# Order Structure of the Figà-Talamanca-Herz Algebra

Authors: Marzieh Shams Yousefi; Massoud Amini; Fereshteh Sady
Journal: ISRN Geometry
 (2011)

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## Abstract

We study the interplay between the order structure and thep-operator space structure of Figà-Talamanca-Herz algebra Ap(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.


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## Body

## 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 G(1.1)A(G)={w=(λξ,η):ξ,η∈L2(G)}.
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(Ĝ), where Ĝ 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 Lp(G). In [3], Herz extended the notion to an arbitrary group, leading to the commutative Banach algebra Ap(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 Ap(G) has a bounded approximate identity [4]. The p-analog, Bp(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 Bp(G).This paper investigates the order structure ofAp(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 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 ofp-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 𝕄n,m(E). We put simply 𝕄n,m:=𝕄n,m(ℂ). The space 𝕄n:=𝕄n,n is equipped with the operator norm |·|n from its canonical action on n-dimensional Lp-space, ℓpn. Clearly, 𝕄n acts on 𝕄n(E) by matrix multiplication. For a square matrix a=(aij)∈𝕄n, we have (1.2)‖a‖B(lpn)=sup⁡{(∑i=1n|∑j=1naijxj|p)1/p:xj∈C,∑j=1n|xj|p≤1}.Definition 1.1.
LetE 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 𝕄n(E) satisfying
(1.3)‖λ⋅x⋅μ‖≤|λ|‖x‖n|μ|,‖x⊕y‖n+m=max⁡{‖x‖n,‖y‖m},
for each λ∈𝕄m,n,μ∈𝕄n,m,x∈𝕄n(E), and y∈𝕄m(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(ℓpn,ℓpm) and ℬ(ℓpm,ℓpn), respectively.
The vector spaceE equipped with a p-matricial norm (∥·∥n)n=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.
LetE and F be p-operator spaces, and let T∈ℬ(E,F), then for each n∈ℕ,
(1.4)T(n):Mn(E)⟶Mn(F),T(n)([xij])=[T(xij)]
is the nth amplification of T. The map T is called p-completely bounded if
(1.5)‖T‖pcb:=sup⁡‖T(n)‖<∞.
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 𝒞ℬp(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 (1.6)Mn(CBp(E,F))=CBp(E,Mn(F))(n∈N).Figà-Talamanca-Herz algebras are our main examples ofp-operator spaces, studied in [13, 14]. For any function f:G→ℂ we define f̃:G→ℂ by f̃(x)=f(x-1)¯,  x∈G. The Figà-Talamanca-Herz algebra Ap(G) consists of those functions f:G→ℂ for which there are sequences (ξn)n=1∞ and (ηn)n=1∞ in Lq(G) and Lp(G), respectively, such that f=∑n=1∞ξn*η̃n and (1.7)∑n=1∞‖ξn‖q‖ηn‖p<∞.The norm∥f∥Ap(G) of f∈Ap(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 C0(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).Letλp:G→ℬ(Lp(G)) be the left regular representation of G on Lp(G), defined by λp(s)(f)(t)=f(s-1t). Then λp can be lifted to a representation of L1(G) on Lp(G). The algebra of pseudomeasures PMp(G) is defined as the w*-closure of λp(L1(G)) in ℬ(Lp(G)). There is a canonical duality PMp(G)≅Ap(G)*  via(1.8)〈ξ*η̃,T〉:=〈ξ,T(η)〉(ξ∈Lp(G),η∈Lq(G),T∈PMp(G)).
In particular, PM2(G) is the group von Neumann algebra VN(G). If the map Λp from the projective tensor product Lq(G)⊗̂γLp(G) to C0(G) is defined by(1.9)Λp(g⊗f)(s)=〈g,λp(s)f〉,
for g∈Lq(G),f∈Lp(G), and s∈G, then Ap(G) is isometrically isomorphic to Lq(G)⊗̂γLp(G)/kerΛp and (1.10)Ap(G)*={T∈B(Lp(G)):T|ker⁡Λp=0}.


