Order Structure of the Figà-Talamanca-Herz Algebra

Abstract

We study the interplay between the order structure and the 𝑝-operator space structure of Figà-Talamanca-Herz algebra 𝐴_𝑝(𝐺) of a locally compact group 𝐺. 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 𝑝-completely positive linear map from Figà-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 𝑞∈(1,∞) is the conjugate of 𝑝, that is, 1/𝑝+1/𝑞=1. The Fourier algebra 𝐴(𝐺) consists of all coefficient functions of the left regular representation 𝜆 of 𝐺𝐴(𝐺)=𝑤=(𝜆𝜉,𝜂)∶𝜉,𝜂∈𝐿₂(𝐺).(1.1) This is a Banach algebra with the norm ‖𝑤‖_𝐴(𝐺)=inf{‖𝜉‖₂‖𝜂‖₂∶𝑤=(𝜆𝜉,𝜂)} [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 𝐵(𝐺) [1]. This is the linear span of the set 𝑃(𝐺) of all positive definite continuous functions on 𝐺.

In [2], Figà-Talamanca introduced a natural generalization of the Fourier algebra, for a compact abelian group 𝐺, by replacing 𝐿₂(𝐺) by 𝐿_𝑝(𝐺). In [3], Herz extended the notion to an arbitrary group, leading to the commutative Banach algebra 𝐴_𝑝(𝐺), called the Figà-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 𝐴_𝑝(𝐺) has a bounded approximate identity [4]. The 𝑝-analog, 𝐵_𝑝(𝐺) of the Fourier-Stieltjes algebra is defined as the multiplier algebra of 𝐴_𝑝(𝐺), by some authors in [5, 6]. In this paper, we follow [7] for the definition of 𝐵_𝑝(𝐺).

This paper investigates the order structure of 𝐴_𝑝(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 𝐴_𝑝(𝐺), then we show that for locally compact amenable groups 𝐺₁ and 𝐺₂, if 𝐴_𝑝(𝐺₁) and 𝐴_𝑝(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 Cannière 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 𝑛∈ℕ,  𝑝∈(1,∞), and let 𝐸 be a vector space. We denote the vector space of 𝑛×𝑚 matrices with entries from 𝐸 by 𝕄_𝑛,𝑚(𝐸). We put simply 𝕄_𝑛,𝑚∶=𝕄_𝑛,𝑚(ℂ). The space 𝕄_𝑛∶=𝕄_𝑛,𝑛 is equipped with the operator norm |⋅|_𝑛 from its canonical action on 𝑛-dimensional 𝐿_𝑝-space, ℓ^𝑛𝑝. Clearly, 𝕄_𝑛 acts on 𝕄_𝑛(𝐸) by matrix multiplication. For a square matrix 𝑎=(𝑎_𝑖𝑗)∈𝕄_𝑛, we have ‖𝑎‖_𝐵(ℓ^𝑛𝑝₎=sup⎧⎪⎨⎪⎩⎛⎜⎜⎝^𝑛_𝑖=1|||||_𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗|||||^𝑝⎞⎟⎟⎠^1/𝑝∶𝑥_𝑗∈ℂ,^𝑛_𝑗=1||𝑥_𝑗||^𝑝≤1⎫⎪⎬⎪⎭.(1.2)

Definition 1.1. Let 𝐸 be a vector space. A 𝑝-matricial norm on 𝐸 is a family (‖⋅‖_𝑛)^∞𝑛=1 such that for each 𝑛∈ℕ,‖⋅‖_𝑛 is a norm on 𝕄_𝑛(𝐸) satisfying ‖𝜆⋅𝑥⋅𝜇‖≤||𝜆||‖𝑥‖_𝑛||𝜇||,‖𝑥⊕𝑦‖_𝑛+𝑚=max‖𝑥‖_𝑛,‖𝑦‖_𝑚,(1.3) for each 𝜆∈𝕄_𝑚,𝑛,𝜇∈𝕄_𝑛,𝑚,𝑥∈𝕄_𝑛(𝐸), and 𝑦∈𝕄_𝑚(𝐸). Here, 𝜆⋅𝑥⋅𝜇 is the obvious matrix product, and |𝜆| and |𝜇| are the norms of 𝜆 and 𝜇 as the members of ℬ(ℓ^𝑛𝑝,ℓ^𝑚𝑝) and ℬ(ℓ^𝑚𝑝,ℓ^𝑛𝑝), respectively.
The vector space 𝐸 equipped with a 𝑝-matricial norm (‖⋅‖_𝑛)^∞𝑛=1 is called a 𝑝-matricial normed space. If moreover, each (𝕄_𝑛(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 𝑇∈ℬ(𝐸,𝐹), then for each 𝑛∈ℕ, 𝑇^(𝑛)∶𝕄_𝑛(𝐸)⟶𝕄_𝑛(𝐹),𝑇^(𝑛)𝑥_𝑖𝑗=𝑇𝑥_𝑖𝑗(1.4) is the 𝑛th amplification of 𝑇. The map 𝑇 is called 𝑝-completely bounded if ‖𝑇‖_𝑝cb∶=sup‖‖𝑇^(𝑛)‖‖<∞.(1.5) If ‖𝑇‖_𝑝cb≤1, we say that 𝑇 is a 𝑝-complete contraction, and if 𝑇^(𝑛) is an isometry, for each 𝑛∈ℕ, we call 𝑇 a 𝑝-complete isometry.

By [13, Section 4], the collection 𝒞ℬ_𝑝(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 𝕄_𝑛𝒞ℬ_𝑝(𝐸,𝐹)=𝒞ℬ_𝑝𝐸,𝕄_𝑛(𝐹)(𝑛∈ℕ).(1.6)

Figà-Talamanca-Herz algebras are our main examples of 𝑝-operator spaces, studied in [13, 14]. For any function 𝑓∶𝐺→ℂ we define 𝑓∶𝐺→ℂ by 𝑓(𝑥)=𝑓(𝑥⁻¹),  𝑥∈𝐺. The Figà-Talamanca-Herz algebra 𝐴_𝑝(𝐺) consists of those functions 𝑓∶𝐺→ℂ for which there are sequences (𝜉_𝑛)^∞𝑛=1 and (𝜂_𝑛)^∞𝑛=1 in 𝐿_𝑞(𝐺) and 𝐿_𝑝(𝐺), respectively, such that 𝑓=∑^∞𝑛=1𝜉_𝑛∗𝜂̃_𝑛 and ^∞_𝑛=1‖‖𝜉_𝑛‖‖_𝑞‖‖𝜂_𝑛‖‖_𝑝<∞.(1.7)

The norm ‖𝑓‖_{𝐴𝑝(𝐺)} of 𝑓∈𝐴_𝑝(𝐺) is defined as the infimum of the above sums over all possible representations of 𝑓. Then 𝐴_𝑝(𝐺) is a Banach space which is embedded contractively in 𝐶₀(𝐺). It was shown by Herz that 𝐴_𝑝(𝐺) is a Banach algebra under pointwise multiplication. When 𝑝=2, we get the Fourier algebra 𝐴(𝐺).

