assume towards a contradiction that π‘‡βˆΆπΈβ†’πΉ is a positive linear map that is not continuous. Then 𝑇 is unbounded on the unit ball π‘ˆ of 𝐸, and hence, on π‘ˆ+∢=π‘ˆβˆ©πΈ+ since π‘ˆβŠ†π‘ˆ+βˆ’π‘ˆ++𝑖(π‘ˆ+βˆ’π‘ˆ+). This implies that there exists a sequence {π‘₯𝑛}βˆžπ‘›=1 in π‘ˆ+ such that ‖𝑇π‘₯𝑛‖β‰₯𝑛3, for each π‘›βˆˆβ„•. Since 𝐸+ is closed, βˆ‘π‘₯π‘§βˆΆ=𝑛/𝑛2 is in 𝐸+. Hence, 𝑇𝑧β‰₯𝑇π‘₯𝑛/𝑛2>0, for each π‘›βˆˆβ„•. Therefore, ‖𝑇𝑧‖β‰₯𝑛, for each π‘›βˆˆβ„•, which is impossible.

We note that the above proposition remains true for a Banach space 𝐸 with a closed positive cone 𝐸+ satisfying the following property: for each π‘₯∈𝐸, there is a sequence {π‘₯𝑛} converging to π‘₯, such that π‘₯𝑛=π‘₯1π‘›βˆ’π‘₯2𝑛+𝑖(π‘₯3π‘›βˆ’π‘₯4𝑛) with π‘₯π‘–π‘›βˆˆπΈ+ and β€–π‘₯𝑖𝑛‖≀‖π‘₯𝑛‖.

A linear map 𝑇between two ordered spaces is called an order isomorphism if 𝑇 is one-to-one and onto, and moreover 𝑇 and π‘‡βˆ’1 are both positive maps.

Theorem 3.2. Let 𝐺1 and 𝐺2 be amenable locally compact groups with identities 𝑒1 and 𝑒2, respectively, and let π‘‡βˆΆπ΄π‘(𝐺1)→𝐴𝑝(𝐺2) be an order and algebra isomorphism. Then (i)π‘‡βˆ—(πœ†π‘’2)=πœ†π‘’1, where πœ†π‘’π‘– is the evaluation homomorphism at 𝑒𝑖,  𝑖=1,2,(ii)there exists an isomorphism or anti-isomorphism πœ‘ from 𝐺2 onto 𝐺1 such that 𝑇(𝑓)=π‘“βˆ˜πœ‘ for all π‘“βˆˆπ΄π‘(𝐺1).

Proof. (i) Consider the adjoint map π‘‡βˆ—βˆΆπ΄π‘(𝐺2)βˆ—β†’π΄π‘(𝐺1)βˆ—, which is clearly an order isomorphism. Since 𝑇 is an algebra isomorphism, π‘‡βˆ—(πœ†π‘’2) is a multiplicative linear functional on 𝐴𝑝(𝐺1), and since for each locally compact group 𝐺, any (non zero) multiplicative linear functional on 𝐴𝑝(𝐺)is an evaluation homomorphism at some point of 𝐺, it follows that, π‘‡βˆ—(πœ†π‘’2)=πœ†π‘₯, for some π‘₯∈𝐺1. We note that πœ†π‘₯ is positive because πœ†π‘’2 is a positive element of 𝐴𝑝(𝐺)βˆ—, hence, π‘₯=𝑒1 by [20, Proposition 4.3].
(ii) We first note that 𝐴𝑝(𝐺𝑖)+=𝑃(𝐺𝑖)∩𝐴(𝐺𝑖),  𝑖=1,2, by Theorem 2.7. Hence 𝑇 maps 𝑃(G1)∩𝐴(𝐺1) onto 𝑃(𝐺2)∩𝐴(𝐺2), and since for 𝑖=1,2,  𝑃(𝐺𝑖)∩𝐴(𝐺𝑖) spans 𝐴(𝐺𝑖), it follows that the restriction map 𝑇1=𝑇|𝐴(𝐺1) is an order and algebra isomorphism from 𝐴(𝐺1) onto 𝐴(𝐺2). Now, [9, Theorem 3.2] implies that there exists an isomorphism or anti-isomorphism πœ“βˆΆπΊ2→𝐺1 such that 𝑇1(𝑓)=π‘“βˆ˜πœ“ for all π‘“βˆˆπ΄(𝐺1). Now, the result follows from the density of 𝐴(𝐺1) in 𝐴𝑝(𝐺1) and continuity of 𝑇.