### 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].IfE is a real vector space, then the complexification Eℂ 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, Eℂ can be written as E+iE.For real vector spacesE and F with complexifications Eℂ and Fℂ, every ℝ-linear map T:E→F has a unique ℂ-linear extension T̃ given by T̃(z):=Tx+iTy, for z=x+iy∈Eℂ, where x,y∈E. The map T̃ is the canonical extension of T (usually again denoted by T).A real vector spaceE, 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 subsetE+:={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 (1.11)E+*={f∈E*:f(p)≥0(p∈E+)}.A complex vector space is called anordered 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 mapi:B→A is an isometry,(ii)
each elementa∈A can be written as a=a1-a2+i(a3-a4), where a1,…,a4 are positive in A and ∥aj∥≤∥a∥, for j=1,…,4.Banach lattices,C*-algebras, and their duals are Banach ordered spaces.Given ordered spacesEℂ and Fℂ, a ℂ-linear map T:Eℂ→Fℂ is positive if T(E)⊆F and the restriction T|E:E→F is positive.


### 1.3. Order Structure ofp-Operator Spaces

Each operator spaceE 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 QSLp 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 ofℰ=Lp(X), where X is a measure space and 1<p<∞. We say that T∈ℬ(Lp(X)) is positive if 〈Tf,f〉≥0 for each f∈Lp(X)∩Lq(X), where the pairing is the canonical dual action of Lq(X) on Lp(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 𝕄n(ℬ(Lp(X))) for n≥1. We say T∈𝕄n(ℬ(Lp(X)))≅ℬ(Lpn(X)) is positive if ∑i,j=1n〈Tfi,fj〉≥0, for each f1,…,fn∈Lp(X)∩Lq(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]∈𝕄n(ℬ(Lp(X))*), we say T is positive, if for each m∈ℕ and ϕ=[ϕij]∈𝕄m(ℬ(Lp(X)))+, the natural matrix action 〈ϕ,T〉 is positive.


## 1.1.p-Operator Spaces

In this section, we give a brief introduction to the notion ofp-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 𝕄n,m(E). We put simply 𝕄n,m:=𝕄n,m(ℂ). The space 𝕄n:=𝕄n,n is equipped with the operator norm |·|n from its canonical action on n-dimensional Lp-space, ℓpn. Clearly, 𝕄n acts on 𝕄n(E) by matrix multiplication. For a square matrix a=(aij)∈𝕄n, we have (1.2)‖a‖B(lpn)=sup⁡{(∑i=1n|∑j=1naijxj|p)1/p:xj∈C,∑j=1n|xj|p≤1}.Definition 1.1.
LetE 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 𝕄n(E) satisfying
(1.3)‖λ⋅x⋅μ‖≤|λ|‖x‖n|μ|,‖x⊕y‖n+m=max⁡{‖x‖n,‖y‖m},
for each λ∈𝕄m,n,μ∈𝕄n,m,x∈𝕄n(E), and y∈𝕄m(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(ℓpn,ℓpm) and ℬ(ℓpm,ℓpn), respectively.
The vector spaceE equipped with a p-matricial norm (∥·∥n)n=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.
LetE and F be p-operator spaces, and let T∈ℬ(E,F), then for each n∈ℕ,
(1.4)T(n):Mn(E)⟶Mn(F),T(n)([xij])=[T(xij)]
is the nth amplification of T. The map T is called p-completely bounded if
(1.5)‖T‖pcb:=sup⁡‖T(n)‖<∞.
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 𝒞ℬp(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 (1.6)Mn(CBp(E,F))=CBp(E,Mn(F))(n∈N).Figà-Talamanca-Herz algebras are our main examples ofp-operator spaces, studied in [13, 14]. For any function f:G→ℂ we define f̃:G→ℂ by f̃(x)=f(x-1)¯,  x∈G. The Figà-Talamanca-Herz algebra Ap(G) consists of those functions f:G→ℂ for which there are sequences (ξn)n=1∞ and (ηn)n=1∞ in Lq(G) and Lp(G), respectively, such that f=∑n=1∞ξn*η̃n and (1.7)∑n=1∞‖ξn‖q‖ηn‖p<∞.The norm∥f∥Ap(G) of f∈Ap(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 C0(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).Letλp:G→ℬ(Lp(G)) be the left regular representation of G on Lp(G), defined by λp(s)(f)(t)=f(s-1t). Then λp can be lifted to a representation of L1(G) on Lp(G). The algebra of pseudomeasures PMp(G) is defined as the w*-closure of λp(L1(G)) in ℬ(Lp(G)). There is a canonical duality PMp(G)≅Ap(G)*  via(1.8)〈ξ*η̃,T〉:=〈ξ,T(η)〉(ξ∈Lp(G),η∈Lq(G),T∈PMp(G)).
In particular, PM2(G) is the group von Neumann algebra VN(G). If the map Λp from the projective tensor product Lq(G)⊗̂γLp(G) to C0(G) is defined by(1.9)Λp(g⊗f)(s)=〈g,λp(s)f〉,
for g∈Lq(G),f∈Lp(G), and s∈G, then Ap(G) is isometrically isomorphic to Lq(G)⊗̂γLp(G)/kerΛp and (1.10)Ap(G)*={T∈B(Lp(G)):T|ker⁡Λp=0}.


