P-Tensor Product for Group C*-Algebras

Abstract

In this paper, we introduce new tensor products ${\displaystyle \stackrel{p}{\otimes }}(1\le p\le +\infty )$ on $C_{{\ell }_p}^{*}(\Gamma )\otimes C_{{\ell }_p}^{*}(\Gamma )$ and ${\displaystyle \stackrel{c_0}{\otimes }}$ on $C_{c_0}^{*}(\Gamma )\otimes C_{c_0}^{*}(\Gamma )$ for any discrete group $\Gamma$. We obtain that for $1\le p<+\infty$ $C_{{\ell }_p}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{{\ell }_p}^{*}(\Gamma )=C_{{\ell }_p}^{*}(\Gamma ){\displaystyle \stackrel{p}{\otimes }}{C}_{{\ell }_p}^{*}(\Gamma )$ if and only if $\Gamma$ is amenable; $C_{c_0}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{c_0}^{*}(\Gamma )=C_{c_0}^{*}(\Gamma ){\displaystyle \stackrel{c_0}{\otimes }}{C}_{c_0}^{*}(\Gamma )$ if and only if $\Gamma$ has Haagerup property. In particular, for the free group with two generators ${\mathbb{F}}_2$ we show that $C_{{\ell }_p}^{*}\left({\mathbb{F}}_2\right){\displaystyle \stackrel{p}{\otimes }}{C}_{{\ell }_p}^{*}\left({\mathbb{F}}_2\right)\ncong C_{{\ell }_q}^{*}\left({\mathbb{F}}_2\right){\displaystyle \stackrel{q}{\otimes }}{C}_{{\ell }_q}^{*}\left({\mathbb{F}}_2\right)$ for $2\le q<p\le +\infty$.

1. Introduction

When $\mathcal{A}$ and $\mathcal{B}$ are C${}^{*}$-algebras, it can happen that numerous different norms make $\mathcal{A}\odot \mathcal{B}$ into a pre-C${}^{*}$-algebra. In other words, $\mathcal{A}\odot \mathcal{B}$ may carry more than one C${}^{*}$-norms. For example, the spatial (or minimal) tensor product norm ${\Vert ·\Vert }_{min}$ and the maximal tensor product ${\Vert ·\Vert }_{max}$ are always C${}^{*}$-norms on $\mathcal{A}\odot \mathcal{B}$. As the names suggest, the spatial (minimal) tensor norm is the smallest C${}^{*}$-norm one can place on $\mathcal{A}\odot \mathcal{B}$ and the maximal is the largest. In general these norms do not agree. In 1995, Junge and Pisier [1] proved that

$$\mathbb{B}\left({\ell }_2\right)\stackrel{max}{\otimes }\mathbb{B}\left({\ell }_2\right)\ne \mathbb{B}\left({\ell }_2\right)\stackrel{min}{\otimes }\mathbb{B}\left({\ell }_2\right).$$

In 2014, Ozawa and Pisier [2] demonstrated that $\mathbb{B}\left(\mathcal{H}\right)\otimes \mathbb{B}\left(\mathcal{H}\right)$ admits $2^{{\aleph }_0}$ distinct C${}^{*}$-norms. Ozawa and Pisier also showed that $C_{\lambda }^{*}\left({\mathbb{F}}_n\right)\otimes C_{\lambda }^{*}\left({\mathbb{F}}_n\right)$ admits $2^{{\aleph }_0}$ distinct C${}^{*}$-norms where ${\mathbb{F}}_n$ is the noncommutative free group on $n\ge 2$ generators. Recently, Wiersma generalized Ozawa and Pisier’s result. In [3], Wiersma proved that $C_{\lambda }^{*}\left({\Gamma }_1\right)\otimes C_{\lambda }^{*}\left({\Gamma }_2\right)$ and $C^{*}\left({\Gamma }_1\right)\otimes C_{\lambda }^{*}\left({\Gamma }_2\right)$ admit $2^{{\aleph }_0}$ distinct C${}^{*}$-norms where ${\Gamma }_1$ and ${\Gamma }_2$ are discrete groups containing copies of noncommutative free groups. In the other respect, Kirchberg [4] proved the following striking theorem:

$$C^{*}\left(\mathbb{F}\right)\stackrel{max}{\otimes }\mathbb{B}\left(\mathcal{H}\right)=C^{*}\left(\mathbb{F}\right)\stackrel{min}{\otimes }\mathbb{B}\left(\mathcal{H}\right)$$

for any free group $\mathbb{F}$. Kirchberg’s famous QWEP conjecture is one of the most important open problems in the theory of operator algebras. Kirchberg showed that QWEP conjecture is equivalent to

$$C^{*}\left({\mathbb{F}}_2\right)\stackrel{max}{\otimes }{C}^{*}\left({\mathbb{F}}_2\right)=C^{*}\left({\mathbb{F}}_2\right)\stackrel{min}{\otimes }{C}^{*}\left({\mathbb{F}}_2\right).$$

Brown and Guentner introduced a new $C^{*}$-completion of the group ring of a countable discrete group $\Gamma$ in [5]. In the following, we first recall some results in [5].

Let $\Gamma$ be a countable discrete group and $\pi$ be a unitary representation of $\Gamma$ on a Hilbert space $\mathcal{H}$. For $\xi ,\eta \in \mathcal{H}$, we denote the matrix coefficient of $\pi$ by

$${\pi }_{\xi ,\eta }\left(s\right)=⟨\pi \left(s\right)\xi |\eta ⟩.$$

It is clear that ${\pi }_{\xi ,\eta }\in {\ell }_{\infty }\left(\Gamma \right)$.

Let D be an algebraic two-side ideal of ${\ell }_{\infty }\left(\Gamma \right)$. If there exists a dense subspace ${\mathcal{H}}_0$ of $\mathcal{H}$ such that ${\pi }_{\xi ,\eta }\in D$ for all $\xi ,\eta \in {\mathcal{H}}_0$, then $\pi$ is called $D-$representation. If D is invariant under the left and right translation of $\Gamma$ on ${\ell }_{\infty }(\Gamma )$, then it is said to be translation invariant. In this paper, we always assume that D is a non-zero translation invariant ideal of ${\ell }_{\infty }(\Gamma )$. For each $p\in [1,+\infty )$, we denote the norm on ${\ell }_p(\Gamma )$ by

$${\left|f\right|}_p=(\sum _{s\in \Gamma }|f^p\left(s\right){\left|\right)}^{\frac{1}{p}}\mathrm{for}f\in {\ell }_p(\Gamma ).$$

We denote by $c_0(\Gamma )$ the functions of ${\ell }_{\infty }(\Gamma )$ with vanishing at infinity. It is clear that ${\ell }_p(\Gamma )$ and $c_0(\Gamma )$ are non-trivial translation invariant ideals of ${\ell }_{\infty }(\Gamma )$.

The $C^{*}$-algebra $C_D^{*}\left(\Gamma \right)$ is the $C^{*}$-completion of the group ring $\mathbb{C}\Gamma$ by ${∥·∥}_D$, where for $\forall f\in \mathbb{C}\Gamma$,

$${∥f∥}_D=sup\left\{∥\pi \left(f\right)∥\right.:\pi isaD-representation\}.$$

We denote by $C^{*}(\Gamma )$ the full group $C^{*}$-algebra and by ${C_{\lambda }}^{*}(\Gamma )$ the reduced group $C^{*}$-algebra, where $C^{*}(\Gamma )$ is the completion of $C(\Gamma )$ with respect to the norm

$${∥x∥}_u=sup\left\{∥\pi \left(x\right)∥\right.:\pi isacyclicrepresentation\}.$$

and ${C_{\lambda }}^{*}(\Gamma )$ is the completion of $C(\Gamma )$ with the norm

$${∥x∥}_r=sup\left\{∥\lambda \left(x\right)∥\right.:\pi isaleftregularrepresentation\}.$$

In [5], the following results are obtained:

(1) $C^{*}\left(\Gamma \right)={C_{l_{\infty }}}^{*}\left(\Gamma \right)$ and ${C_{\lambda }}^{*}\left(\Gamma \right)={C_{c_c}}^{*}\left(\Gamma \right);$ Where $C_c(\Gamma )$ is the function of finitely supported functions on $\Gamma$.

(2) ${C_{l_p}}^{*}\left(\Gamma \right)={C_{\lambda }}^{*}\left(\Gamma \right)$ for every $p\in [1,2];$

(3) $C^{*}\left(\Gamma \right)={C_D}^{*}\left(\Gamma \right)$ if and only if there exists a sequence $\left\{h_n\right\}$ of positive definite functions in D such that $h_n\to 1$;

(4) $\Gamma$ is amenable if and only if $C^{*}\left(\Gamma \right)={C_{c_c}}^{*}(\Gamma )$;

(5) $\Gamma$ has the Haagerup property if and only if $C^{*}\left(\Gamma \right)={C_{c_0}}^{*}\left(\Gamma \right)$.