Proposition 3.3. Let 𝐺 and 𝐻 be locally compact groups, and let π‘‡βˆΆπ΄(𝐺)β†’VN(𝐻) be a completely positive linear map. Then there are infinitely many π‘›βˆˆβ„• such that ‖𝑇(𝑛)‖≀𝑛2.

Proof. We first note that 𝑇 is continuous by Proposition 3.1. Now, assume on the contrary that there exists 𝑛0βˆˆβ„• such that for each 𝑛β‰₯𝑛0,‖𝑇(𝑛)β€–>𝑛2. Fix 𝑛β‰₯𝑛0. Then there is some π‘₯∈(𝕄𝑛(𝐴(𝐺)))1, the unit ball of 𝕄𝑛(𝐴(𝐺)), such that ‖𝑇(𝑛)(π‘₯)β€–>𝑛2. Since 𝕄𝑛(𝐴(𝐺))=π’žβ„¬πœŽ(VN(𝐺),𝕄𝑛), where π’žβ„¬πœŽ(VN(𝐺),𝕄𝑛) is the algebra of all π‘€βˆ—-continuous, completely bounded linear maps from VN(𝐺) to 𝕄𝑛, and since 𝕄𝑛 is an injective von Neumann algebra, by [7, Corollary 2.6] or [4, Theorem 2.1], it follows that π‘₯ 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 π‘₯. Considering π‘¦βˆΆ=(π‘₯+π‘₯𝑑)/2 instead of π‘₯, without loss of generality, we may assume that π‘₯ is positive, β€–π‘₯‖≀1,π‘₯=π‘₯𝑑, and ‖𝑇(𝑛)(π‘₯)β€–β‰₯𝑛2/8. Let π‘‡π‘›β¨βˆΆ=𝑛𝑖=1β¨π‘‡βˆΆπ‘›π‘–=1⨁𝐴(G)→𝑛𝑖=1VN(𝐻) be defined by (𝑧𝑖)↦(𝑇𝑧𝑖). Then clearly, ‖𝑇𝑛⨁‖=‖𝑛𝑖=1𝑇‖=supβ€–(𝑧𝑖)‖≀1(βˆ‘π‘›π‘–=1‖𝑇𝑧𝑖‖2)1/2≀𝑛‖𝑇‖.
Now, 𝑇(𝑛)(π‘₯)=[𝑇(π‘₯𝑖𝑗)] is positive in 𝕄𝑛(VN(𝐻)), hence, [𝑇(π‘₯𝑖𝑗)βˆ—]=[𝑇(π‘₯𝑖𝑗)], and for each 𝑖,𝑗,  𝑇(π‘₯𝑖𝑗) is self-adjoint. Hence, 𝑇π‘₯𝑖𝑗=‖𝑇‖(𝑇π‘₯1𝑖𝑗+𝑇π‘₯2𝑖𝑗)/2, where for each 𝑖,π‘—βˆˆ{1,…,𝑛} and π‘˜=1,2,  𝑇π‘₯π‘˜π‘–π‘— is a unitary operator in ℬ(𝐿2(𝐺)). For each unit vector (𝑓1,𝑓2,…,𝑓𝑛) in ⨁𝑛𝑖=1𝐿2(𝐻), set 𝑇π‘₯π‘˜π‘–π‘—(𝑓𝑖)=π‘’π‘˜π‘–π‘—, β€‰π‘˜=1,2,1≀𝑖,𝑗≀𝑛. Then