## 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].IfE is a real vector space, then the complexification Eℂ 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, Eℂ can be written as E+iE.For real vector spacesE and F with complexifications Eℂ and Fℂ, every ℝ-linear map T:E→F has a unique ℂ-linear extension T̃ given by T̃(z):=Tx+iTy, for z=x+iy∈Eℂ, where x,y∈E. The map T̃ is the canonical extension of T (usually again denoted by T).A real vector spaceE, 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 subsetE+:={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 (1.11)E+*={f∈E*:f(p)≥0(p∈E+)}.A complex vector space is called anordered 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 mapi:B→A is an isometry,(ii)
each elementa∈A can be written as a=a1-a2+i(a3-a4), where a1,…,a4 are positive in A and ∥aj∥≤∥a∥, for j=1,…,4.Banach lattices,C*-algebras, and their duals are Banach ordered spaces.Given ordered spacesEℂ and Fℂ, a ℂ-linear map T:Eℂ→Fℂ is positive if T(E)⊆F and the restriction T|E:E→F is positive.


## 1.3. Order Structure ofp-Operator Spaces

Each operator spaceE 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 QSLp 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 ofℰ=Lp(X), where X is a measure space and 1<p<∞. We say that T∈ℬ(Lp(X)) is positive if 〈Tf,f〉≥0 for each f∈Lp(X)∩Lq(X), where the pairing is the canonical dual action of Lq(X) on Lp(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 𝕄n(ℬ(Lp(X))) for n≥1. We say T∈𝕄n(ℬ(Lp(X)))≅ℬ(Lpn(X)) is positive if ∑i,j=1n〈Tfi,fj〉≥0, for each f1,…,fn∈Lp(X)∩Lq(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]∈𝕄n(ℬ(Lp(X))*), we say T is positive, if for each m∈ℕ and ϕ=[ϕij]∈𝕄m(ℬ(Lp(X)))+, the natural matrix action 〈ϕ,T〉 is positive.