Let 𝜆_𝑝∶𝐺→ℬ(𝐿_𝑝(𝐺)) be the left regular representation of 𝐺 on 𝐿_𝑝(𝐺), defined by 𝜆_𝑝(𝑠)(𝑓)(𝑡)=𝑓(𝑠⁻¹𝑡). Then 𝜆_𝑝 can be lifted to a representation of 𝐿₁(𝐺) on 𝐿_𝑝(𝐺). The algebra of pseudomeasures PM_𝑝(𝐺) is defined as the 𝑤^∗-closure of 𝜆_𝑝(𝐿₁(𝐺)) in ℬ(𝐿_𝑝(𝐺)). There is a canonical duality PM_𝑝(𝐺)≅𝐴_𝑝(𝐺)^∗ via⟨𝜉∗𝜂̃,𝑇⟩∶=⟨𝜉,𝑇(𝜂)⟩𝜉∈𝐿_𝑝(𝐺),𝜂∈𝐿_𝑞(𝐺),𝑇∈PM_𝑝(𝐺).(1.8) In particular, PM₂(𝐺) is the group von Neumann algebra VN(𝐺). If the map Λ_𝑝 from the projective tensor product 𝐿_𝑞(𝐺)⊗^𝛾𝐿_𝑝(𝐺) to 𝐶₀(𝐺) is defined byΛ_𝑝(𝑔⊗𝑓)(𝑠)=𝑔,𝜆_𝑝(𝑠)𝑓,(1.9) for 𝑔∈𝐿_𝑞(𝐺),𝑓∈𝐿_𝑝(𝐺), and 𝑠∈𝐺, then 𝐴_𝑝(𝐺) is isometrically isomorphic to 𝐿_𝑞(𝐺)⊗^𝛾𝐿_𝑝(𝐺)/kerΛ_𝑝 and 𝐴_𝑝(𝐺)^∗=𝑇∈ℬ𝐿_𝑝(𝐺)∶𝑇|_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 𝐸_ℂ of 𝐸 is the additive group 𝐸×𝐸 with scalar multiplication defined by (𝛼,𝛽)(𝑥,𝑦)∶=(𝛼𝑥−𝛽𝑦,𝛽𝑥+𝛼𝑦), for (𝛼,𝛽)=𝛼+𝑖𝛽∈ℂ. Each 𝑧∈𝐸×𝐸 is uniquely represented as 𝑧=𝑥+𝑖𝑦, where 𝑥,𝑦∈𝐸. Thus, 𝐸_ℂ can be written as 𝐸+𝑖𝐸.

For real vector spaces 𝐸 and 𝐹 with complexifications 𝐸_ℂ and 𝐹_ℂ, every ℝ-linear map 𝑇∶𝐸→𝐹 has a unique ℂ-linear extension 𝑇 given by 𝑇(𝑧)∶=𝑇𝑥+𝑖𝑇𝑦, for 𝑧=𝑥+𝑖𝑦∈𝐸_ℂ, where 𝑥,𝑦∈𝐸. 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)𝑥≤𝑦 implies 𝑥+𝑧≤𝑦+𝑧, for all 𝑥,𝑦,𝑧∈𝐸(ii)𝑥≤𝑦 implies 𝛼𝑥≤𝛼𝑦, for all 𝑥,𝑦∈𝐸 and 𝛼∈ℝ₊.

The subset 𝐸₊∶={𝑥∈𝐸∶0≤𝑥} is called the positive cone of the real ordered space 𝐸. In general, a cone is a subset 𝑃 of 𝐸 such that 𝑥+𝑦∈𝑃 and 𝛼𝑥∈𝑃, for 𝑥,𝑦∈𝑃 and 𝛼∈ℝ₊. Then 𝑃 defines an order structure on 𝐸 by 𝑥≤𝑦 if and only if 𝑦−𝑥∈𝑃. A cone 𝑃 is called proper if 𝑃∩(−𝑃)={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 𝐸^∗. The positive cone of 𝐸^∗ is defined as 𝐸^∗+=𝑓∈𝐸^∗∶𝑓(𝑝)≥0𝑝∈𝐸₊.(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 𝐵 which is also a real Banach space, such that:(i)the inclusion map 𝑖∶𝐵→𝐴 is an isometry,(ii)each element 𝑎∈𝐴 can be written as 𝑎=𝑎₁−𝑎₂+𝑖(𝑎₃−𝑎₄), where 𝑎₁,…,𝑎₄ are positive in 𝐴 and ‖𝑎_𝑗‖≤‖𝑎‖, for 𝑗=1,…,4.

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

Given ordered spaces 𝐸_ℂ and 𝐹_ℂ, a ℂ-linear map 𝑇∶𝐸_ℂ→𝐹_ℂ is positive if 𝑇(𝐸)⊆𝐹 and the restriction 𝑇|_𝐸∶𝐸→𝐹 is positive.