βŽ›βŽœβŽœβŽξ“π‘—β€–β€–β€–β€–ξ“π‘–π‘‡π‘₯1𝑖𝑗+𝑇π‘₯2𝑖𝑗2𝑓𝑖‖‖‖‖2⎞⎟⎟⎠1/2=βŽ›βŽœβŽœβŽξ“π‘—β€–β€–β€–β€–ξ“π‘–π‘’1𝑖𝑗+𝑒2𝑖𝑗2β€–β€–β€–β€–2⎞⎟⎟⎠1/2=12𝑗𝑖𝑒1𝑖𝑗+𝑒2𝑖𝑗,𝑖𝑒1𝑖𝑗+𝑒2𝑖𝑗ξƒͺ1/2≀12𝑖,𝑗𝑒1𝑖,𝑗+𝑒2𝑖,𝑗,𝑒1𝑖𝑗+𝑒2𝑖𝑗+4𝑛2ξƒͺ1/2=12𝑖,𝑗‖‖𝑒1𝑖,𝑗+𝑒2𝑖,𝑗‖‖2+4𝑛2ξƒͺ1/2≀‖‖𝑇𝑛2ξ€·π‘₯𝑖𝑗‖‖.+𝑛(3.1) Hence, we have ‖‖𝑇(𝑛)ξ€·π‘₯𝑖𝑗‖‖‖‖𝑇≀‖𝑇‖𝑛2ξ€·π‘₯𝑖𝑗‖‖+𝑛.(3.2) On the other hand, β€–π‘₯‖𝕄𝑛(𝐴(𝐺))=supπ‘†βˆˆπ•„π‘›(VN(𝐺))1β€–β€–ξ€Ίπ‘†π‘˜π‘™ξ€·π‘₯𝑖𝑗‖‖=supπ‘†βˆˆπ•„π‘›(VN(𝐺))1𝑖,𝑗,π‘˜,𝑙||π‘†π‘˜π‘™ξ€·π‘₯𝑖𝑗||2ξƒͺ1/2.(3.3) By the Hahn-Banach theorem, for each π‘₯𝑖𝑗, there is some π‘†π‘–π‘—βˆˆVN(𝐺)1 such that 𝑆𝑖𝑗(π‘₯)=β€–π‘₯𝑖𝑗‖. Now, put 𝑆=[𝑆𝑖𝑗], then ‖𝑆‖𝕄𝑛(VN(𝐺))≀𝑛. Hence, 𝑆/π‘›βˆˆπ•„π‘›(VN(𝐺))1 and (𝑆𝑖𝑗/𝑛)(π‘₯𝑖𝑗)=β€–π‘₯𝑖𝑗‖/𝑛. Therefore, β€–π‘₯‖𝕄𝑛(𝐴(𝐺))=supπ‘…βˆˆπ•„π‘›(VN(𝐺))1β€–β€–ξ€Ίπ‘…π‘˜π‘™ξ€·π‘₯𝑖𝑗‖‖β‰₯‖‖‖π‘₯𝑖𝑗𝑛‖‖‖2ξ‚Ά1/2=1𝑛‖‖π‘₯𝑖𝑗‖‖21/2.(3.4) This means that the norm of π‘₯ as an element of ⨁𝑛2𝑖=1𝐴(𝐺) is less than or equal to 𝑛. Therefore, by inequality (3.2), 𝑛2‖‖𝑇(8‖𝑇‖)βˆ’π‘›β‰€π‘›2ξ€·π‘₯𝑖𝑗‖‖≀‖‖π‘₯‖𝑇‖𝑖𝑗‖‖⨁𝑛2𝑖=1𝐴(𝐺)≀𝑛‖𝑇‖.(3.5) Hence, 𝑇 is unbounded, which is a contradiction.

Let 1<π‘ξ…ž<∞, and let π‘Œ be an arbitrary measure space. For subspaces β„³βŠ†β„¬(𝐿𝑝(𝑋)) and π’©βŠ†β„¬(𝐿𝑝′(π‘Œ)), we say that a linear map π‘‡βˆΆβ„³β†’π’© is (𝑝,π‘ξ…ž)-completely positive, if for all π‘›βˆˆβ„•, the 𝑛th amplification 𝑇(𝑛)βˆΆπ•„π‘›(β„³)→𝕄𝑛(𝒩),  𝑇(𝑛)[π‘₯𝑖𝑗]=[𝑇(π‘₯𝑖𝑗)] of 𝑇 is a positive map. For simplicity in the case where 𝑝=π‘ξ…ž, we call such maps 𝑝-completely positive.