## 2. Order Structure of Somep-Operator Spaces

### 2.1. Thep-Fourier Stieltjes Algebra

The Fourier-Stieltjes algebraB(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 Bp(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 ofG 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 spaceℰ is called (i)
anLp-space if it is of the form Lp(X), for some measure space X,(ii)
aQSLp-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 byRepp(G), the collection of all (equivalence classes of) representations of G on QSLp-spaces.Definition 2.3.
LetG 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
(2.1)f(x)=〈π(x)ξ,ϕ〉(x∈G),
where ξ∈ℰ and ϕ∈ℰ*. Define
(2.2)Bp(G):={f:G⟶C:fisacoefficientfunctionofsome(π,E)∈Repp(G)}.
For f∈Bp(G), put ∥f∥:=inf{∥ξ∥∥ϕ∥:f(·)=〈π(·)ξ,ϕ〉}.Using a suitable definition of tensor product ofQSLp-spaces, it is shown in [7] that Bp(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 Bp(G) isometrically if G is amenable [7, Corollary 5.3].Acompatible 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 (Ap(G),Aq(G)) and (Lp(G),Lq(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 Bp(G), where π 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 f∈Pp(G), associated to a representation (π,ℰ), there exist a sequence (πn,ℰn)n=1∞ of cyclic representations of G on closed subspaces ℰn of ℰ∩ℰ*, and {ξn} in ℰn, such that (2.3)f(x)=∑n=1∞〈πn(x)ξn,ξn〉(x∈G).The closed subspaceBpp(G) of Bp(G) is the closure of the set of all functions f∈Bp(G)  of the form f(x)=〈π(x)ξ,η〉,x∈G, for some representation (π,ℰ), where ℰ is an Lp-space, ξ∈ℰ, and η∈ℰ*.Proposition 2.5.
The linear span ofPp(G) is dense in Bpp(G).Proof.
Letu∈Bpp(G) have a representation as u(x)=〈π(x)ξ,η〉,x∈G, where π is a representation on some Lp-space ℰ,ξ∈ℰ and η∈ℰ*. It is clear that ℰ∩ℰ* is dense in both ℰ and ℰ*. Hence, there exist sequences {ξn} and {ηn} in ℰ∩ℰ* converging to ξ and η in ℰ and ℰ*, respectively. Put un(x):=〈π(x)ξn,ηn〉, then
(2.4)(u-un)(x)=〈π(x)ξ,(η-ηn)〉+〈π(x)(ξ-ξn),ηn〉.
By the definition of the norm in Bp(G), we have
(2.5)‖u-un‖≤‖ξ‖p‖η-ηn‖q+‖ξ-ξn‖p‖ηn‖q,
which tends to zero, as n→∞.
Now, it is enough to consider the following decomposition forun,
(2.6)〈π(x)ξn,ηn〉=14∑j=03〈π(x)(ξn+ijηn),ξn+ijηn〉,
where i=-1, and note that each ξn+ijηn, for j=0,1,2,3, belongs to ℰ∩ℰ*. So u belongs to the closed linear span of Pp(G).Each representation(π,ℰ) of G induces a representation of the group algebra L1(G) on ℰ via (2.7)π(f):=∫Gf(x)π(x)(f∈L1(G)).
Let (πp,ℰ) be the p-universal representation of G. The Banach space UPFp(G) of all pseudofunctions on G is the closure of πp(L1(G)) in B(ℰ). By [7, Theorem 6.6], we know that Bp(G) is the dual space of UPFp(G) via the pairing (2.8)〈πp(f),g〉=∫Gf(x)g(x)(f∈L1(G),g∈Bp(G)).We say thatf∈UPFp(G) is positive if ϕg(f)≥0, for all g∈Bp(G)+=Pp(G), where ϕg 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 fromUPFp(G) to C*(G).Proof.
Since each Hilbert space is aQSLp-space [3] and for each f∈L1(G),∥f∥Rep2≤∥f∥Repp, where ∥f∥Repp=supπ∈Repp(G),∥π∥≤1∥π(f)∥, it follows that the identity map
(2.9)i:(L1(G),‖⋅‖Repp)⟶(L1(G),‖⋅‖Rep2)
is continuous. Thus, it induces a continuous map
(2.10)ic:UPFp(G)⟶C*(G).
Consider the conjugate map
(2.11)(ic)*:B(G)⟶Bp(G).
Since for each u∈B(G) and f∈L1(G),
(2.12)〈(ic)*(u),f〉=〈u∘i,f〉=〈u,f〉,(ic)* is the inclusion map. By the definition of positivity in Bp(G) and B(G),(ic)* is a positive map. Therefore (ic)** is positive, and so is ic=(ic)**∣UPFp(G).


### 2.2. Figà-Talamanca-Herz Algebra

In this section, we study the order structure of the Figà-Talamanca-Herz algebraAp(G). Since Ap(G) is the set of coefficient functions of the left regular representation of G on Lp(G), we have Ap(G)⊆Bpp(G). We define the positive cone of Ap(G) as the closure in Ap(G), of the set of all function of the form f=∑i=1nξi*ξ̃i, for a sequence (ξi) in Lp(G)∩Lq(G), and denote it by Ap(G)+. It is clear that Ap(G)+ is contained in Bp(G)+. Since Cc(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∈PMp(G) is positive as an element of B(Lp(G)) if and only if it is positive as an element of Ap(G)+*, where Ap(G)+* is the dual cone induced by the positive cone Ap(G)+. Also, since Ap(G)+ is closed, u∈Ap(G)+ if and only if for each T∈PMp(G)+,  T(u)≥0 [19].Theorem 2.7.
LetG be a locally compact amenable group. Then Ap(G)+=P(G)∩A(G)={u∈A(G):∥u∥Ap(G)=u(e)}, where e is the identity of G. In particular, Ap(G)+ is a proper cone.Proof.
SinceLp(G)∩Lq(G)⊆L2(G), it follows that Ap(G)+⊆P(G)∩A(G)¯∥·∥Ap(G). But when G is amenable, the identity map i:A(G)→Ap(G) is an embedding with ∥·∥Ap(G)≤∥·∥A(G) [3, Theorem C]. For each u∈P(G)∩A(G), we have
(2.13)‖u‖A(G)=u(e)≤‖u‖∞≤‖u‖Ap(G)≤‖u‖A(G),
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 Ap(G) and consequently Ap(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)∩Cc(G) of the form {fi*f̃i},fi∈Cc(G),i∈ℕ converging to v in A(G). Since for each u∈A(G),∥u∥Ap(G)≤∥u∥A(G), it follows that {fi*f̃i} converges to v in Ap(G). Clearly, {fi*f̃i} is contained in Ap(G)+ and since Ap(G)+ is closed, v∈Ap(G)+.
The second equality follows immediately from the above-mentioned fact that for eachu∈P(G)∩A(G),  ∥u∥A(G)=∥u∥Ap(G).