1.3. Order Structure of 𝑝-Operator Spaces

Each operator space 𝐸 can be embedded in ℬ(ℋ), for some Hilbert space ℋ, by Ruan's Theorem [10, Theorem 2.3.5], and the order structure of 𝐸 is induced by ℬ(ℋ) [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(ℰ) for some QS𝐿_𝑝 space ℰ, which again induces an order structure on 𝐸. The main challenge is of course that the powerful methods from 𝐶^∗-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 ℰ=𝐿_𝑝(𝑋), where 𝑋 is a measure space and 1<𝑝<∞. We say that 𝑇∈ℬ(𝐿_𝑝(𝑋)) is positive if ⟨𝑇𝑓,𝑓⟩≥0 for each 𝑓∈𝐿_𝑝(𝑋)∩𝐿_𝑞(𝑋), where the pairing is the canonical dual action of 𝐿_𝑞(𝑋) on 𝐿_𝑝(𝑋). For 𝑝=2, this order is the natural order on the 𝐶^∗-algebra ℬ(𝐿₂(𝑋)). We also put an order on subspaces of 𝕄_𝑛(ℬ(𝐿_𝑝(𝑋))) for 𝑛≥1. We say 𝑇∈𝕄_𝑛(ℬ(𝐿_𝑝(𝑋)))≅ℬ(𝐿^𝑛𝑝(𝑋)) is positive if ∑^{𝑛𝑖,𝑗=1}⟨𝑇𝑓_𝑖,𝑓_𝑗⟩≥0, for each 𝑓₁,…,𝑓_𝑛∈𝐿_𝑝(𝑋)∩𝐿_𝑞(𝑋). It is easy to see that these orders define proper cones on ℬ(𝐿_𝑝(𝑋)) and 𝕄_𝑛(ℬ(𝐿_𝑝(𝑋))), respectively. Also for 𝑇=[𝑇_𝑖𝑗]∈𝕄_𝑛(ℬ(𝐿_𝑝(𝑋))_∗), we say 𝑇 is positive, if for each 𝑚∈ℕ and 𝜙=[𝜙_𝑖𝑗]∈𝕄_𝑚(ℬ(𝐿_𝑝(𝑋)))₊, the natural matrix action ⟨𝜙,𝑇⟩ is positive.

2. Order Structure of Some 𝑝-Operator Spaces

2.1. The 𝑝-Fourier Stieltjes Algebra

The Fourier-Stieltjes algebra 𝐵(𝐺) was defined by Eymard as the algebra of coefficient functions 𝑥↦⟨𝜋(𝑥)𝜉,𝜂⟩ of unitary representations 𝜋 of 𝐺 on a Hilbert space ℋ, where 𝜉,𝜂∈ℋ [1]. In this section, we consider the 𝑝-analog 𝐵_𝑝(𝐺) of the Fourier-Stieltjes algebra, introduced by Runde [7] and study its order structure given by the 𝑝-analog 𝑃_𝑝(𝐺) of positive definite continuous functions.

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

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

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

Definition 2.3. Let 𝐺 be a locally compact group and let (𝜋,ℰ) be a representation of 𝐺. A coefficient function of (𝜋,ℰ) is a function 𝑓∶𝐺→ℂ of the form 𝑓(𝑥)=⟨𝜋(𝑥)𝜉,𝜙⟩(𝑥∈𝐺),(2.1) where 𝜉∈ℰ and 𝜙∈ℰ^∗. Define 𝐵_𝑝(𝐺)∶=𝑓∶𝐺⟶ℂ∶𝑓isacoefficientfunctionofsome(𝜋,𝐸)∈Rep_𝑝(𝐺).(2.2) For 𝑓∈𝐵_𝑝(𝐺), put ‖𝑓‖∶=inf{‖𝜉‖‖𝜙‖∶𝑓(⋅)=⟨𝜋(⋅)𝜉,𝜙⟩}.

Using a suitable definition of tensor product of QS𝐿_𝑝-spaces, it is shown in [7] that 𝐵_𝑝(𝐺) is a commutative unital Banach algebra with the pointwise multiplication which contains 𝐴_𝑝(𝐺) contractively as a closed ideal. Also we know that 𝐴_𝑝(𝐺) can be embedded in 𝐵_𝑝(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 (ℰ₀,ℰ₁) of Banach spaces such that both ℰ₀ and ℰ₁ are embedded continuously in some (Hausdorff) topological vector space. In this case, the intersection ℰ₀∩ℰ₁ is again a Banach space under the norm ‖⋅‖_(ℰ0,ℰ1)=max{‖⋅‖_ℰ0,‖⋅‖_ℰ1}. For example, the pairs (𝐴_𝑝(𝐺),𝐴_𝑞(𝐺)) and (𝐿_𝑝(𝐺),𝐿_𝑞(𝐺)) are compatible couples.

Definition 2.4. Let (𝜋,ℰ) be a representation of 𝐺 such that (ℰ,ℰ^∗) is a compatible couple. Then a 𝜋-positive definite function on 𝐺 is a function which has a representation as 𝑓(𝑥)=⟨𝜋(𝑥)𝜉,𝜉⟩(𝑥∈𝐺), where 𝜉∈ℰ∩ℰ^∗. We call each element in the closure of the set of all 𝜋-positive definite functions on 𝐺 in 𝐵_𝑝(𝐺), where 𝜋 is a representation of 𝐺 on an 𝐿_𝑝-space, a 𝑝-positive definite function on 𝐺, and the set of all 𝑝-positive definite functions on 𝐺 will be denoted by 𝑃_𝑝(𝐺).

It follows from [7, Lemma 4.3] and the definition of 𝑃_𝑝(𝐺) that for each 𝑓∈𝑃_𝑝(𝐺), associated to a representation (𝜋,ℰ), there exist a sequence (𝜋_𝑛,ℰ_𝑛)^∞𝑛=1 of cyclic representations of 𝐺 on closed subspaces ℰ_𝑛 of ℰ∩ℰ^∗, and {𝜉_𝑛} in ℰ_𝑛, such that 𝑓(𝑥)=^∞_𝑛=1⟨𝜋_𝑛(𝑥)𝜉_𝑛,𝜉_𝑛⟩(𝑥∈𝐺).(2.3)

The closed subspace 𝐵^𝑝𝑝(𝐺) of 𝐵_𝑝(𝐺) is the closure of the set of all functions 𝑓∈𝐵_𝑝(𝐺) of the form 𝑓(𝑥)=⟨𝜋(𝑥)𝜉,𝜂⟩,𝑥∈𝐺, for some representation (𝜋,ℰ), where ℰ is an 𝐿_𝑝-space, 𝜉∈ℰ, and 𝜂∈ℰ^∗.

Proposition 2.5. The linear span of 𝑃_𝑝(𝐺) is dense in 𝐵^𝑝𝑝(𝐺).