Recall that for a locally compact group 𝐺, 𝐴𝑝(𝐺) has the natural 𝑝-operator space structure as the predual of PM𝑝(𝐺) [13]. Now, we define an order structure on 𝕄𝑛(𝐴𝑝(𝐺)) as follows. Let π‘š,π‘›βˆˆβ„•, we say that 𝑇=[𝑇𝑖𝑗]βˆˆπ•„π‘›(𝐴𝑝(𝐺)) is positive if for every 𝑝-completely positive and 𝑝-completely bounded linear map πœ™βˆΆPM𝑝(𝐺)β†’π•„π‘š, the natural action βŸ¨πœ™,π‘‡βŸ© is a positive scalar matrix.

For β„³=𝐴𝑝(𝐺)orPM𝑝(𝐺) and 𝒩=𝐴𝑝′(𝐺) or PM𝑝′(𝐺), we say that a linear map π‘‡βˆΆβ„³β†’π’© is (𝑝,π‘ξ…ž)-completely positive if for each positive integer 𝑛, the 𝑛th amplification π‘‡π‘›βˆΆπ•„π‘›(β„³)→𝕄𝑛(𝒩) of 𝑇 is a positive map. If this is the case for 𝑝=𝑝′, we say that 𝑇 is 𝑝- completely positive.

Let 𝑝β‰₯2. Then since β€–β‹…β€–2 and ‖⋅‖𝑝 are equivalent norms on ℂ𝑛, for each π‘›βˆˆβ„•, there exists a positive scalar 𝛼𝑛 such that 𝛼𝑛‖⋅‖2≀‖⋅‖𝑝≀‖⋅‖2 on ℂ𝑛. For the case where 1<𝑝≀2, for each π‘›βˆˆβ„•, we choose 𝛽𝑛>0 such that 𝛽𝑛‖⋅‖≀‖⋅‖2≀‖⋅‖𝑝 on ℂ𝑛.

Proposition 3.4. Let 𝐺 be an amenable locally compact group, let π‘βˆˆ(1,∞) and let π‘‡βˆΆπ΄π‘(𝐺)β†’PMp(𝐺) be a linear map. Let 𝑇1=𝑇|𝐴(𝐺) be the restriction of 𝑇 to 𝐴(𝐺). (i)If 𝑇 is 𝑝-completely positive, then 𝑇1 is a completely positive linear map from 𝐴(𝐺) to VN(𝐺). (ii)If 𝑝β‰₯2 and 𝑇 is a bounded linear map, then for each π‘›βˆˆβ„•, we have ‖𝑇1(𝑛)‖≀‖𝑇(𝑛)β€–.(iii)If 1<𝑝≀2 and 𝑇 is bounded, then for each π‘›βˆˆβ„•, we have ‖𝑇(𝑛)‖≀𝛽2𝑛‖𝑇1(𝑛)β€–.(iv)If 1<𝑝≀2 and 𝑇 is a 𝑝-completely positive map, then there are infinitely many π‘›βˆˆβ„• such that ‖𝑇(𝑛)‖≀𝛽2𝑛𝑛2.