## 2.1. Thep-Fourier Stieltjes Algebra

The Fourier-Stieltjes algebraB(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 Bp(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 ofG 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 spaceℰ is called (i)
anLp-space if it is of the form Lp(X), for some measure space X,(ii)
aQSLp-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 byRepp(G), the collection of all (equivalence classes of) representations of G on QSLp-spaces.Definition 2.3.
LetG 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
(2.1)f(x)=〈π(x)ξ,ϕ〉(x∈G),
where ξ∈ℰ and ϕ∈ℰ*. Define
(2.2)Bp(G):={f:G⟶C:fisacoefficientfunctionofsome(π,E)∈Repp(G)}.
For f∈Bp(G), put ∥f∥:=inf{∥ξ∥∥ϕ∥:f(·)=〈π(·)ξ,ϕ〉}.Using a suitable definition of tensor product ofQSLp-spaces, it is shown in [7] that Bp(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 Bp(G) isometrically if G is amenable [7, Corollary 5.3].Acompatible 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 (Ap(G),Aq(G)) and (Lp(G),Lq(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 Bp(G), where π 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 f∈Pp(G), associated to a representation (π,ℰ), there exist a sequence (πn,ℰn)n=1∞ of cyclic representations of G on closed subspaces ℰn of ℰ∩ℰ*, and {ξn} in ℰn, such that (2.3)f(x)=∑n=1∞〈πn(x)ξn,ξn〉(x∈G).The closed subspaceBpp(G) of Bp(G) is the closure of the set of all functions f∈Bp(G)  of the form f(x)=〈π(x)ξ,η〉,x∈G, for some representation (π,ℰ), where ℰ is an Lp-space, ξ∈ℰ, and η∈ℰ*.Proposition 2.5.
The linear span ofPp(G) is dense in Bpp(G).Proof.
Letu∈Bpp(G) have a representation as u(x)=〈π(x)ξ,η〉,x∈G, where π is a representation on some Lp-space ℰ,ξ∈ℰ and η∈ℰ*. It is clear that ℰ∩ℰ* is dense in both ℰ and ℰ*. Hence, there exist sequences {ξn} and {ηn} in ℰ∩ℰ* converging to ξ and η in ℰ and ℰ*, respectively. Put un(x):=〈π(x)ξn,ηn〉, then
(2.4)(u-un)(x)=〈π(x)ξ,(η-ηn)〉+〈π(x)(ξ-ξn),ηn〉.
By the definition of the norm in Bp(G), we have
(2.5)‖u-un‖≤‖ξ‖p‖η-ηn‖q+‖ξ-ξn‖p‖ηn‖q,
which tends to zero, as n→∞.
Now, it is enough to consider the following decomposition forun,
(2.6)〈π(x)ξn,ηn〉=14∑j=03〈π(x)(ξn+ijηn),ξn+ijηn〉,
where i=-1, and note that each ξn+ijηn, for j=0,1,2,3, belongs to ℰ∩ℰ*. So u belongs to the closed linear span of Pp(G).Each representation(π,ℰ) of G induces a representation of the group algebra L1(G) on ℰ via (2.7)π(f):=∫Gf(x)π(x)(f∈L1(G)).
Let (πp,ℰ) be the p-universal representation of G. The Banach space UPFp(G) of all pseudofunctions on G is the closure of πp(L1(G)) in B(ℰ). By [7, Theorem 6.6], we know that Bp(G) is the dual space of UPFp(G) via the pairing (2.8)〈πp(f),g〉=∫Gf(x)g(x)(f∈L1(G),g∈Bp(G)).We say thatf∈UPFp(G) is positive if ϕg(f)≥0, for all g∈Bp(G)+=Pp(G), where ϕg 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 fromUPFp(G) to C*(G).Proof.
Since each Hilbert space is aQSLp-space [3] and for each f∈L1(G),∥f∥Rep2≤∥f∥Repp, where ∥f∥Repp=supπ∈Repp(G),∥π∥≤1∥π(f)∥, it follows that the identity map
(2.9)i:(L1(G),‖⋅‖Repp)⟶(L1(G),‖⋅‖Rep2)
is continuous. Thus, it induces a continuous map
(2.10)ic:UPFp(G)⟶C*(G).
Consider the conjugate map
(2.11)(ic)*:B(G)⟶Bp(G).
Since for each u∈B(G) and f∈L1(G),
(2.12)〈(ic)*(u),f〉=〈u∘i,f〉=〈u,f〉,(ic)* is the inclusion map. By the definition of positivity in Bp(G) and B(G),(ic)* is a positive map. Therefore (ic)** is positive, and so is ic=(ic)**∣UPFp(G).