Proof. Let 𝑢∈𝐵^𝑝𝑝(𝐺) have a representation as 𝑢(𝑥)=⟨𝜋(𝑥)𝜉,𝜂⟩,𝑥∈𝐺, where 𝜋 is a representation on some 𝐿_𝑝-space ℰ,𝜉∈ℰ and 𝜂∈ℰ^∗. It is clear that ℰ∩ℰ^∗ is dense in both ℰ and ℰ^∗. Hence, there exist sequences {𝜉_𝑛} and {𝜂_𝑛} in ℰ∩ℰ^∗ converging to 𝜉 and 𝜂 in ℰ and ℰ^∗, respectively. Put 𝑢_𝑛(𝑥)∶=⟨𝜋(𝑥)𝜉_𝑛,𝜂_𝑛⟩, then 𝑢−𝑢_𝑛(𝑥)=𝜋(𝑥)𝜉,𝜂−𝜂_𝑛+𝜋(𝑥)𝜉−𝜉_𝑛,𝜂_𝑛.(2.4) By the definition of the norm in 𝐵_𝑝(𝐺), we have ‖‖𝑢−𝑢_𝑛‖‖≤‖𝜉‖_𝑝‖‖𝜂−𝜂_𝑛‖‖_𝑞+‖‖𝜉−𝜉_𝑛‖‖_𝑝‖‖𝜂_𝑛‖‖_𝑞,(2.5) which tends to zero, as 𝑛→∞.
Now, it is enough to consider the following decomposition for 𝑢_𝑛, ⟨𝜋(𝑥)𝜉_𝑛,𝜂_𝑛⟩=14³_𝑗=0𝜋(𝑥)𝜉_𝑛+𝑖^𝑗𝜂_𝑛,𝜉_𝑛+𝑖^𝑗𝜂_𝑛,(2.6) where 𝑖=√−1, and note that each 𝜉_𝑛+𝑖^𝑗𝜂_𝑛, for 𝑗=0,1,2,3, belongs to ℰ∩ℰ^∗. So 𝑢 belongs to the closed linear span of 𝑃_𝑝(𝐺).

Each representation (𝜋,ℰ) of 𝐺 induces a representation of the group algebra 𝐿₁(𝐺) on ℰ via 𝜋(𝑓)∶=_𝐺𝑓(𝑥)𝜋(𝑥)𝑓∈𝐿₁(𝐺).(2.7) Let (𝜋_𝑝,ℰ) be the 𝑝-universal representation of 𝐺. The Banach space UPF_𝑝(𝐺) of all pseudofunctions on 𝐺 is the closure of 𝜋_𝑝(𝐿₁(𝐺)) in 𝐵(ℰ). By [7, Theorem 6.6], we know that 𝐵_𝑝(𝐺) is the dual space of UPF_𝑝(𝐺) via the pairing 𝜋_𝑝(𝑓),𝑔=_𝐺𝑓(𝑥)𝑔(𝑥)𝑓∈𝐿₁(𝐺),𝑔∈𝐵_𝑝(𝐺).(2.8)

We say that 𝑓∈UPF_𝑝(𝐺) is positive if 𝜙_𝑔(𝑓)≥0, for all 𝑔∈𝐵_𝑝(𝐺)₊=𝑃_𝑝(𝐺), where 𝜙_𝑔 is the corresponding linear functional of 𝑔. Let 𝐶^∗(𝐺) be the full group 𝐶^∗-algebra of 𝐺, which is the enveloping 𝐶^∗-algebra of 𝐿₁(𝐺) [1].

Proposition 2.6. There exists a positive contraction from UPF_𝑝(𝐺) to 𝐶^∗(𝐺).

Proof. Since each Hilbert space is a QS𝐿_𝑝-space [3] and for each 𝑓∈𝐿₁(𝐺),‖𝑓‖_Rep2≤‖𝑓‖_Rep𝑝, where ‖𝑓‖_Rep𝑝=sup_{𝜋∈Rep𝑝(𝐺),‖𝜋‖≤1}‖𝜋(𝑓)‖, it follows that the identity map 𝑖∶𝐿₁(𝐺),‖⋅‖_Rep𝑝⟶𝐿¹(𝐺),‖⋅‖_Rep2(2.9) is continuous. Thus, it induces a continuous map 𝑖_𝑐∶UPF_𝑝(𝐺)⟶𝐶^∗(𝐺).(2.10) Consider the conjugate map 𝑖_𝑐^∗∶𝐵(𝐺)⟶𝐵_𝑝(𝐺).(2.11) Since for each 𝑢∈𝐵(𝐺) and 𝑓∈𝐿¹(𝐺), 𝑖_𝑐^∗(𝑢),𝑓=⟨𝑢∘𝑖,𝑓⟩=⟨𝑢,𝑓⟩,(2.12)(𝑖_𝑐)^∗ is the inclusion map. By the definition of positivity in 𝐵_𝑝(𝐺) and 𝐵(𝐺),(𝑖_𝑐)^∗ is a positive map. Therefore (𝑖_𝑐)^∗∗ is positive, and so is 𝑖_𝑐=(𝑖_𝑐)^∗∗∣_{UPF𝑝(𝐺)}.

2.2. Figà-Talamanca-Herz Algebra

In this section, we study the order structure of the Figà-Talamanca-Herz algebra 𝐴_𝑝(𝐺). Since 𝐴_𝑝(𝐺) is the set of coefficient functions of the left regular representation of 𝐺 on 𝐿_𝑝(𝐺), we have 𝐴_𝑝(𝐺)⊆𝐵^𝑝𝑝(𝐺). We define the positive cone of 𝐴_𝑝(𝐺) as the closure in 𝐴_𝑝(𝐺), of the set of all function of the form 𝑓=∑^𝑛𝑖=1𝜉_𝑖∗𝜉̃_𝑖, for a sequence (𝜉_𝑖) in 𝐿_𝑝(𝐺)∩𝐿_𝑞(𝐺), and denote it by 𝐴_𝑝(𝐺)₊. It is clear that 𝐴_𝑝(𝐺)₊ is contained in 𝐵_𝑝(𝐺)₊. Since 𝐶_𝑐(𝐺)∩𝑃(𝐺)=𝐴(𝐺)∩𝑃(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, 𝑇∈PM_𝑝(𝐺) is positive as an element of 𝐵(𝐿_𝑝(𝐺)) if and only if it is positive as an element of 𝐴_𝑝(𝐺)^∗+, where 𝐴_𝑝(𝐺)^∗+ is the dual cone induced by the positive cone 𝐴_𝑝(𝐺)₊. Also, since 𝐴_𝑝(𝐺)₊ is closed, 𝑢∈𝐴_𝑝(𝐺)₊ if and only if for each 𝑇∈PM_𝑝(𝐺)₊,  𝑇(𝑢)≥0 [19].