In this paper, we introduce new tensor products ${\displaystyle \stackrel{p}{\otimes }}(1\le p\le +\infty )$ on $C_{{\ell }_p}^{*}(\Gamma )\otimes C_{{\ell }_p}^{*}(\Gamma )$ and ${\displaystyle \stackrel{c_0}{\otimes }}$ on $C_{c_0}^{*}(\Gamma )\otimes C_{c_0}^{*}(\Gamma )$ for any discrete group $\Gamma$. We obtain that for $1\le p<+\infty$, $C_{{\ell }_p}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{{\ell }_p}^{*}(\Gamma )=C_{{\ell }_p}^{*}(\Gamma ){\displaystyle \stackrel{p}{\otimes }}{C}_{{\ell }_p}^{*}(\Gamma )$ if and only if $\Gamma$ is amenable; $C_{c_0}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{c_0}^{*}(\Gamma )=C_{c_0}^{*}(\Gamma ){\displaystyle \stackrel{c_0}{\otimes }}{C}_{c_0}^{*}(\Gamma )$ if and only if $\Gamma$ has Haagerup property. In last section, for the free group with two generators ${\mathbb{F}}_2$ we show that $C_{{\ell }_p}^{*}\left({\mathbb{F}}_2\right){\displaystyle \stackrel{p}{\otimes }}{C}_{{\ell }_p}^{*}\left({\mathbb{F}}_2\right)\ncong C_{{\ell }_q}^{*}\left({\mathbb{F}}_2\right){\displaystyle \stackrel{q}{\otimes }}{C}_{{\ell }_q}^{*}\left({\mathbb{F}}_2\right)$ for $2\le q<p\le +\infty$.

2. Amenability and Haagerup Property

Definition 1.

For a discrete group Γ and $1\le p\le +\infty$, we define

$$C_{{\ell }_p}^{*}(\Gamma )\stackrel{p}{\otimes }{C}_{{\ell }_p}^{*}(\Gamma )\triangleq C_{{\ell }_p}^{*}(\Gamma \times \Gamma ).$$

We need to check that ${\displaystyle \stackrel{p}{\otimes }}$ is a $C^{*}$-tensor product of $C_{{\ell }_p}^{*}(\Gamma )$ and $C_{{\ell }_p}^{*}(\Gamma )$. First we will show that the map $x\to x\otimes e$ from $C_{{\ell }_p}^{*}(\Gamma )$ into $C_{{\ell }_p}^{*}(\Gamma \times \Gamma )$ is isometric, where e is the unit of $\Gamma$. For $x={\displaystyle \sum _{s\in \Gamma }}{a}_{s}s\in \mathbb{C}\Gamma$ and the unit e of $\Gamma$, $x\otimes e\in \mathbb{C}(\Gamma )\otimes \mathbb{C}(\Gamma )\subseteq \mathbb{C}(\Gamma \times \Gamma )$. We compute

$$\begin{array}{lll}{\Vert x\otimes e\Vert }_{{\ell }_p}& =& sup\{\Vert \pi (x\otimes e)\Vert |\pi :\Gamma \times \Gamma \to \mathcal{B}\left(\mathcal{H}\right)is{\ell }_p(\Gamma \times \Gamma )-\mathrm{representation}\}\\ & =& sup\{\Vert \pi ({\displaystyle \sum _{s\in \Gamma }}{a}_{s}s\otimes e)\Vert |\pi :\Gamma \times \Gamma \to \mathcal{B}\left(\mathcal{H}\right)is{\ell }_p(\Gamma \times \Gamma )-\mathrm{representation}\}\\ & =& sup\{\Vert {\displaystyle \sum _{s\in \Gamma }}{a}_s\pi (s\otimes e)\Vert |\pi :\Gamma \times \Gamma \to \mathcal{B}\left(\mathcal{H}\right)is{\ell }_p(\Gamma \times \Gamma )-\mathrm{representation}\}\\ & \le & sup\{\Vert {\displaystyle \sum _{s\in \Gamma }}{a}_s\sigma \left(s\right)\Vert |\sigma \mathrm{is}{\ell }_p(\Gamma )-\mathrm{representation}\}\\ & \le & sup\{\Vert \sigma \left(x\right)\Vert |\sigma \mathrm{is}{\ell }_p(\Gamma )-\mathrm{representation}\}\\ & =& {\Vert x\Vert }_{{\ell }_p},\end{array}$$

since it is easy to check that $s\to \pi (s\otimes e)$ is an ${\ell }_p(\Gamma )-$representation.

Conversely, we have

$$\begin{array}{lll}{\Vert x\otimes e\Vert }_{{\ell }_p}& =& sup\{\Vert \pi (x\otimes e)\Vert |\pi :\Gamma \times \Gamma \to \mathcal{B}\left(\mathcal{H}\right)is{\ell }_p(\Gamma \times \Gamma )-\mathrm{representation}\}\\ & =& sup\{\Vert {\displaystyle \sum _{s\in \Gamma }}{a}_s\pi (s\otimes e)\Vert |\pi :\Gamma \times \Gamma \to \mathcal{B}\left(\mathcal{H}\right)is{\ell }_p(\Gamma \times \Gamma )-\mathrm{representation}\}\\ & \ge & sup\{\Vert {\displaystyle \sum _{s\in \Gamma }}{a}_s{\sigma }_s\otimes {\sigma }_e\Vert |\sigma \mathrm{is}{\ell }_p(\Gamma )-\mathrm{representation}\}\\ & =& sup\{\Vert {\displaystyle \sum _{s\in \Gamma }}{a}_s{\sigma }_s\Vert |\sigma \mathrm{is}{\ell }_p(\Gamma )-\mathrm{representation}\}\\ & =& {\Vert x\Vert }_{{\ell }_p},\hfill \end{array}$$

since it is routine to show that $(s,t)\in \Gamma \times \Gamma \to {\sigma }_s\otimes {\sigma }_t\in \mathcal{B}(\mathcal{H}\otimes \mathcal{H})$ is an ${\ell }_p(\Gamma \times \Gamma )$-representation. Under this identification, we have

$$\mathbb{C}(\Gamma \times \Gamma )\subseteq C_{{\ell }_p}^{*}(\Gamma )\odot C_{{\ell }_p}^{*}(\Gamma )\subseteq C_{{\ell }_p}^{*}(\Gamma \times \Gamma ).$$

This implies that Definition 1 is well defined.

If $1\le p\le 2$, it follows from Proposition 2.11 in [5] that

$$\begin{array}{lll}{C}_{\lambda }^{*}(\Gamma ){\displaystyle \stackrel{p}{\otimes }}{C}_{\lambda }^{*}(\Gamma )& =& C_{{\ell }_p}^{*}(\Gamma ){\displaystyle \stackrel{p}{\otimes }}{C}_{{\ell }_p}^{*}(\Gamma )\\ & =& C_{{\ell }_p}^{*}(\Gamma \times \Gamma )\\ & =& C_{\lambda }^{*}(\Gamma \times \Gamma )\\ & =& C_{\lambda }^{*}(\Gamma ){\displaystyle \stackrel{min}{\otimes }}{C}_{\lambda }^{*}(\Gamma ).\end{array}$$

This shows that ${\displaystyle \stackrel{p}{\otimes }}={\displaystyle \stackrel{min}{\otimes }}$ for $1\le p\le 2$. If $p=\infty$, we have

$$\begin{array}{lll}{C}^{*}(\Gamma ){\displaystyle \stackrel{\infty }{\otimes }}{C}^{*}(\Gamma )& =& C_{{\ell }_{\infty }}^{*}(\Gamma ){\displaystyle \stackrel{\infty }{\otimes }}{C}_{{\ell }_{\infty }}^{*}(\Gamma )\\ & =& C_{{\ell }_{\infty }}^{*}(\Gamma \times \Gamma )\\ & =& C^{*}(\Gamma \times \Gamma )\\ & =& C^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}^{*}(\Gamma ).\end{array}$$

This shows that

$$\begin{array}{ccc}\hfill C_{\lambda }^{*}(\Gamma ){\displaystyle \stackrel{p}{\otimes }}{C}_{\lambda }^{*}(\Gamma )& =& C_{\lambda }^{*}(\Gamma ){\displaystyle \stackrel{min}{\otimes }}{C}_{\lambda }^{*}(\Gamma ).\hfill \end{array}$$

Theorem 1.

For $1\le p<+\infty$, $C_{{\ell }_p}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{{\ell }_p}^{*}(\Gamma )=C_{{\ell }_p}^{*}(\Gamma ){\displaystyle \stackrel{p}{\otimes }}{C}_{{\ell }_p}^{*}(\Gamma )$ if and only if Γ is amenable.

Proof. 