## 2.2. Figà-Talamanca-Herz Algebra

In this section, we study the order structure of the Figà-Talamanca-Herz algebraAp(G). Since Ap(G) is the set of coefficient functions of the left regular representation of G on Lp(G), we have Ap(G)⊆Bpp(G). We define the positive cone of Ap(G) as the closure in Ap(G), of the set of all function of the form f=∑i=1nξi*ξ̃i, for a sequence (ξi) in Lp(G)∩Lq(G), and denote it by Ap(G)+. It is clear that Ap(G)+ is contained in Bp(G)+. Since Cc(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∈PMp(G) is positive as an element of B(Lp(G)) if and only if it is positive as an element of Ap(G)+*, where Ap(G)+* is the dual cone induced by the positive cone Ap(G)+. Also, since Ap(G)+ is closed, u∈Ap(G)+ if and only if for each T∈PMp(G)+,  T(u)≥0 [19].Theorem 2.7.
LetG be a locally compact amenable group. Then Ap(G)+=P(G)∩A(G)={u∈A(G):∥u∥Ap(G)=u(e)}, where e is the identity of G. In particular, Ap(G)+ is a proper cone.Proof.
SinceLp(G)∩Lq(G)⊆L2(G), it follows that Ap(G)+⊆P(G)∩A(G)¯∥·∥Ap(G). But when G is amenable, the identity map i:A(G)→Ap(G) is an embedding with ∥·∥Ap(G)≤∥·∥A(G) [3, Theorem C]. For each u∈P(G)∩A(G), we have
(2.13)‖u‖A(G)=u(e)≤‖u‖∞≤‖u‖Ap(G)≤‖u‖A(G),
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 Ap(G) and consequently Ap(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)∩Cc(G) of the form {fi*f̃i},fi∈Cc(G),i∈ℕ converging to v in A(G). Since for each u∈A(G),∥u∥Ap(G)≤∥u∥A(G), it follows that {fi*f̃i} converges to v in Ap(G). Clearly, {fi*f̃i} is contained in Ap(G)+ and since Ap(G)+ is closed, v∈Ap(G)+.
The second equality follows immediately from the above-mentioned fact that for eachu∈P(G)∩A(G),  ∥u∥A(G)=∥u∥Ap(G).