Theorem 2.7. Let 𝐺 be a locally compact amenable group. Then 𝐴_𝑝(𝐺)₊=𝑃(𝐺)∩𝐴(𝐺)={𝑢∈𝐴(𝐺)∶‖𝑢‖_{𝐴𝑝(𝐺)}=𝑢(𝑒)}, where 𝑒 is the identity of 𝐺. In particular, 𝐴_𝑝(𝐺)₊ is a proper cone.

Proof. Since 𝐿_𝑝(𝐺)∩𝐿_𝑞(𝐺)⊆𝐿₂(𝐺), it follows that 𝐴_𝑝(𝐺)₊⊆𝑃(𝐺)∩𝐴(𝐺)^{‖⋅‖𝐴𝑝(𝐺)}. But when 𝐺 is amenable, the identity map 𝑖∶𝐴(𝐺)→𝐴_𝑝(𝐺) is an embedding with ‖⋅‖_{𝐴𝑝(𝐺)}≤‖⋅‖_𝐴(𝐺) [3, Theorem C]. For each 𝑢∈𝑃(𝐺)∩𝐴(𝐺), we have ‖𝑢‖_𝐴(𝐺)=𝑢(𝑒)≤‖𝑢‖_∞≤‖𝑢‖_{𝐴𝑝(𝐺)}≤‖𝑢‖_𝐴(𝐺),(2.13) where ‖⋅‖_∞ is the supremum norm on 𝐴(𝐺), that is, ‖𝑢‖_𝐴(𝐺)=‖𝑢‖_{𝐴𝑝(𝐺)}. In particular, 𝑃(𝐺)∩𝐴(𝐺) is closed in 𝐴_𝑝(𝐺) and consequently 𝐴_𝑝(𝐺)₊⊆𝑃(𝐺)∩𝐴(𝐺). Now, let 𝑣 be an arbitrary element of 𝐴(𝐺)∩𝑃(𝐺). Then there exists a sequence in 𝑃(𝐺)∩𝐶_𝑐(𝐺) of the form {𝑓_𝑖∗𝑓_𝑖},𝑓_𝑖∈𝐶_𝑐(𝐺),𝑖∈ℕ converging to 𝑣 in 𝐴(𝐺). Since for each 𝑢∈𝐴(𝐺),‖𝑢‖_{𝐴𝑝(𝐺)}≤‖𝑢‖_𝐴(𝐺), it follows that {𝑓_𝑖∗𝑓_𝑖} converges to 𝑣 in 𝐴_𝑝(𝐺). Clearly, {𝑓_𝑖∗𝑓_𝑖} is contained in 𝐴_𝑝(𝐺)₊ and since 𝐴_𝑝(𝐺)₊ is closed, 𝑣∈𝐴_𝑝(𝐺)₊.
The second equality follows immediately from the above-mentioned fact that for each 𝑢∈𝑃(𝐺)∩𝐴(𝐺),  ‖𝑢‖_𝐴(𝐺)=‖𝑢‖_{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 Figà-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 𝑥,𝑦∈𝐹, 0≤𝑥≤𝑦 implies ‖𝑥‖≤‖𝑦‖. Then every positive linear map from 𝐸 into 𝐹 is continuous.

Proof. Assume towards a contradiction that 𝑇∶𝐸→𝐹 is a positive linear map that is not continuous. Then 𝑇 is unbounded on the unit ball 𝑈 of 𝐸, and hence, on 𝑈₊∶=𝑈∩𝐸₊ since 𝑈⊆𝑈₊−𝑈₊+𝑖(𝑈₊−𝑈₊). This implies that there exists a sequence {𝑥_𝑛}^∞𝑛=1 in 𝑈₊ such that ‖𝑇𝑥_𝑛‖≥𝑛³, for each 𝑛∈ℕ. Since 𝐸₊ is closed, 𝑧∶=∑𝑥_𝑛/𝑛² is in 𝐸₊. Hence, 𝑇𝑧≥𝑇𝑥_𝑛/𝑛²>0, for each 𝑛∈ℕ. Therefore, ‖𝑇𝑧‖≥𝑛, for each 𝑛∈ℕ, 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 𝑥∈𝐸, there is a sequence {𝑥_𝑛} converging to 𝑥, such that 𝑥_𝑛=𝑥^1𝑛−𝑥^2𝑛+𝑖(𝑥^3𝑛−𝑥^4𝑛) with 𝑥^𝑖𝑛∈𝐸₊ and ‖𝑥^𝑖𝑛‖≤‖𝑥_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 𝑇⁻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 𝑇∶𝐴_𝑝(𝐺₁)→𝐴_𝑝(𝐺₂) be an order and algebra isomorphism. Then (i)𝑇^∗(𝜆_𝑒2)=𝜆_𝑒1, where 𝜆_𝑒𝑖 is the evaluation homomorphism at 𝑒_𝑖,  𝑖=1,2,(ii)there exists an isomorphism or anti-isomorphism 𝜑 from 𝐺₂ onto 𝐺₁ such that 𝑇(𝑓)=𝑓∘𝜑 for all 𝑓∈𝐴_𝑝(𝐺₁).