Suppose that $\Gamma$ is amenable, ${\Vert ·\Vert }_{min}={\Vert ·\Vert }_{max}$ on $\mathbb{C}(\Gamma )$. Since

$${\Vert ·\Vert }_{min}\le {\Vert ·\Vert }_{{\ell }_p}\le {\Vert ·\Vert }_{max}$$

on $\mathbb{C}(\Gamma )$, we have ${\Vert ·\Vert }_{min}={\Vert ·\Vert }_p={\Vert ·\Vert }_{max}$ on $\mathbb{C}(\Gamma )$. This implies that $C_{\lambda }^{*}(\Gamma )=C_{{\ell }_p}^{*}(\Gamma )=C^{*}(\Gamma )$. Thus

$$C_{{\ell }_p}^{*}(\Gamma )\stackrel{max}{\otimes }{C}_{{\ell }_p}^{*}(\Gamma )=C^{*}(\Gamma )\stackrel{max}{\otimes }{C}^{*}(\Gamma )=C^{*}(\Gamma \times \Gamma ).$$

Since $\Gamma \times \Gamma$ is also amenable, it follows from the Definition 1 that

$$C_{{\ell }_p}^{*}(\Gamma )\stackrel{p}{\otimes }{C}_{{\ell }_p}^{*}(\Gamma )=C_{{\ell }_p}^{*}(\Gamma \times \Gamma )=C^{*}(\Gamma \times \Gamma ).$$

Therefore

$$C_{{\ell }_p}^{*}(\Gamma )\stackrel{max}{\otimes }{C}_{{\ell }_p}^{*}(\Gamma )=C_{{\ell }_p}^{*}(\Gamma )\stackrel{p}{\otimes }{C}_{{\ell }_p}^{*}(\Gamma ).$$

Conversely, we suppose that

$$C_{{\ell }_p}^{*}(\Gamma )\stackrel{max}{\otimes }{C}_{{\ell }_p}^{*}(\Gamma )=C_{{\ell }_p}^{*}(\Gamma )\stackrel{p}{\otimes }{C}_{{\ell }_p}^{*}(\Gamma )=C_{{\ell }_p}^{*}(\Gamma \times \Gamma ).$$

Then $C_{{\ell }_p}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{{\ell }_p}^{*}(\Gamma )$ has a faithful ${\ell }_p(\Gamma \times \Gamma )$-representation $\pi :C_{{\ell }_p}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{{\ell }_p}^{*}(\Gamma )\to \mathcal{B}\left(\mathcal{H}\right)$ and by taking an infinite direct sum if necessary, we can assume $\pi \left(C_{{\ell }_p}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{{\ell }_p}^{*}(\Gamma )\right)$ contains no compact operators. By Glimm’s Lemma [6], for any state $\phi$ of $\pi \left(C_{{\ell }_p}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{{\ell }_p}^{*}(\Gamma )\right)$, there exist orthonormal vectors $v_n\in \mathcal{H}$ such that

$$⟨\pi \left(x\right)v_n|v_n⟩\to \phi \left(\pi \left(x\right)\right),\forall x\in C_{{\ell }_p}^{*}(\Gamma )\stackrel{max}{\otimes }{C}_{{\ell }_p}^{*}(\Gamma )=C_{{\ell }_p}^{*}(\Gamma \times \Gamma ).$$

Choose $\phi$ the trivial state, we have

$$⟨\pi \left(x\right)v_n|v_n⟩\to 1,\forall x\in C_{{\ell }_p}^{*}(\Gamma )\stackrel{max}{\otimes }{C}_{{\ell }_p}^{*}(\Gamma )=C_{{\ell }_p}^{*}(\Gamma \times \Gamma ).$$

In particular,

$$⟨{\pi }_{s,t}{v}_n|v_n⟩\to 1,\forall s,t\in \Gamma .$$

Since $\pi$ is a ${\ell }_p(\Gamma \times \Gamma )$-representation, we can approximate the $v_n$’s with vectors having associated matrix coefficients in ${\ell }_p(\Gamma \times \Gamma )$. Thus we may assume that ${\pi }_{v_n,v_n}\in {\ell }_p(\Gamma \times \Gamma )$ for each n, where ${\pi }_{v_n,v_n}(s,t)=⟨{\pi }_{s,t}{v}_n|v_n⟩$. Since ${\pi }_{v_n,v_n}$ are positive definite functions in ${\ell }_p(\Gamma \times \Gamma )$ tending pointwise to one, it follows from the Remark 2.13 in [5] that $\Gamma \times \Gamma$ is amenable and so is $\Gamma$. □

Theorem 2.

For $1\le p<+\infty$, $C_{{\ell }_p}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{{\ell }_p}^{*}(\Gamma )=C_{{\ell }_p}^{*}(\Gamma ){\displaystyle \stackrel{min}{\otimes }}{C}_{{\ell }_p}^{*}(\Gamma )$ if and only if Γ is amenable.

Proof. 

Suppose that $\Gamma$ is amenable, we have

$$C^{*}(\Gamma )=C_{{\ell }_p}^{*}(\Gamma )=C_{\lambda }^{*}(\Gamma )$$

and

$$C^{*}(\Gamma \times \Gamma )=C_{{\ell }_p}^{*}(\Gamma \times \Gamma )=C_{\lambda }^{*}(\Gamma \times \Gamma ).$$

Thus

$$C_{{\ell }_p}^{*}(\Gamma )\stackrel{max}{\otimes }{C}_{{\ell }_p}^{*}(\Gamma )=C^{*}(\Gamma )\stackrel{max}{\otimes }{C}^{*}(\Gamma )=C^{*}(\Gamma \times \Gamma )$$

and

$$C_{{\ell }_p}^{*}(\Gamma )\stackrel{min}{\otimes }{C}_{{\ell }_p}^{*}(\Gamma )=C_{\lambda }^{*}(\Gamma )\stackrel{min}{\otimes }{C}_{\lambda }^{*}(\Gamma )=C_{\lambda }^{*}(\Gamma \times \Gamma ).$$

Therefore

$$C_{{\ell }_p}^{*}(\Gamma )\stackrel{max}{\otimes }{C}_{{\ell }_p}^{*}(\Gamma )=C_{{\ell }_p}^{*}(\Gamma )\stackrel{min}{\otimes }{C}_{{\ell }_p}^{*}(\Gamma ).$$

Conversely, suppose that $C_{{\ell }_p}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{{\ell }_p}^{*}(\Gamma )=C_{{\ell }_p}^{*}(\Gamma ){\displaystyle \stackrel{min}{\otimes }}{C}_{{\ell }_p}^{*}(\Gamma )$. Since

$${\Vert ·\Vert }_{min}\le {\Vert ·\Vert }_{{\ell }_p}\le {\Vert ·\Vert }_{max}$$

on the algebraic tensor product $C_{{\ell }_p}^{*}(\Gamma )\odot C_{{\ell }_p}^{*}(\Gamma )$,

$$C_{{\ell }_p}^{*}(\Gamma )\stackrel{max}{\otimes }{C}_{{\ell }_p}^{*}(\Gamma )=C_{{\ell }_p}^{*}(\Gamma )\stackrel{p}{\otimes }{C}_{{\ell }_p}^{*}(\Gamma ).$$

It follows from Theorem 1 that $\Gamma$ is amenable. □

Corollary 1.

For free group ${\mathbb{F}}_n(2\le n\le +\infty )$, we have

$$C_{{\ell }_p}^{*}\left({\mathbb{F}}_n\right)\stackrel{max}{\otimes }{C}_{{\ell }_p}^{*}\left({\mathbb{F}}_n\right)\ne C_{{\ell }_p}^{*}\left({\mathbb{F}}_n\right)\stackrel{min}{\otimes }{C}_{{\ell }_p}^{*}\left({\mathbb{F}}_n\right)\forall 1\le p<+\infty .$$

It is well known that the famous QWEP conjecture is equivalent to

$$C^{*}\left({\mathbb{F}}_2\right)\stackrel{max}{\otimes }{C}^{*}\left({\mathbb{F}}_2\right)=C^{*}\left({\mathbb{F}}_2\right)\stackrel{min}{\otimes }{C}^{*}\left({\mathbb{F}}_2\right).$$

From Proposition 2.10 in [5], $C^{*}(\Gamma )=C_{{\ell }_{\infty }}^{*}(\Gamma )$. Compare with Corollary 1, maybe we can get some ideas about QWEP.

Definition 2.

For a discrete group Γ, we define

$$C_{c_0}^{*}(\Gamma )\stackrel{c_0}{\otimes }{C}_{c_0}^{*}(\Gamma )\triangleq C_{c_0}^{*}(\Gamma \times \Gamma ).$$

By a similar argument after Definition 1, we can show that Definition 2 is well defined also.

Theorem 3.

$C_{c_0}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{c_0}^{*}(\Gamma )=C_{c_0}^{*}(\Gamma ){\displaystyle \stackrel{c_0}{\otimes }}{C}_{c_0}^{*}(\Gamma )$ if and only if Γ has Haagerup property.

Proof. 