## 3. Order Maps betweenp-Operator Spaces

In this section, we study the positive maps between variousp-operator spaces. Also, we give the general form of the algebra and order isomorphisms between Figà-Talamanca-Herz algebras.Proposition 3.1.
LetE 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 thatT: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 {xn}n=1∞ in U+ such that ∥Txn∥≥n3, for each n∈ℕ. Since E+ is closed, z:=∑xn/n2 is in E+. Hence, Tz≥Txn/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 spaceE with a closed positive cone E+ satisfying the following property: for each x∈E, there is a sequence {xn} converging to x, such that  xn=xn1-xn2+i(xn3-xn4) with xni∈E+ and ∥xni∥≤∥xn∥.A linear mapTbetween 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.
LetG1 and G2 be amenable locally compact groups with identities e1 and e2, respectively, and let T:Ap(G1)→Ap(G2) be an order and algebra isomorphism. Then(i)
T*(λe2)=λe1, where λei is the evaluation homomorphism at ei,  i=1,2,(ii)
there exists an isomorphism or anti-isomorphismφ from G2 onto G1 such that T(f)=f∘φ for all f∈Ap(G1).Proof.
(i) Consider the adjoint mapT*:Ap(G2)*→Ap(G1)*, which is clearly an order isomorphism. Since T is an algebra isomorphism, T*(λe2) 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, T*(λe2)=λx, for some x∈G1. We note that λx is positive because λe2 is a positive element of Ap(G)*, hence, x=e1 by [20, Proposition 4.3].
(ii) We first note thatAp(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 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 Ap(G1) and continuity of T.Proposition 3.3.
LetG 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 thatT 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∈(𝕄n(A(G)))1, the unit ball of 𝕄n(A(G)), such that ∥T(n)(x)∥>n2. Since 𝕄n(A(G))=𝒞ℬσ(VN(G),𝕄n), where 𝒞ℬσ(VN(G),𝕄n) is the algebra of all w*-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 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+xt)/2 instead of x, without loss of generality, we may assume that x is positive, ∥x∥≤1,x=xt, and ∥T(n)(x)∥≥n2/8. Let Tn:=⨁i=1nT:⨁i=1nA(G)→⨁i=1nVN(H) be defined by (zi)↦(Tzi). Then clearly, ∥Tn∥=∥⨁i=1nT∥=sup∥(zi)∥≤1(∑i=1n∥Tzi∥2)1/2≤n∥T∥.
Now,T(n)(x)=[T(xij)] is positive in 𝕄n(VN(H)), hence, [T(xij)*]=[T(xij)], and for each i,j,  T(xij) is self-adjoint. Hence, Txij=∥T∥(Txij1+Txij2)/2, where for each i,j∈{1,…,n} and k=1,2,  Txijk is a unitary operator in ℬ(L2(G)). For each unit vector (f1,f2,…,fn) in ⨁i=1nL2(H), set Txijk(fi)=eijk,  k=1,2,1≤i,j≤n. Then (3.1)(∑j‖∑iTxij1+Txij22(fi)‖2)1/2=(∑j‖∑ieij1+eij22‖2)1/2=12(∑j〈∑ieij1+eij2,∑ieij1+eij2〉)1/2≤12(∑i,j〈ei,j1+ei,j2,eij1+eij2〉+4n2)1/2=12(∑i,j‖ei,j1+ei,j2‖2+4n2)1/2≤(‖Tn2(xij)‖+n).
Hence, we have
(3.2)‖T(n)(xij)‖≤‖T‖(‖Tn2(xij)‖+n).
On the other hand,
(3.3)‖x‖Mn(A(G))=sup⁡S∈Mn(VN(G))1‖[Skl(xij)]‖=sup⁡S∈Mn(VN(G))1(∑i,j,k,l|Skl(xij)|2)1/2.
By the Hahn-Banach theorem, for each xij, there is some Sij∈VN(G)1 such that Sij(x)=∥xij∥. Now, put S=[Sij], then ∥S∥𝕄n(VN(G))≤n. Hence, S/n∈𝕄n(VN(G))1 and (Sij/n)(xij)=∥xij∥/n. Therefore,
(3.4)‖x‖Mn(A(G))=sup⁡R∈Mn(VN(G))1‖[Rkl(xij)]‖≥(∑‖xijn‖2)1/2=1n(∑‖xij‖2)1/2.
This means that the norm of x as an element of ⨁i=1n2A(G) is less than or equal to n. Therefore, by inequality (3.2),
(3.5)(n2(8‖T‖))-n≤‖Tn2(xij)‖≤‖T‖‖(xij)‖⨁i=1n2A(G)≤n‖T‖.