Proof. (i) Consider the adjoint map 𝑇^∗∶𝐴_𝑝(𝐺₂)^∗→𝐴_𝑝(𝐺₁)^∗, which is clearly an order isomorphism. Since 𝑇 is an algebra isomorphism, 𝑇^∗(𝜆_𝑒2) is a multiplicative linear functional on 𝐴_𝑝(𝐺₁), and since for each locally compact group 𝐺, any (non zero) multiplicative linear functional on 𝐴_𝑝(𝐺)is an evaluation homomorphism at some point of 𝐺, it follows that, 𝑇^∗(𝜆_𝑒2)=𝜆_𝑥, for some 𝑥∈𝐺₁. We note that 𝜆_𝑥 is positive because 𝜆_𝑒2 is a positive element of 𝐴_𝑝(𝐺)^∗, hence, 𝑥=𝑒₁ by [20, Proposition 4.3].
(ii) We first note that 𝐴_𝑝(𝐺_𝑖)₊=𝑃(𝐺_𝑖)∩𝐴(𝐺_𝑖),  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(𝐺_𝑖)∩𝐴(𝐺_𝑖) spans 𝐴(𝐺_𝑖), 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 𝜓∶𝐺₂→𝐺₁ such that 𝑇₁(𝑓)=𝑓∘𝜓 for all 𝑓∈𝐴(𝐺₁). Now, the result follows from the density of 𝐴(𝐺₁) in 𝐴_𝑝(𝐺₁) and continuity of 𝑇.

Proposition 3.3. Let 𝐺 and 𝐻 be locally compact groups, and let 𝑇∶𝐴(𝐺)→VN(𝐻) be a completely positive linear map. Then there are infinitely many 𝑛∈ℕ 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 𝑛₀∈ℕ such that for each 𝑛≥𝑛₀,‖𝑇^(𝑛)‖>𝑛². Fix 𝑛≥𝑛₀. Then there is some 𝑥∈(𝕄_𝑛(𝐴(𝐺)))₁, the unit ball of 𝕄_𝑛(𝐴(𝐺)), such that ‖𝑇^(𝑛)(𝑥)‖>𝑛². Since 𝕄_𝑛(𝐴(𝐺))=𝒞ℬ^𝜎(VN(𝐺),𝕄_𝑛), where 𝒞ℬ^𝜎(VN(𝐺),𝕄_𝑛) is the algebra of all 𝑤^∗-continuous, completely bounded linear maps from VN(𝐺) to 𝕄_𝑛, and since 𝕄_𝑛 is an injective von Neumann algebra, by [7, Corollary 2.6] or [4, Theorem 2.1], it follows that 𝑥 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 𝑥. Considering 𝑦∶=(𝑥+𝑥^𝑡)/2 instead of 𝑥, without loss of generality, we may assume that 𝑥 is positive, ‖𝑥‖≤1,𝑥=𝑥^𝑡, and ‖𝑇^(𝑛)(𝑥)‖≥𝑛²/8. Let 𝑇^𝑛∶=⨁^𝑛𝑖=1𝑇∶⨁^𝑛𝑖=1𝐴(G)→⨁^𝑛𝑖=1VN(𝐻) be defined by (𝑧_𝑖)↦(𝑇𝑧_𝑖). Then clearly, ‖𝑇^𝑛‖=‖⨁^𝑛𝑖=1𝑇‖=sup_{‖(𝑧𝑖)‖≤1}(∑^𝑛𝑖=1‖𝑇𝑧_𝑖‖²)^1/2≤𝑛‖𝑇‖.
Now, 𝑇^(𝑛)(𝑥)=[𝑇(𝑥_𝑖𝑗)] is positive in 𝕄_𝑛(VN(𝐻)), hence, [𝑇(𝑥_𝑖𝑗)^∗]=[𝑇(𝑥_𝑖𝑗)], and for each 𝑖,𝑗,  𝑇(𝑥_𝑖𝑗) is self-adjoint. Hence, 𝑇𝑥_𝑖𝑗=‖𝑇‖(𝑇𝑥^1𝑖𝑗+𝑇𝑥^2𝑖𝑗)/2, where for each 𝑖,𝑗∈{1,…,𝑛} and 𝑘=1,2,  𝑇𝑥^𝑘𝑖𝑗 is a unitary operator in ℬ(𝐿₂(𝐺)). For each unit vector (𝑓₁,𝑓₂,…,𝑓_𝑛) in ⨁^𝑛𝑖=1𝐿₂(𝐻), set 𝑇𝑥^𝑘𝑖𝑗(𝑓_𝑖)=𝑒^𝑘𝑖𝑗,  𝑘=1,2,1≤𝑖,𝑗≤𝑛. Then ⎛⎜⎜⎝_𝑗‖‖‖‖_𝑖𝑇𝑥^1𝑖𝑗+𝑇𝑥^2𝑖𝑗2𝑓^𝑖‖‖‖‖²⎞⎟⎟⎠^1/2=⎛⎜⎜⎝_𝑗‖‖‖‖_𝑖𝑒^1𝑖𝑗+𝑒^2𝑖𝑗2‖‖‖‖²⎞⎟⎟⎠^1/2=12_𝑗_𝑖𝑒^1𝑖𝑗+𝑒^2𝑖𝑗,_𝑖𝑒^1𝑖𝑗+𝑒^2𝑖𝑗^1/2≤12_𝑖,𝑗𝑒^1𝑖,𝑗+𝑒^2𝑖,𝑗,𝑒^1𝑖𝑗+𝑒^2𝑖𝑗+4𝑛²^1/2=12_𝑖,𝑗‖‖𝑒^1𝑖,𝑗+𝑒^2𝑖,𝑗‖‖²+4𝑛²^1/2≤‖‖𝑇^𝑛2𝑥_𝑖𝑗‖‖+𝑛.(3.1) Hence, we have ‖‖𝑇^(𝑛)𝑥_𝑖𝑗‖‖≤‖𝑇‖‖‖𝑇^𝑛2𝑥_𝑖𝑗‖‖+𝑛.(3.2) On the other hand, ‖𝑥‖_{𝕄𝑛(𝐴(𝐺))}=sup_{𝑆∈𝕄𝑛(VN(𝐺))1}‖‖𝑆_𝑘𝑙𝑥_𝑖𝑗‖‖=sup_{𝑆∈𝕄𝑛(VN(𝐺))1}_{𝑖,𝑗,𝑘,𝑙}||𝑆_𝑘𝑙𝑥_𝑖𝑗||²^1/2.(3.3) By the Hahn-Banach theorem, for each 𝑥_𝑖𝑗, there is some 𝑆_𝑖𝑗∈VN(𝐺)₁ such that 𝑆_𝑖𝑗(𝑥)=‖𝑥_𝑖𝑗‖. Now, put 𝑆=[𝑆_𝑖𝑗], then ‖𝑆‖_{𝕄𝑛(VN(𝐺))}≤𝑛. Hence, 𝑆/𝑛∈𝕄_𝑛(VN(𝐺))₁ and (𝑆_𝑖𝑗/𝑛)(𝑥_𝑖𝑗)=‖𝑥_𝑖𝑗‖/𝑛. Therefore, ‖𝑥‖_{𝕄𝑛(𝐴(𝐺))}=sup_{𝑅∈𝕄𝑛(VN(𝐺))1}‖‖𝑅_𝑘𝑙𝑥_𝑖𝑗‖‖≥‖‖‖𝑥^𝑖𝑗𝑛‖‖‖²^1/2=1𝑛‖‖𝑥^𝑖𝑗‖‖²^1/2.(3.4) This means that the norm of 𝑥 as an element of ⨁^𝑛2_𝑖=1𝐴(𝐺) is less than or equal to 𝑛. Therefore, by inequality (3.2), (8‖𝑛𝑇²‖)−𝑛≤‖‖𝑇^𝑛2𝑥_𝑖𝑗‖‖≤‖𝑇‖‖‖𝑥_𝑖𝑗‖‖_⨁^𝑛2𝑖=1_𝐴(𝐺)≤𝑛‖𝑇‖.(3.5) Hence, 𝑇 is unbounded, which is a contradiction.