The proof is similar to the argument in Theorem 1. Suppose that $\Gamma$ has Haagerup property. It is well known that $\Gamma \times \Gamma$ also has Haagerup property. Thus it follows from Corollary 3.4 in [5] that we have

$$C_{c_0}^{*}(\Gamma )\stackrel{max}{\otimes }{C}_{c_0}^{*}(\Gamma )=C^{*}(\Gamma )\stackrel{max}{\otimes }{C}^{*}(\Gamma )=C^{*}(\Gamma \times \Gamma )$$

and

$$C_{c_0}^{*}(\Gamma )\stackrel{c_0}{\otimes }{C}_{c_0}^{*}(\Gamma )=C_{c_0}^{*}(\Gamma \times \Gamma )=C^{*}(\Gamma \times \Gamma ).$$

So $C_{c_0}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{c_0}^{*}(\Gamma )=C_{c_0}^{*}(\Gamma ){\displaystyle \stackrel{c_0}{\otimes }}{C}_{c_0}^{*}(\Gamma )$.

Conversely, suppose that

$$C_{c_0}^{*}(\Gamma )\stackrel{max}{\otimes }{C}_{c_0}^{*}(\Gamma )=C_{c_0}^{*}(\Gamma )\stackrel{c_0}{\otimes }{C}_{c_0}^{*}(\Gamma )=C_{c_0}^{*}(\Gamma \times \Gamma ).$$

Then $C_{c_0}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{c_0}^{*}(\Gamma )$ has a faithful $C_0(\Gamma \times \Gamma )$-representation

$$\pi :C_{c_0}^{*}(\Gamma )\stackrel{max}{\otimes }{C}_{c_0}^{*}(\Gamma )\to \mathcal{B}\left(\mathcal{H}\right)$$

and by taking an infinite direct sum if necessary, we can assume $\pi \left(C_{c_0}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{c_0}^{*}(\Gamma )\right)$ contains no compact operators. By Glimm’s Lemma [6], for any state $\phi$ of $\pi \left(C_{c_0}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{c_0}^{*}(\Gamma )\right)$, there exist orthonormal vectors $v_n\in \mathcal{H}$ such that

$$⟨\pi \left(x\right)v_n|v_n⟩\to \phi \left(\pi \left(x\right)\right),\forall x\in C_{c_0}^{*}(\Gamma )\stackrel{max}{\otimes }{C}_{c_0}^{*}(\Gamma )=C_{c_0}^{*}(\Gamma \times \Gamma ).$$

Choose $\phi$ the trivial state, we have

$$⟨\pi \left(x\right)v_n|v_n⟩\to 1,\forall x\in C_{c_0}^{*}(\Gamma )\stackrel{max}{\otimes }{C}_{c_0}^{*}(\Gamma )=C_{c_0}^{*}(\Gamma \times \Gamma ).$$

In particular,

$$⟨{\pi }_{s,t}{v}_n|v_n⟩\to 1,\forall s,t\in \Gamma .$$

Approximating the $v_n$’s with vectors having associated matrix coefficients in $c_0(\Gamma \times \Gamma )$, we may assume that ${\pi }_{v_n,v_n}\in c_0(\Gamma \times \Gamma )$ for each n. Therefore $\left\{{\pi }_{v_n,v_n}\right\}$ is a sequence of positive definite functions in $c_0(\Gamma \times \Gamma )$ tending pointwise to one, this implies that $\Gamma \times \Gamma$ has Haagerup property and so does $\Gamma$. □

Corollary 2.

If $C_{c_0}^{*}(\Gamma ){\displaystyle \stackrel{max}{\otimes }}{C}_{c_0}^{*}(\Gamma )=C_{c_0}^{*}(\Gamma ){\displaystyle \stackrel{min}{\otimes }}{C}_{c_0}^{*}(\Gamma )$, then Γ has Haagerup property.

3. P-Tensor Product on ${\mathbb{F}}_2$

In this section, we mainly consider the p-tensor product ${\displaystyle \stackrel{p}{\otimes }}$ on the free group with two generators ${\mathbb{F}}_2$.

We recall that a function $\phi$: $\Gamma \to C$ is said to be positive definite if the matrix

$${\left[\phi \left(s^{-1}t\right)\right]}_{s,t\in \mathbb{F}}\in M_{\mathbb{F}}\left(C\right)$$

is positive for every finite set $\mathbb{F}\subset \Gamma$.

Proposition 1.

Let ${\mathbb{F}}_2$ be the free group with two generators. Then there exists a $p\in (2,\infty )$ such that

$$C^{*}\left({\mathbb{F}}_2\right)\stackrel{max}{\otimes }{C}^{*}\left({\mathbb{F}}_2\right)\ne C_{{\ell }_p}^{*}\left({\mathbb{F}}_2\right)\stackrel{p}{\otimes }{C}_{{\ell }_p}^{*}\left({\mathbb{F}}_2\right)\ne C_{\lambda }^{*}\left({\mathbb{F}}_2\right)\stackrel{min}{\otimes }{C}_{\lambda }^{*}\left({\mathbb{F}}_2\right).$$

Proof. 

Since ${\mathbb{F}}_2\times {\mathbb{F}}_2$ is not amenable, by Prop 2.12 in [5]$C^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)\ne C_{{\ell }_p}^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$ for any $p\in \left[1,+\infty \right)$. Since $C^{*}\left({\mathbb{F}}_2\right){\displaystyle \stackrel{max}{\otimes }}{C}^{*}\left({\mathbb{F}}_2\right)=C^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$ and $C_{{\ell }_p}^{*}\left({\mathbb{F}}_2\right){\displaystyle \stackrel{p}{\otimes }}{C}_{{\ell }_p}^{*}\left({\mathbb{F}}_2\right)=C_{{\ell }_p}^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$, we have for any $p\in \left[1,+\infty \right)$$C^{*}\left({\mathbb{F}}_2\right){\displaystyle \stackrel{max}{\otimes }}{C}^{*}\left({\mathbb{F}}_2\right)\ne C_{{\ell }_p}^{*}\left({\mathbb{F}}_2\right){\displaystyle \stackrel{p}{\otimes }}{C}_{{\ell }_p}^{*}\left({\mathbb{F}}_2\right)$.

Since $C_{{\ell }_p}^{*}\left({\mathbb{F}}_2\right){\displaystyle \stackrel{p}{\otimes }}{C}_{{\ell }_p}^{*}\left({\mathbb{F}}_2\right)=C_{{\ell }_p}^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$ and $C_{\lambda }^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)=C_{\lambda }^{*}\left({\mathbb{F}}_2\right){\displaystyle \stackrel{min}{\otimes }}{C}_{\lambda }^{*}\left({\mathbb{F}}_2\right)$, we only need to find some $p\in \left(2,+\infty \right)$ with $C_{{\ell }_p}^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)\ne C_{\lambda }^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$. Let $S=\left\{a,b,a^{-1},b^{-1}\right\}\subseteq {\mathbb{F}}_2$ be the standard generating set and let $\left|·\right|$ denote the corresponding word length. A well known result of [7] states that for every $n\in \mathbb{N}$,

$$h_n\left(s\right):=e^{-\frac{\left|s\right|}{n}}$$

is positive definite function on ${\mathbb{F}}_2$ and clearly $h_n\to 1$ pointwise. Now for $\left(s,t\right)\in {\mathbb{F}}_2\times {\mathbb{F}}_2$, we define

$${\phi }_n\left(\left(s,t\right)\right):=h_n\left(s\right)=e^{-\frac{\left|s\right|}{n}}$$

and

$${\psi }_n\left(\left(s,t\right)\right):=h_n\left(t\right)=e^{-\frac{\left|t\right|}{n}}.$$

For any ${\alpha }_i\in \mathbb{C}$ and $\left(s_i,t_i\right)\in {\mathbb{F}}_2\times {\mathbb{F}}_2$,$1\le i\le n$, we have

$$\begin{array}{c}{\displaystyle \sum _{i,j}}{\alpha }_i{\overline{\alpha }}_j{\phi }_n({\left(s_j,t_j\right)}^{-1}\left(s_i,t_i\right))\hfill \\ ={\displaystyle \sum _{i,j}}{\alpha }_i{\overline{\alpha }}_j{\phi }_n(\left(s_j^{-1}{s}_i,t_j^{-1}{t}_i\right))\hfill \\ ={\displaystyle \sum _{i,j}}{\alpha }_i{\overline{\alpha }}_j{h}_n\left(s_j^{-1}{s}_i\right)\ge 0.\hfill \end{array}$$

So each ${\phi }_n$ is a positive definite function on ${\mathbb{F}}_2\times {\mathbb{F}}_2$, (Similarly ${\psi }_n$ is a positive definite function). Fixing n, we have ${\phi }_n\in {\ell }_{p_n}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$ for sufficiently large $p_n$. Let ${\pi }_n:C_{{\ell }_{p_n}}^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)\to B\left({\mathcal{H}}_n\right)$ be the GNS presentations related to ${\phi }_n$, and let ${\xi }_n\in {\mathcal{H}}_n$ be the canonical cyclic vector. Since ${\phi }_n\left(\left(s,t\right)\right)\to 1$, we see that $∥{\pi }_n\left(\left(s,t\right)\right){\xi }_n-{\xi }_n∥\to 0$ for all $\left(s,t\right)\in {\mathbb{F}}_2\times {\mathbb{F}}_2$. Hence the trivial representation is contained in the direct sum representation $\oplus {\pi }_n$ weakly. If for each n, $C_{{\ell }_{p_n}}^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)=C_{\lambda }^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right),\oplus {\pi }_n$ would be defined on $C_{\lambda }^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$. Since ${\mathbb{F}}_2\times {\mathbb{F}}_2$ is not amenable, the trivial representation cannot be contained in any representation of $C_{\lambda }^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$ weakly. This is a contradiction. Therefore for some n, $C_{{\ell }_{p_n}}^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)\ne C_{\lambda }^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$ □

In the paper [8], Okayasu give a characterization of positive definite functions on a free group with finite generators, which can be extended to the positive linear functionals on the free group $C^{*}$-algebras associated with the ideal ${\ell }_p$. This is a generalization of Haagerup’s famous characterization for the case of the reduced free group $C^{*}$-algebra. The strategy in these two papers also works for the group ${\mathbb{F}}_2\times {\mathbb{F}}_2$.