Proof. (i) Let 𝑇 be 𝑝-completely positive. Let π‘›βˆˆβ„• and [π‘₯𝑖𝑗]βˆˆπ•„π‘›(𝐴(𝐺)). Since 𝕄𝑛(𝐴(𝐺))+=π’žπ’«πœŽ(VN(𝐺),𝕄𝑛), where π’žπ’«πœŽ(VN(𝐺),𝕄𝑛) is the set of all π‘€βˆ—-continuous completely positive linear maps from VN(𝐺) to 𝕄𝑛, it follows that [π‘₯𝑖𝑗]βˆˆπ•„π‘›(𝐴(𝐺))+ if and only if for each [𝑇𝑖𝑗]βˆˆπ•„π‘›[VN(𝐺)]+, we have [π‘‡π‘˜π‘™(π‘₯𝑖𝑗)]βˆˆπ•„π‘›2+. Since 𝐺 is amenable, the embedding π‘–βˆΆπ΄(𝐺)→𝐴𝑝(𝐺) is norm decreasing [3]. Therefore, each element [𝑆𝑖𝑗] in 𝕄𝑛[PM𝑝(𝐺)] can be considered as an element of 𝕄𝑛[VN(𝐺)]. Moreover, [𝑆𝑖𝑗]βˆˆπ•„π‘›[PM𝑝(𝐺)]+ if and only if ⟨[𝑆𝑖𝑗][𝑓],[𝑓]⟩β‰₯0, for all [𝑓]=(𝑓1,…,𝑓𝑛) with 𝑓1,…,π‘“π‘›βˆˆπΆπ‘(𝐺). It is clear that if [𝑆𝑖𝑗]βˆˆπ•„π‘›[PM𝑝(𝐺)]+, then it also belongs to 𝕄𝑛[VN(𝐺)]+. Therefore, π‘–βˆΆπ΄(𝐺)→𝐴𝑝(𝐺) is a (2,𝑝)-completely positive map. This implies that 𝑇1∢𝐴(𝐺)β†’VN(𝐺) is a completely positive linear map.
(ii) As we noted before, the embedding π‘–βˆΆπ΄(𝐺)→𝐴𝑝(𝐺) is a norm decreasing map with dense range. So, PM𝑝(𝐺) can be considered as a subspace of VN(𝐺). We first show that for each π‘›βˆˆβ„•, 𝑖(𝑛) is continuous and has dense range. Let [π‘Žπ‘–π‘—]βˆˆπ•„π‘›(𝐴(𝐺)), then [𝑆(π‘Žπ‘–π‘—)]∈𝐡(ℓ𝑛𝑝), for each π‘†βˆˆPM𝑝(𝐺) andβ€–β€–ξ€Ίπ‘Žπ‘–π‘—ξ€»β€–β€–π•„π‘›(𝐴𝑝(𝐺))=supπ‘†βˆˆPM𝑝(𝐺)1β€–β€–ξ€Ίπ‘†ξ€·π‘Žπ‘–π‘—β€–β€–ξ€Έξ€»=supπ‘†βˆˆPM𝑝(𝐺)1⎧βŽͺ⎨βŽͺβŽ©βŽ›βŽœβŽœβŽsup𝑛𝑖=1|||||𝑛𝑗=1π‘†ξ€·π‘Žπ‘–π‘—ξ€Έπ‘₯𝑗|||||π‘βŽžβŽŸβŽŸβŽ 1/𝑝,ξ€·π‘₯π‘—ξ€Έβˆˆβ„‚π‘›,𝑛𝑗=1||π‘₯𝑗||π‘βŽ«βŽͺ⎬βŽͺβŽ­β‰€1≀1𝛼𝑛supπ‘†βˆˆPM𝑝(𝐺)1⎧βŽͺ⎨βŽͺβŽ©βŽ›βŽœβŽœβŽsup𝑛𝑖=1|||||𝑛𝑗=1π‘†ξ€·π‘Žπ‘–π‘—ξ€Έπ‘₯𝑗|||||2⎞⎟⎟⎠1/2,ξ€·π‘₯π‘—ξ€Έβˆˆβ„‚π‘›,𝑛𝑗=1||π‘₯𝑗||2⎫βŽͺ⎬βŽͺ⎭=1≀1𝛼𝑛supπ‘†βˆˆPM𝑝(𝐺)1β€–β€–π‘†ξ€·π‘Žπ‘–π‘—ξ€Έβ€–β€–π΅(β„“2)≀1π›Όπ‘›β€–β€–ξ€Ίπ‘Žπ‘–π‘—ξ€»β€–β€–π•„π‘›(𝐴(𝐺)),(3.6) which