Let 1<𝑝^<∞, and let 𝑌 be an arbitrary measure space. For subspaces ℳ⊆ℬ(𝐿_𝑝(𝑋)) and 𝒩⊆ℬ(𝐿_𝑝′(𝑌)), we say that a linear map 𝑇∶ℳ→𝒩 is (𝑝,𝑝^)-completely positive, if for all 𝑛∈ℕ, the 𝑛th amplification 𝑇^(𝑛)∶𝕄_𝑛(ℳ)→𝕄_𝑛(𝒩),  𝑇^(𝑛)[𝑥_𝑖𝑗]=[𝑇(𝑥_𝑖𝑗)] of 𝑇 is a positive map. For simplicity in the case where 𝑝=𝑝^, we call such maps 𝑝-completely positive.

Recall that for a locally compact group 𝐺, 𝐴_𝑝(𝐺) has the natural 𝑝-operator space structure as the predual of PM_𝑝(𝐺) [13]. Now, we define an order structure on 𝕄_𝑛(𝐴_𝑝(𝐺)) as follows. Let 𝑚,𝑛∈ℕ, we say that 𝑇=[𝑇_𝑖𝑗]∈𝕄_𝑛(𝐴_𝑝(𝐺)) is positive if for every 𝑝-completely positive and 𝑝-completely bounded linear map 𝜙∶PM_𝑝(𝐺)→𝕄_𝑚, the natural action ⟨𝜙,𝑇⟩ is a positive scalar matrix.

For ℳ=𝐴_𝑝(𝐺)orPM_𝑝(𝐺) and 𝒩=𝐴_𝑝′(𝐺) or PM_𝑝′(𝐺), we say that a linear map 𝑇∶ℳ→𝒩 is (𝑝,𝑝^)-completely positive if for each positive integer 𝑛, the 𝑛th amplification 𝑇^𝑛∶𝕄_𝑛(ℳ)→𝕄_𝑛(𝒩) of 𝑇 is a positive map. If this is the case for 𝑝=𝑝′, we say that 𝑇 is 𝑝- completely positive.

Let 𝑝≥2. Then since ‖⋅‖₂ and ‖⋅‖_𝑝 are equivalent norms on ℂ^𝑛, for each 𝑛∈ℕ, there exists a positive scalar 𝛼_𝑛 such that 𝛼_𝑛‖⋅‖₂≤‖⋅‖_𝑝≤‖⋅‖₂ on ℂ^𝑛. For the case where 1<𝑝≤2, for each 𝑛∈ℕ, we choose 𝛽_𝑛>0 such that 𝛽_𝑛‖⋅‖≤‖⋅‖₂≤‖⋅‖_𝑝 on ℂ^𝑛.

Proposition 3.4. Let 𝐺 be an amenable locally compact group, let 𝑝∈(1,∞) and let 𝑇∶𝐴_𝑝(𝐺)→PM_p(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 𝑝≥2 and 𝑇 is a bounded linear map, then for each 𝑛∈ℕ, we have ‖𝑇_1(𝑛)‖≤‖𝑇^(𝑛)‖.(iii)If 1<𝑝≤2 and 𝑇 is bounded, then for each 𝑛∈ℕ, we have ‖𝑇^(𝑛)‖≤𝛽^2𝑛‖𝑇_1(𝑛)‖.(iv)If 1<𝑝≤2 and 𝑇 is a 𝑝-completely positive map, then there are infinitely many 𝑛∈ℕ such that ‖𝑇^(𝑛)‖≤𝛽^2𝑛𝑛².