For non negative integers $k_1,k_2$, we define

$$W_{(k_1,k_2)}=\{(s_1,s_2)\in {\mathbb{F}}_2\times {\mathbb{F}}_2\left|\right|s_1|=k_1\mathrm{and}|s_2|=k_2\}.$$

${\chi }_{(k_1,k_2)}$ denotes the characteristic function on $W_{(k_1,k_2)}$.

Lemma 1.

Let $q\in [1,2]$. Let $k_i,l_i$ and $m_i(i=1,2)$ be non-negative integers. Let f and g be functions on ${\mathbb{F}}_2\times {\mathbb{F}}_2$ such that $suppf\subseteq W_{(k_1,k_2)}$ and $suppg\subseteq W_{(l_1,l_2)}$ respectively. If $|k_i-l_i|\le m_i\le k_i+l_i$ and $k_i+l_i-m_i$ is even, then

$$|(f*g)·{\chi }_{(m_1,m_2)}{|}_q\le {\left|f\right|}_q·{\left|g\right|}_q$$

and if $(m_1,m_2)$ is any other values, then

$$|(f*g)·{\chi }_{(m_1,m_2)}{|}_q=0.$$

Proof. 

Note that

$$\begin{array}{lll}\left(f*g\right)\left(\left(s_1,s_2\right)\right)& =& {\displaystyle \sum _{\genfrac{}{}{0pt}{}{\left(t_1,t_2\right),\left(u_1,u_2\right)\in {\mathbb{F}}_2\times {\mathbb{F}}_2\hfill }{s_i=t_i{u}_i\hfill }}}f\left(\left(t_1,t_2\right)\right)·g\left(\left(u_1,u_2\right)\right)\\ & =& {\displaystyle \sum _{\genfrac{}{}{0pt}{}{\left|t_i\right|=k_i,\left|u_i\right|=l_i\hfill }{s_i=t_i{u}_i\hfill }}}f\left(\left(t_1,t_2\right)\right)·g\left(\left(u_1,u_2\right)\right).\end{array}$$

Since the possible values of $\left|t_i{u}_i\right|$ are $\left|k_i-l_i\right|,\left|k_i-l_i\right|+2,\dots ,k_i+l_i$, we have

$${\left|\left(f*g\right){\chi }_{\left(m_1,m_2\right)}\right|}_q=0$$

for any other $(m_1,m_2)$. We only consider the $q\ne 1$ ($q=1$ is similar and trivial). First, we assume that $m_i=k_i+l_i(i=1,2)$. In this case, if $\left|s_i\right|=m_i$, then $s_i$ can be uniquely written as a product $t_i{u}_i$ with $t_i=\left|k_i\right|$ and $u_i=\left|l_i\right|$. Hence

$$\left(f*g\right)\left(\left(s_1,s_2\right)\right)=f\left(\left(t_1,t_2\right)\right)·g\left(\left(u_1,u_2\right)\right).$$

Therefore

$$\begin{array}{lll}{\left|\left(f*g\right){\chi }_{\left(m_1,m_2\right)}\right|}_q^q& =& {\displaystyle \sum _{\left(s_1,s_2\right)\in {\mathbb{F}}_2\times {\mathbb{F}}_2}}|\left(f*g\right)\left(\left(s_1,s_2\right)\right)·{\chi }_{\left(m_1,m_2\right)}\left(\left(s_1,s_2\right)\right){|}^q\\ & =& {\displaystyle \sum _{\genfrac{}{}{0pt}{}{\left|t_i{u}_i\right|=k_i+l_i\hfill }{\left|t_i\right|=k_i,\left|u_i\right|=l_i\hfill }}}{\left|f\left(\left(t_1,t_2\right)\right)\right|}^q·{\left|g\left(\left(u_1,u_2\right)\right)\right|}^q\\ & \le & {\displaystyle \sum _{\left|t_i\right|=k_i,\left|u_i\right|=l_i}}{\left|f\left(\left(t_1,t_2\right)\right)\right|}^q·{\left|g\left(\left(u_1,u_2\right)\right)\right|}^q\\ & =& {\left|f\right|}_q^q·{\left|g\right|}_q^q.\end{array}$$

Next we assume that $m_i=\left|k_i-l_i\right|,\dots ,k_i+l_i-2$. In these cases, we have $m_i=k_i+l_i-2j_i$, for $1\le j_i\le min\left(k_i,l_i\right),(i=1,2)$. Let $s_i=t_i{u}_i$ with $\left|s_i\right|=m_i$,$\left|t_i\right|=k_i$, and $|u_i|=l_i$. Then $s_i$ can be uniquely written as a product ${t^{\prime }}_i{u^{\prime }}_i$ such that $t_i={t^{\prime }}_i{v}_i,u_i=v_i^{-1}{u^{\prime }}_i$ with $\left|{t^{\prime }}_i\right|=k_i-j_i,\left|{u^{\prime }}_i\right|=l_i-j_i$, and $\left|v_i\right|=\left|v_i^{-1}\right|=j_i$. We define

$$\begin{array}{c}{f}^{\prime }\left(\left(t_1,t_2\right)\right)={\left({\displaystyle \sum _{\left|v_i\right|=j_i}}{\left|f\left(\left(t_1{v}_1,t_2{v}_2\right)\right)\right|}^q\right)}^{\frac{1}{q}},if\left|t_i\right|=k_i-j_i,\hfill \end{array}$$

and $f^{\prime }\left(\left(t_1,t_2\right)\right)=0$ otherwise. Similarly, we define

$$\begin{array}{c}{g}^{\prime }\left(\left(u_1,u_2\right)\right)={\left({{\displaystyle \sum _{\left|v_i\right|=j_i}}\left|g\left(\left(v_1^{-1}{u}_1,v_2^{-1}{u}_2\right)\right)\right|}^q\right)}^{\frac{1}{q}},if\left|u_i\right|=l_i-j_i,\hfill \end{array}$$

and $g^{\prime }\left(\left(u_1,u_2\right)\right)=0$, otherwise. Note that $\mathrm{supp}{f}^{\prime }\subseteq W_{\left(k_1-j_1,k_2-j_2\right)}$, and $\mathrm{supp}{g}^{\prime }\subseteq W_{\left(l_1-j_1,l_2-j_2\right)}$. Moreover,

$${\left|f^{\prime }\right|}_q^q=\sum _{\left|t_i\right|=k_i-j_i}\left({\sum _{\left|v_i\right|=j_i}\left|f\left(\left(t_1{v}_1,t_2{v}_2\right)\right)\right|}^q\right)={\left|f\right|}_q^q,$$

and ${\left|g^{\prime }\right|}_q^q={\left|g\right|}_q^q$. Take a real number p with $\frac{1}{p}+\frac{1}{q}=1$. Since $1<q\le 2,2\le p<+\infty$, so $q\le p$ in general. Owing to Hölder inequality, we have

$$\begin{array}{lll}\left|\left(f*g\right)\left(\left(s_1,s_2\right)\right)\right|& =& \left|{\displaystyle \sum _{\genfrac{}{}{0pt}{}{\left|t_i\right|=k_i,\left|u_i\right|=l_i\hfill }{s_i=t_i{u}_i\hfill }}}f\left(\left(t_1,t_2\right)\right)·g\left(\left(u_1,u_2\right)\right)\right|\\ & =& \left|{\displaystyle \sum _{\left|v_i\right|=j_i}}f\left(\left({t^{\prime }}_1{v}_1,{t^{\prime }}_2{v}_2\right)\right)·g\left(\left(v_1^{-1}{u^{\prime }}_1,v_2^{-1}{u^{\prime }}_2\right)\right)\right|\\ & \le & {\left|{\displaystyle \sum _{\left|v_i\right|=j_i}}{\left|f\left(\left({t^{\prime }}_1{v}_1,{t^{\prime }}_2{v}_2\right)\right)\right|}^q\right|}^{\frac{1}{q}}·{\left|\sum _{\left|v_i\right|=j_i}{\left|g\left(\left(v_1^{-1}{u^{\prime }}_1,v_2^{-1}{u^{\prime }}_2\right)\right)\right|}^p\right|}^{\frac{1}{p}}\\ & \le & {\left|{\displaystyle \sum _{\left|v_i\right|=j_i}}{\left|f\left(\left({t^{\prime }}_1{v}_1,{t^{\prime }}_2{v}_2\right)\right)\right|}^q\right|}^{\frac{1}{q}}·{\left|\sum _{\left|v_i\right|=j_i}{\left|g\left(\left(v_1^{-1}{u^{\prime }}_1,v_2^{-1}{u^{\prime }}_2\right)\right)\right|}^q\right|}^{\frac{1}{q}}\\ & =& f^{\prime }\left({t^{\prime }}_1,{t^{\prime }}_2\right)·g\left({u^{\prime }}_1,{u^{\prime }}_2\right)=\left(f^{\prime }*g^{\prime }\right)\left(\left(s_1,s_2\right)\right),\end{array}$$