shows that 𝑖(𝑛) is continuous. Consider now an element [π‘Žπ‘–π‘—]βˆˆπ•„π‘›(𝐴𝑝(𝐺)). Then for each 𝑖,𝑗, there exists a sequence (π‘Žπ‘šπ‘–π‘—)π‘š in 𝐴(𝐺) converging to π‘Žπ‘–π‘— in 𝐴𝑝(𝐺). Hence, β€–β€–ξ€Ίπ‘Žπ‘šπ‘–π‘—ξ€»βˆ’ξ€Ίπ‘Žπ‘–π‘—ξ€»β€–β€–π•„π‘›(𝐴𝑝(𝐺))=β€–β€–ξ€Ίπ‘Žπ‘šπ‘–π‘—βˆ’π‘Žπ‘–π‘—ξ€»β€–β€–π•„π‘›(𝐴𝑝(𝐺))=supπ‘†βˆˆPM𝑝(𝐺)1⎧βŽͺ⎨βŽͺβŽ©βŽ›βŽœβŽœβŽsup𝑛𝑖=1|||||𝑛𝑖=1ξ€·π‘†ξ€·π‘Žπ‘–π‘—ξ€Έξ€·π‘Žβˆ’π‘†π‘šπ‘–π‘—π‘₯𝑗|||||π‘βŽžβŽŸβŽŸβŽ 1/𝑝,ξ€·π‘₯π‘—ξ€Έβˆˆβ„‚π‘›,𝑛𝑗=1||π‘₯𝑗||π‘βŽ«βŽͺ⎬βŽͺ⎭,≀1(3.7) which clearly converges to zero as π‘šβ†’βˆž. This implies that 𝑖(𝑛) has dense range. Hence, for each π‘›βˆˆβ„•, since 𝛼𝑛‖⋅‖𝑀𝑛(VN(𝐺))≀‖⋅‖𝑀𝑛(PM𝑝(𝐺)) we have 𝛼𝑛‖‖𝑇(𝑛)β€–β€–=𝛼𝑛‖‖𝑇π‘₯β‹…sup𝑖𝑗‖‖𝑀𝑛(PM𝑝(𝐺))βˆΆξ€Ίπ‘₯π‘–π‘—ξ€»βˆˆπ•„π‘›ξ€·π΄π‘ξ€Έξ€Ίπ‘₯(𝐺),‖𝑖𝑗‖𝕄𝑛(𝐴𝑝(𝐺))≀1β‰₯𝛼𝑛𝛼⋅sup𝑛‖‖𝑇1ξ€·π‘₯𝑖𝑗‖‖𝑀𝑛(VN(𝐺))βˆΆξ€Ίπ‘₯π‘–π‘—ξ€»βˆˆπ•„π‘›β€–β€–ξ€Ίπ‘₯(𝐴(𝐺)),𝑖𝑗‖‖𝕄𝑛(𝐴𝑝(𝐺))𝛼≀1β‰₯sup𝑛‖‖𝑇1ξ€·π‘₯𝑖𝑗‖‖𝑀𝑛(VN(𝐺))βˆΆξ€Ίπ‘₯π‘–π‘—ξ€»βˆˆπ•„π‘›β€–β€–ξ€Ίπ‘₯(𝐴(𝐺)),𝑖𝑗‖‖𝕄𝑛(𝐴(𝐺))≀1=𝛼𝑛‖‖𝑇1(𝑛)β€–β€–,(3.8) that is, ‖𝑇1(𝑛)‖≀‖𝑇(𝑛)β€–.(iii)Using the weak-star density of PM𝑝(𝐺)1 in VN(𝐺)1, the same proof as in (ii) can be applied.(iv)Let π‘‡βˆΆπ΄π‘(𝐺)β†’PM𝑝(𝐺) be a 𝑝-completely positive linear map. By part (i), the restriction map 𝑇1 is a completely positive map. Hence, by Proposition 3.3, there are infinitely many π‘›βˆˆβ„• such that ‖𝑇1(𝑛)‖≀𝑛2. Now, since by part (iii), for each π‘›βˆˆβ„•,  ‖𝑇(𝑛)‖≀𝛽2𝑛‖𝑇1(𝑛)β€– the statement is clear.


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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.

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