Hence, T is unbounded, which is a contradiction.Let1<p′<∞, and let Y be an arbitrary measure space. For subspaces ℳ⊆ℬ(Lp(X)) and 𝒩⊆ℬ(Lp′(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):𝕄n(ℳ)→𝕄n(𝒩),  T(n)[xij]=[T(xij)] 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 groupG, 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 m,n∈ℕ, we say that T=[Tij]∈𝕄n(Ap(G)) is positive if for every p-completely positive and p-completely bounded linear map ϕ:PMp(G)→𝕄m, the natural action 〈ϕ,T〉 is a positive scalar matrix.Forℳ=Ap(G)orPMp(G) and 𝒩=Ap′(G) or PMp′(G), we say that a linear map T:ℳ→𝒩 is (p,p′)-completely positive if for each positive integer n, the nth amplification Tn:𝕄n(ℳ)→𝕄n(𝒩) of T is a positive map. If this is the case for p=p′, we say that T is p-completely positive.Letp≥2. Then since ∥·∥2 and ∥·∥p are equivalent norms on ℂn, for each n∈ℕ, there exists a positive scalar αn such that αn∥·∥2≤∥·∥p≤∥·∥2 on ℂn. For the case where 1<p≤2, for each n∈ℕ, we choose βn>0 such that βn∥·∥≤∥·∥2≤∥·∥p on ℂn.Proposition 3.4.
LetG be an amenable locally compact group, let p∈(1,∞) and let T:Ap(G)→PMp(G) be a linear map. Let T1=T|A(G) be the restriction of T to A(G).(i)
IfT is p-completely positive, then T1 is a completely positive linear map from A(G) to VN(G).(ii)
Ifp≥2 and T is a bounded linear map, then for each n∈ℕ, we have ∥T1(n)∥≤∥T(n)∥.(iii)
If1<p≤2 and T is bounded, then for each n∈ℕ, we have ∥T(n)∥≤βn2∥T1(n)∥.(iv)
If1<p≤2 and T is a p-completely positive map, then there are infinitely many n∈ℕ such that ∥T(n)∥≤βn2n2.Proof.
(i) LetT be p-completely positive. Let n∈ℕ and [xij]∈𝕄n(A(G)). Since 𝕄n(A(G))+=𝒞𝒫σ(VN(G),𝕄n), where 𝒞𝒫σ(VN(G),𝕄n) is the set of all w*-continuous completely positive linear maps from VN(G) to 𝕄n, it follows that [xij]∈𝕄n(A(G))+ if and only if for each [Tij]∈𝕄n[VN(G)]+, we have [Tkl(xij)]∈𝕄n2+. Since G is amenable, the embedding i:A(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]∈𝕄n[PMp(G)]+ if and only if 〈[Sij][f],[f]〉≥0, for all [f]=(f1,…,fn) with f1,…,fn∈Cc(G). It is clear that if [Sij]∈𝕄n[PMp(G)]+, then it also belongs to 𝕄n[VN(G)]+. Therefore, i:A(G)→Ap(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 embeddingi:A(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 n∈ℕ, i(n) is continuous and has dense range. Let [aij]∈𝕄n(A(G)), then [S(aij)]∈B(ℓpn), for each S∈PMp(G) and(3.6)‖[aij]‖Mn(Ap(G))=sup⁡S∈PMp(G)1‖[S(aij)]‖=sup⁡S∈PMp(G)1sup⁡{(∑i=1n|∑j=1nS(aij)xj|p)1/p,(xj)∈Cn,∑j=1n|xj|p≤1}≤1αnsup⁡S∈PMp(G)1sup⁡{(∑i=1n|∑j=1nS(aij)xj|2)1/2,(xj)∈Cn,∑j=1n|xj|2≤1}=1αnsup⁡S∈PMp(G)1‖S(aij)‖B(l2)≤1αn‖[aij]‖Mn(A(G)),
which shows that i(n) is continuous. Consider now an element [aij]∈𝕄n(Ap(G)). Then for each i,j, there exists a sequence (aijm)m in A(G) converging to aij in Ap(G). Hence,
(3.7)‖[aijm]-[aij]‖Mn(Ap(G))=‖[aijm-aij]‖Mn(Ap(G))=sup⁡S∈PMp(G)1sup⁡{(∑i=1n|∑i=1n(S(aij)-S(aijm))xj|p)1/p,(xj)∈Cn,∑j=1n|xj|p≤1},
which clearly converges to zero as m→∞. This implies that i(n) has dense range. Hence, for each n∈ℕ, since αn∥·∥Mn(VN(G))≤∥·∥Mn(PMp(G)) we have
(3.8)αn‖T(n)‖=αn⋅sup⁡{‖[T(xij)]‖Mn(PMp(G)):[xij]∈Mn(Ap(G)),∥[xij]∥Mn(Ap(G))≤1}≥αn⋅sup⁡{αn‖[T1(xij)]‖Mn(VN(G)):[xij]∈Mn(A(G)),‖[xij]‖Mn(Ap(G))≤1}≥sup⁡{αn‖[T1(xij)]‖Mn(VN(G)):[xij]∈Mn(A(G)),‖[xij]‖Mn(A(G))≤1}=αn‖T1(n)‖,
that is, ∥T1(n)∥≤∥T(n)∥.(iii)
Using the weak-star density ofPMp(G)1 in VN(G)1, the same proof as in (ii) can be applied.(iv)
LetT:Ap(G)→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 n∈ℕ such that ∥T1(n)∥≤n2. Now, since by part (iii), for each n∈ℕ,  ∥T(n)∥≤βn2∥T1(n)∥ the statement is clear.



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*Source: 531023-2011-06-22.xml*