where $s_i={t^{\prime }}_i{u^{\prime }}_i$ and $|s_i|=k_i+l_i-2j_i=|{t^{\prime }}_i|+|{u^{\prime }}_i|$. Therefore, $\left|\left(f*g\right)·{\chi }_{\left(m_1,m_2\right)}\right|\le \left(f^{\prime }*g^{\prime }\right)·{\chi }_{\left(m_1,m_2\right)}$. Since $(k_i-j_i)+(l_i-j_i)=m_i$, it follows from the first part of the proof that

$$\begin{array}{lll}\left|\left(f*g\right){\chi }_{\left(m_1,m_2\right)}\right|& \le & \left|\left(f^{\prime }*g^{\prime }\right)·{\chi }_{\left(m_1,m_2\right)}\right|\\ & \le & {\left|f^{\prime }\right|}_q·{\left|g^{\prime }\right|}_q\\ & =& {\left|f\right|}_q·{\left|g\right|}_q.\end{array}$$

At last, we assume that $m_1=k_1+l_1$ and $m_2=|k_2-l_2|,\dots ,k_2+l_2-2$; or $m_1=|k_1-l_1|,\dots ,k_1+l_1-2$ and $m_2=k_2+l_2$. We only need to consider the first case.In this case, $m_1=k_1+l_1$, and $m_2=k_2+l_2-2j_2$ for $1\le j_2\le min(k_2,l_2)$. Then $s_1$ can be uniquely written as a product $t_1{u}_1$ with $|t_1|=k_1$ and $|u_1|=l_1$. Let $s_2=t_2{u}_2$ with $|s_2|=m_2$, $|t_2|=k_2,\left|u_2\right|=l_2$. Then $s_2$ can be uniquely written as a product ${t^{\prime }}_2{u^{\prime }}_2$ such that $t_2={t^{\prime }}_2{v}_2,u_2=v_2^{-1}{u^{\prime }}_2$, with $|{t^{\prime }}_2|=k_2-j_2,\left|{u^{\prime }}_2\right|=l_2-j_2$ and $|v_2|=|v_2^{-1}|=j_2$. The following proof is almost the same as the second part with $j_1=0$. □

Lemma 2.

Let $k_1,k_2$ be non-negative integers. Let $1\le q\le p\le \infty$ with $\frac{1}{p}+\frac{1}{q}=1$. If a unitary representation $\pi :{\mathbb{F}}_2\times {\mathbb{F}}_2\to \mathcal{U}\left(\mathcal{H}\right)$ has a cyclic vector ξ such that ${\pi }_{\xi ,\xi }\in {\ell }_p({\mathbb{F}}_2\times {\mathbb{F}}_2)$ then

$$\Vert \pi \left(f\right)\Vert \le (k_1+k_2+2)·{\left|f\right|}_q,$$

for $f\in C_c({\mathbb{F}}_2\times {\mathbb{F}}_2)$ with $suppf\subseteq W_{(k_1,k_2)}$.

Proof. 

We only consider $1\le q\le 2$ and $2\le p\le +\infty$ with $\frac{1}{p}+\frac{1}{q}=1$. We consider the norm ${\left|{\left(f^{*}*f\right)}^{\left(*2n\right)}\right|}_q$. Write $f_{2j-1}=f^{*}$ and $f_{2j}=f$ for $j=1,\dots ,2n$. Then

$${\left(f^{*}*f\right)}^{\left(*2n\right)}=f_1*f_2*\cdots *f_{4n}.$$

We also denote $g=f_2*\cdots *f_{4n}$. So we have

$${\left(f^{*}*f\right)}^{\left(*2n\right)}=f_1*g.$$

Since $f^{*}\left((s_1,s_2)\right)=\overline{f}\left((s_1^{-1},s_2^{-1})\right)$, $\mathrm{supp}{f}_j\subseteq W_{\left(k_1,k_2\right)}$, for $j=1,2,\dots ,4n$ and $g\in c_c\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$. Put $g_{\left(l_1,l_2\right)}=g{\mathcal{X}}_{\left(l_1,l_2\right)}$. Then $\mathrm{supp}{g}_{\left(l_1,l_2\right)}\subseteq W_{\left(l_1,l_2\right)}$ and

$${\left|g\right|}_q^q=\sum _{l_1,l_2=0}^{+\infty }{\left|g_{\left(l_1,l_2\right)}\right|}_q^q.$$

Clearly, ${\left|g_{\left(l_1,l_2\right)}\right|}_q=0$ for all but finitely many $l_1,l_2$. Moreover set

$$h=f_1*g=\sum _{l_1,l_2=0}^{+\infty }{f}_1*g_{\left(l_1,l_2\right)},$$

and $h_{\left(m_1,m_2\right)}=h{\chi }_{\left(m_1,m_2\right)}$. Then $h\in c_c({\mathbb{F}}_2\times {\mathbb{F}}_2)$ and

$${\left|h\right|}_q^q=\sum _{m_1,m_2=0}^{+\infty }{\left|h_{\left(m_1,m_2\right)}\right|}_q^q.$$

${\left|h_{\left(m_1,m_2\right)}\right|}_q=0$ for all but finitely many $m_1,m_2$. By Lemma 1,

$${\left|\left(f_1*g_{\left(l_1,l_2\right)}\right)·{\chi }_{\left(m_1,m_2\right)}\right|}_q\le {\left|f_1\right|}_q·{\left|g_{\left(l_1,l_2\right)}\right|}_q$$

in the case where $|k_i-l_i|\le m_i\le k_i+l_i$, and $k_i+l_i-m_i$ is even, and $|(f_1*g)·{\chi }_{\left(m_1,m_2\right)}{|}_q=0$ for any other values of $m_i$. Hence,

$$\begin{array}{lll}{\left|h_{\left(m_1,m_2\right)}\right|}_q& =& {\left|{\displaystyle \sum _{l_1,l_2=0}^{+\infty }}\left(f_1*g_{\left(l_1,l_2\right)}\right)·{\chi }_{\left(m_1,m_2\right)}\right|}_q\\ & \le & {\displaystyle \sum _{l_1,l_2=0}^{+\infty }}{\left|\left(f_1*g_{\left(l_1,l_2\right)}\right)·{\chi }_{\left(m_1,m_2\right)}\right|}_q\\ & \le & {\left|f_1\right|}_q·{\displaystyle \sum _{\genfrac{}{}{0pt}{}{l_i=\left|m_i-k_i\right|\hfill }{m_i+k_i-l_{i}even\hfill }}^{m_i+k_i}}{\left|g_{\left(l_1,l_2\right)}\right|}_q.\end{array}$$

By writing $l_i=m_i+k_i-2j_i$, we have

$$\begin{array}{lll}{\left|h\left(m_1,m_2\right)\right|}_q& \le & {\left|f_1\right|}_q·{\displaystyle \sum _{j_1=0}^{min\left(m_1,k_1\right)}}{\displaystyle \sum _{j_2=0}^{min\left(m_2,k_2\right)}}{\left|g_{\left(m_1+k_1-2j_1,m_2+k_2-2j_2\right)}\right|}_q\\ \le & {\left|f_1\right|}_q·{\left({\displaystyle \sum _{j_1,j_2}}{\left|g_{\left(m_1+k_1-2j_1,m_2+k_2-2j_2\right)}\right|}_q^q\right)}^{\frac{1}{q}}·{\left({\displaystyle \sum _{j_1,j_2}}{1}^p\right)}^{\frac{1}{p}}\\ & \le & {\left(k_1+k_2+2\right)}^{\frac{1}{p}}·{\left|f_1\right|}_q·{\left({\displaystyle \sum _{j_1,j_2}}{\left|g_{\left(m_1+k_1-2j_1,m_2+k_2-2j_2\right)}\right|}_q^q\right)}^{\frac{1}{q}}.\end{array}$$