Proof. (i) Let 𝑇 be 𝑝-completely positive. Let 𝑛∈ℕ and [𝑥_𝑖𝑗]∈𝕄_𝑛(𝐴(𝐺)). Since 𝕄_𝑛(𝐴(𝐺))₊=𝒞𝒫^𝜎(VN(𝐺),𝕄_𝑛), where 𝒞𝒫^𝜎(VN(𝐺),𝕄_𝑛) is the set of all 𝑤^∗-continuous completely positive linear maps from VN(𝐺) to 𝕄_𝑛, it follows that [𝑥_𝑖𝑗]∈𝕄_𝑛(𝐴(𝐺))₊ if and only if for each [𝑇_𝑖𝑗]∈𝕄_𝑛[VN(𝐺)]₊, we have [𝑇_𝑘𝑙(𝑥_𝑖𝑗)]∈𝕄_𝑛2+. Since 𝐺 is amenable, the embedding 𝑖∶𝐴(𝐺)→𝐴_𝑝(𝐺) is norm decreasing [3]. Therefore, each element [𝑆_𝑖𝑗] in 𝕄_𝑛[PM_𝑝(𝐺)] can be considered as an element of 𝕄_𝑛[VN(𝐺)]. Moreover, [𝑆_𝑖𝑗]∈𝕄_𝑛[PM_𝑝(𝐺)]₊ if and only if ⟨[𝑆_𝑖𝑗][𝑓],[𝑓]⟩≥0, for all [𝑓]=(𝑓₁,…,𝑓_𝑛) with 𝑓₁,…,𝑓_𝑛∈𝐶_𝑐(𝐺). It is clear that if [𝑆_𝑖𝑗]∈𝕄_𝑛[PM_𝑝(𝐺)]₊, then it also belongs to 𝕄_𝑛[VN(𝐺)]₊. Therefore, 𝑖∶𝐴(𝐺)→𝐴_𝑝(𝐺) is a (2,𝑝)-completely positive map. This implies that 𝑇₁∶𝐴(𝐺)→VN(𝐺) is a completely positive linear map.
(ii) As we noted before, the embedding 𝑖∶𝐴(𝐺)→𝐴_𝑝(𝐺) is a norm decreasing map with dense range. So, PM_𝑝(𝐺) can be considered as a subspace of VN(𝐺). We first show that for each 𝑛∈ℕ, 𝑖^(𝑛) is continuous and has dense range. Let [𝑎_𝑖𝑗]∈𝕄_𝑛(𝐴(𝐺)), then [𝑆(𝑎_𝑖𝑗)]∈𝐵(ℓ^𝑛𝑝), for each 𝑆∈PM_𝑝(𝐺) and‖‖𝑎_𝑖𝑗‖‖_{𝕄𝑛(𝐴𝑝(𝐺))}==ssuupp_{𝑆∈PM𝑝(𝐺)1𝑆∈PM𝑝(𝐺)1}‖‖𝑆𝑎_𝑖𝑗‖‖sup⎧⎪⎨⎪⎩⎛⎜⎜⎝^𝑛_𝑖=1|||||_𝑛_𝑗=1𝑆𝑎_𝑖𝑗𝑥_𝑗|||||^𝑝⎞⎟⎟⎠^1/𝑝,𝑥_𝑗∈ℂ^𝑛,^𝑛_𝑗=1≤1||𝑥_𝑗||^𝑝≤1⎫⎪⎬⎪⎭𝛼_𝑛sup_{𝑆∈PM𝑝(𝐺)1}sup⎧⎪⎨⎪⎩⎛⎜⎜⎝^𝑛_𝑖=1|||||_𝑛_𝑗=1𝑆𝑎_𝑖𝑗𝑥_𝑗|||||²⎞⎟⎟⎠^1/2,𝑥_𝑗∈ℂ^𝑛,^𝑛_𝑗=1=1||𝑥_𝑗||²≤1⎫⎪⎬⎪⎭𝛼_𝑛sup_{𝑆∈PM𝑝(𝐺)1}‖‖𝑆𝑎_𝑖𝑗‖‖_𝐵(ℓ2)≤1𝛼_𝑛‖‖𝑎_𝑖𝑗‖‖_{𝕄𝑛(𝐴(𝐺))},(3.6) which shows that 𝑖^(𝑛) is continuous. Consider now an element [𝑎_𝑖𝑗]∈𝕄_𝑛(𝐴_𝑝(𝐺)). Then for each 𝑖,𝑗, there exists a sequence (𝑎^𝑚𝑖𝑗)_𝑚 in 𝐴(𝐺) converging to 𝑎_𝑖𝑗 in 𝐴_𝑝(𝐺). Hence, ‖‖𝑎^𝑚𝑖𝑗−𝑎_𝑖𝑗‖‖_{𝕄𝑛(𝐴𝑝(𝐺))}=‖‖𝑎^𝑚𝑖𝑗−𝑎_𝑖𝑗‖‖_{𝕄𝑛(𝐴𝑝(𝐺))}=sup_{𝑆∈PM𝑝(𝐺)1}sup⎧⎪⎨⎪⎩⎛⎜⎜⎝^𝑛_𝑖=1|||||_𝑛_𝑖=1𝑆𝑎_𝑖𝑗−𝑆𝑎^𝑚𝑖𝑗𝑥_𝑗|||||^𝑝⎞⎟⎟⎠^1/𝑝,𝑥_𝑗∈ℂ^𝑛,^𝑛_𝑗=1||𝑥_𝑗||^𝑝≤1⎫⎪⎬⎪⎭,(3.7) which clearly converges to zero as 𝑚→∞. This implies that 𝑖^(𝑛) has dense range. Hence, for each 𝑛∈ℕ, since 𝛼_𝑛‖⋅‖_{𝑀𝑛(VN(𝐺))}≤‖⋅‖_{𝑀𝑛(PM𝑝(𝐺))} we have 𝛼_𝑛‖‖𝑇^(𝑛)‖‖=≥𝛼𝛼_𝑛𝑛⋅sup‖‖𝑇𝑥_𝑖𝑗‖‖_{𝑀𝑛(PM𝑝(𝐺))}∶𝑥_𝑖𝑗∈𝕄_𝑛𝐴_𝑝(𝐺),‖𝑥_𝑖𝑗‖_{𝕄𝑛(𝐴𝑝(𝐺))}≤1≥sup⋅sup𝛼_𝑛𝛼_𝑛‖‖𝑇₁𝑥_𝑖𝑗‖‖_{𝑀𝑛(VN(𝐺))}∶𝑥_𝑖𝑗∈𝕄_𝑛(𝐴(𝐺)),‖‖𝑥_𝑖𝑗‖‖_{𝕄𝑛(𝐴𝑝(𝐺))}≤1=𝛼_𝑛‖‖𝑇₁𝑥_𝑖𝑗‖‖_{𝑀𝑛(VN(𝐺))}∶𝑥_𝑖𝑗∈𝕄_𝑛(𝐴(𝐺)),‖‖𝑥_𝑖𝑗‖‖_{𝕄𝑛(𝐴(𝐺))}≤1‖‖𝑇_1(𝑛)‖‖,(3.8) that is, ‖𝑇_1(𝑛)‖≤‖𝑇^(𝑛)‖.(iii)Using the weak-star density of PM_𝑝(𝐺)₁ in VN(𝐺)₁, the same proof as in (ii) can be applied.(iv)Let 𝑇∶𝐴_𝑝(𝐺)→PM_𝑝(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 𝑛∈ℕ such that ‖𝑇_1(𝑛)‖≤𝑛². Now, since by part (iii), for each 𝑛∈ℕ,  ‖𝑇^(𝑛)‖≤𝛽^2𝑛‖𝑇_1(𝑛)‖ the statement is clear.