Therefore,

$$\begin{array}{lll}{\left|h\right|}_q^q& =& {\displaystyle \sum _{m_1,m_2=0}^{+\infty }}{\left|h\left(m_1,m_2\right)\right|}_q^q\\ & \le & {\left(k_1+k_2+2\right)}^{\frac{q}{p}}·{\left|f_1\right|}_q^q·{\displaystyle \sum _{m_1,m_2=0}^{+\infty }}{\displaystyle \sum _{j_1=0}^{min\left(m_1,k_1\right)}}{\displaystyle \sum _{j_2=0}^{min\left(m_2,k_2\right)}}{\left|g_{\left(m_1+k_1-2j_1,m_2+k_2-2j_2\right)}\right|}_q^q\\ & =& {\left(k_1+k_2+2\right)}^{\frac{q}{p}}·{\left|f_1\right|}_q^q·{\displaystyle \sum _{j_1=0}^{k_1}}{\displaystyle \sum _{m_1=j_1}^{+\infty }}{\displaystyle \sum _{j_2=0}^{k_2}}{\displaystyle \sum _{m_2=j_2}^{+\infty }}{\left|g_{\left(m_1+k_1-2j_1,m_2+k_2-2j_2\right)}\right|}_q^q\\ & =& {\left(k_1+k_2+2\right)}^{\frac{q}{p}}·{\left|f_1\right|}_q^q·{\displaystyle \sum _{j_1=0}^{k_1}}{\displaystyle \sum _{l_1=k_1-j_1}^{+\infty }}{\displaystyle \sum _{j_2=0}^{+\infty }}{\displaystyle \sum _{l_2=k_2-j_2}^{+\infty }}{\left|g_{\left(l_1{l}_2\right)}\right|}_q^q\\ & \le & {\left(k_1+k_2+2\right)}^{\frac{q}{p}}·{\left|f_1\right|}_q^q·{\displaystyle \sum _{j_1=0}^{k_1}}{\displaystyle \sum _{j_2=0}^{k_2}}{\left|g\right|}_q^q\\ & \le & {\left(k_1+k_2+2\right)}^{\frac{q}{p}+1}·{\left|f_1\right|}_q^q·{\left|g\right|}_q^q\end{array}$$

Hence ${\left|f_1*g\right|}_q\le \left(k_1+k_2+2\right)·{\left|f_1\right|}_q·{\left|g\right|}_q$, i.e.

$${\left|f_1*f_2*\cdots *f_{4n}\right|}_q\le \left(k_1+k_2+2\right)·{\left|f_1\right|}_q·{\left|f_2*\cdots *f_{4n}\right|}_q.$$

Inductively we have

$${\left|{\left(f^{*}*f\right)}^{*2n}\right|}_q\le {\left(k_1+k_2+2\right)}^{4n-1}·{\left|f\right|}_q^{4n}.$$

Therefore, it follows from Lemma 3.2 in [8] that

$$∥\pi \left(f\right)∥\le \underset{n\to +\infty }{liminf}{\left|{\left(f^{*}*f\right)}^{*2n}\right|}_q^{\frac{1}{4n}}\le \left(k_1+k_2+2\right)·{\left|f\right|}_q.$$

□

Theorem 4.

Let $2\le p<\infty$. Let φ be a positve definite function on ${\mathbb{F}}_2\times {\mathbb{F}}_2$. Then the following conditions are equivalent:

(1) φ can be extended to the positive linear functional on $C_{{\ell }_p}^{*}({\mathbb{F}}_2\times {\mathbb{F}}_2)$;

(2) $\underset{k_1,k_2}{sup}{|\phi ·{\chi }_{(k_1,k_2)}|}_p·{(k_1+k_2+2)}^{-1}<\infty$;

(3) The function $(s_1,s_2)\to \phi (s_1,s_2)·(2+|s_1|+|s_2{\left|\right)}^{-1-\frac{2}{p}}$ belongs to ${\ell }_p({\mathbb{F}}_2\times {\mathbb{F}}_2)$;

(4) For any $\alpha \in (0,1)$, the function $(s_1,s_2)\to \phi (s_1,s_2)·{\alpha }^{|s_1|+|s_2|}$ belongs to ${\ell }_p({\mathbb{F}}_2\times {\mathbb{F}}_2)$.

Proof. 

We assume that $\phi \left(\left(e,e\right)\right)=1$.

(1)⇒(2) It follows from (1) that $w_{\phi }$ extends to the station $C_{{\ell }_p}^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$, where

$${\omega }_{\phi }\left(f\right)=\sum _{\left(s_1,s_2\right)\in {\mathbb{F}}_2\times {\mathbb{F}}_2}f\left(\left(s_1,s_2\right)\right)·\phi \left(\left(s_1,s_2\right)\right)\mathrm{for}f\in c_c\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$$

Hence, for $f\in c_c\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$, we have

$$\left|{\omega }_{\phi }\left(f\right)\right|\le {∥f∥}_{{\ell }_p}.$$

Set $f={\left|\phi \right|}^{p-2}·\overline{\phi }·{\chi }_{\left(k_1,k_2\right)}$.

Then

$$\begin{array}{lll}\left|{\omega }_{\phi }\left(f\right)\right|& =& {\displaystyle \sum _{\left(s_1,s_2\right)\in {\mathbb{F}}_2\times {\mathbb{F}}_2}}\phi \left(\left(s_1,s_2\right)\right)·f\left(\left(s_1,s_2\right)\right)\\ & =& {\displaystyle \sum _{\left(s_1,s_2\right)\in {\mathbb{F}}_2\times {\mathbb{F}}_2}}\phi \left(\left(s_1,s_2\right)\right)·{\left|\phi \right|}^{p-2}\left(\left(s_1,s_2\right)\right)·\overline{\phi }\left(\left(s_1,s_2\right)\right)·{\chi }_{\left(k_1,k_2\right)}\left(\left(s_1,s_2\right)\right)\\ & =& {\displaystyle \sum _{\left(s_1,s_2\right)\in {\mathbb{F}}_2\times {\mathbb{F}}_2}}{\left|\phi \right|}^p\left(\left(s_1,s_2\right)\right)·{\chi }_{\left(k_1,k_2\right)}\left(\left(s_1,s_2\right)\right)\\ & =& {\left|\phi ·{\chi }_{\left(k_1,k_2\right)}\right|}_p^p.\end{array}$$

Let $\pi :{\mathbb{F}}_2\times {\mathbb{F}}_2\to U\left(\mathcal{H}\right)$ be an ${\ell }_p-$representation with a dense subspace ${\mathcal{H}}_0$, then

$${∥\pi \left(f\right)∥}^2=\underset{\xi \in H_0,∥\xi ∥=1}{sup}{⟨\pi \left(f^{*}*f\right)\xi \left|\xi \right.⟩}_{\mathcal{H}}.$$

Fix $\xi \in {\mathcal{H}}_0$ with $∥\xi ∥=1$. We denote by $\sigma$ the restriction of $\pi$ onto the subspace

$${\mathcal{H}}_{\sigma }=\overline{span}\left\{\pi \left(\left(s_1,s_2\right)\right)\xi \left|\left(s_1,s_2\right)\in {\mathbb{F}}_2\times {\mathbb{F}}_2\right.\right\}\subseteq \mathcal{H}.$$

Then

$${⟨\pi \left(f^{*}*f\right)\xi \left|\xi \right.⟩}_{\mathcal{H}}={⟨\sigma \left(f^{*}*f\right)\xi \left|\xi \right.⟩}_{{\mathcal{H}}_{\sigma }}.$$

Note that $\xi$ is cyclic for $\sigma$ such that ${\sigma }_{\xi ,\xi }\in {\ell }_p\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$. Take a real number q with $\frac{1}{p}+\frac{1}{q}=1$. Since $2\le p<+\infty$, we have $1<q\le 2$. Since $\mathrm{supp}f\subseteq W_{\left(k_1,k_2\right)}$, it follows the Lemma 2 that

$$∥\sigma \left(f\right)∥\le \left(k_1+k_2+2\right)·{\left|f\right|}_q.$$

Hence

$$∥\sigma \left(f^{*}*f\right)∥={∥\sigma \left(f\right)∥}^2\le {\left(k_1+k_2+2\right)}^2·{\left|f\right|}_q^2.$$

Therefore,

$$\begin{array}{lll}{∥f∥}_{l_p}^2& =& sup\left\{\left.∥\pi \left(f\right)∥\right|\pi isan{\ell }_p-representation\right\}\\ & \le & \left(k_1+k_2+2\right)·{\left|f\right|}_q\\ & =& \left(k_1+k_2+2\right)·{\left({\displaystyle \sum _{\left(s_1,s_2\right)\in {\mathbb{F}}_2\times {\mathbb{F}}_2}}{\left|f\left(\left(s_1,s_2\right)\right)\right|}^q\right)}^{\frac{1}{q}}\\ & =& \left(k_1+k_2+2\right)·{\left({\displaystyle \sum _{\left(s_1,s_2\right)\in {\mathbb{F}}_2\times {\mathbb{F}}_2}}{\left|\phi \right|}^{\left(p-1\right)q}\left(\left(s_1,s_2\right)\right)·{\chi }_{\left(k_1,k_2\right)}\left(\left(s_1,s_2\right)\right)\right)}^{\frac{1}{q}}\\ & =& \left(k_1+k_2+2\right)·{\left({\displaystyle \sum _{\left(s_1,s_2\right)\in {\mathbb{F}}_2\times {\mathbb{F}}_2}}{\left|\phi \right|}^p\left(\left(s_1,s_2\right)\right)·{\chi }_{\left(k_1,k_2\right)}\left(\left(s_1,s_2\right)\right)\right)}^{\frac{1}{q}}\\ & =& \left(k_1+k_2+2\right)·{\left|\phi ·{\chi }_{\left(k_1,k_2\right)}\right|}_p^{\frac{p}{q}}\\ & =& \left(k_1+k_2+2\right)·{\left|\phi ·{\chi }_{\left(k_1,k_2\right)}\right|}_p^{p-1}.\end{array}$$

Since $f={\left|\phi \right|}^{p-2}·\overline{\phi }·{\chi }_{\left(k_1,k_2\right)}$, we have

$$\begin{array}{lll}\left|{\omega }_{\phi }\left(f\right)\right|& =& {\left|\phi ·{\chi }_{\left(k_1,k_2\right)}\right|}_p^p\\ & \le & {∥f∥}_{{\ell }_p}\\ & \le & \left(k_1+k_2+2\right)·{\left|\phi ·{\chi }_{\left(k_1,k_2\right)}\right|}_p^{p-1}.\end{array}$$

Consequently,

$$\left|\phi ·{\chi }_{\left(k_1,k_2\right)}\right|\le k_1+k_2+2.$$

(2)⇒(3)

$$\begin{array}{c}{\displaystyle \sum _{\left(s_1,s_2\right)\in {\mathbb{F}}_2\times {\mathbb{F}}_2}}{\left|\phi \left(\left(s_1,s_2\right)\right)\right|}^p·{\left(2+\left|s_1\right|+\left|s_2\right|\right)}^{-p-2}\hfill \\ ={\displaystyle \sum _{k_1,k_2=0}^{+\infty }}{\displaystyle \sum _{\genfrac{}{}{0pt}{}{\left|s_1\right|=k_1\hfill }{\left|s_2\right|=k_2\hfill }}}{\left|\phi \left(\left(s_1,s_2\right)\right)\right|}^p·{\left(2+k_1+k_2\right)}^{-p-2}\hfill \\ ={\displaystyle \sum _{k_1,k_2=0}^{+\infty }}{\left|\phi ·{\chi }_{\left(k_1,k_2\right)}\right|}_p^p·{\left(2+k_1+k_2\right)}^{-p}·{\left(2+k_1+k_2\right)}^{-2}\hfill \\ \le {\left\{\underset{k_1,k_2}{sup}{\left|\phi ·{\chi }_{\left(k_1,k_2\right)}\right|}_p·{\left(2+k_1+k_2\right)}^{-1}\right\}}^p·{\displaystyle \sum _{k_1,k_2=0}^{+\infty }}{\left(2+k_1+k_2\right)}^{-2}<+\infty .\hfill \\ \hfill \end{array}$$

(3)⇒(4) Obviously.

(4)⇒(1) Set

$${\phi }_{\alpha }:\left(s_1,s_2\right)\mapsto {\alpha }^{\left|s_1\right|},{\psi }_{\alpha }:\left(s_1,s_2\right)\mapsto {\alpha }^{\left|s_2\right|}\mathrm{from}{\mathbb{F}}_2\times {\mathbb{F}}_2\to \mathbb{R}.$$

For any $a_i\in \mathbb{C}$ and $\left(s_{1i},s_{2i}\right)\in {\mathbb{F}}_2\times {\mathbb{F}}_2$, we have

$$\begin{array}{lll}\sum a_i{\overline{a}}_j{\phi }_{\alpha }\left({\left(s_{1j},s_{2j}\right)}^{-1}·\left(s_{1i},s_{2i}\right)\right)& =& \sum a_i{\overline{a}}_j{\phi }_{\alpha }\left(s_{1j}^{-1}{s}_{1i},s_{2j}^{-1}{s}_{2i}\right)\\ & =& \sum a_i{\overline{a}}_j{\alpha }^{\left|s_{1j}^{-1}{s}_{1i}\right|}\ge 0.\end{array}$$

So ${\phi }_{\alpha }$ and similarly ${\psi }_{\alpha }$ are positive definite functions on ${\mathbb{F}}_2\times {\mathbb{F}}_2$. This implies that the function

$${\Phi }_{\alpha }\left((s_1,s_2\right))\stackrel{\Delta }{=}{\alpha }^{\left|s_1\right|+\left|s_2\right|}={\alpha }^{\left|s_1\right|}{\alpha }^{\left|s_2\right|}={\phi }_{\alpha }\left(\left(s_1,s_2\right)\right){\psi }_{\alpha }\left(\left(s_1,s_2\right)\right)$$

is positive definite and ${\Psi }_{\alpha }\left(\left(s_1,s_2\right)\right)=\phi \left(\left(s_1,s_2\right)\right){\alpha }^{\left|s_1\right|+\left|s_2\right|}$ is also positive definite on ${\mathbb{F}}_2\times {\mathbb{F}}_2$. By the GNS construction(The unitary representation via GNS approach refers to the conclusions of appendix C in reference [9]), we obtain the unitary representation ${\sigma }_{\alpha }$ of ${\mathbb{F}}_2\times {\mathbb{F}}_2$ with the cyclic vector ${\xi }_{\alpha }$ such that

$${\Psi }_{\alpha }\left(\left(s_1,s_2\right)\right)=⟨{\sigma }_{\alpha }\left(\left(s_1,s_2\right)\right){\xi }_{\alpha }\left|{\xi }_{\alpha }\right.⟩.$$

Since ${\sigma }_{\alpha }$ is an ${\ell }_p-$ representation, ${\Psi }_{\alpha }$ can be considered as a state on $C_{{\ell }_p}^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$. By taking the $w^{*}-$limit of ${\Psi }_{\alpha }$ as $\alpha \uparrow 1$, we obtain that $\phi$ can be extended to the state of ${C_{l_p}}^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$. □

Corollary 3.

Let $p\in [2,\infty )$ and $\alpha \in (0,1)$. The positive definite function ${\Phi }_{\alpha }(s_1,s_2)={\alpha }^{|s_1|+|s_2|}$ can be extended to the state on $C_{{\ell }_p}^{*}({\mathbb{F}}_2\times {\mathbb{F}}_2)$ if and only if $\alpha <3^{-\frac{1}{p}}$.

Proof. 

Since

$$\begin{array}{lll}{\displaystyle \sum _{k_1,k_2=1}^{+\infty }}{3}^{k_1+k_2-2}{\alpha }^{p\left(k_1+k_2\right)}& =& 3^{-2}{\displaystyle \sum _{k_1,k_2=1}^{+\infty }}{\left(3·{\alpha }^p\right)}^{\left(k_1+k_2\right)}\\ & =& 3^{-2}\left[{\displaystyle \sum _{k_1=1}^{+\infty }}{\left(3·{\alpha }^p\right)}^{k_1}\right]\left[{\displaystyle \sum _{k_2=1}^{+\infty }}{\left(3·{\alpha }^p\right)}^{k_2}\right],\end{array}$$

it follows from Theorem 4 (4) that we have

$$\begin{array}{lll}{\Phi }_{\alpha }\in l_p\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)& \iff & \forall \beta \in (0,1),{\left(\alpha \beta \right)}^{|s_1|+|s_2|}\in {\ell }_p({\mathbb{F}}_2\times {\mathbb{F}}_2)\\ & \iff & {\displaystyle \sum _{(s_1,s_2)\in {\mathbb{F}}_2\times {\mathbb{F}}_2}}{\left(\alpha \beta \right)}^{p\left(\right|s_1|+|s_2\left|\right)}<+\infty \\ & \iff & {\displaystyle \sum _{k_1,k_2=1}^{+\infty }}{\displaystyle \sum _{|s_1|=k_1,\left|s_2\right|=k_2}}{\left(\alpha \beta \right)}^{p\left(\right|s_1|+|s_2\left|\right)}<+\infty \\ & \iff & {\displaystyle \sum _{k_1,k_2=1}^{+\infty }}{3}^{k_1+k_2-2}{\alpha }^{p\left(k_1+k_2\right)}<+\infty \\ & \iff & 3{\alpha }^p<1\\ & \iff & \alpha <3^{-\frac{1}{p}}.\end{array}$$

□

Corollary 4.

For $2\le q<p\le \infty$, the canonical quotient map from $C_{{\ell }_p}^{*}({\mathbb{F}}_2\times {\mathbb{F}}_2)\stackrel{onto}{⟶}{C}_{{\ell }_q}^{*}({\mathbb{F}}_2\times {\mathbb{F}}_2)$ is not injective. So

$$C_{{\ell }_p}^{*}\left({\mathbb{F}}_2\right)\stackrel{p}{\otimes }{C}_{{\ell }_p}^{*}\left({\mathbb{F}}_2\right)\ncong C_{{\ell }_q}^{*}\left({\mathbb{F}}_2\right)\stackrel{q}{\otimes }{C}_{{\ell }_q}^{*}\left({\mathbb{F}}_2\right).$$

Proof. 

If $p=+\infty$ and $C^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)={C_q}^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$, we obtain that ${\mathbb{F}}_2\times {\mathbb{F}}_2$ is amenable by Prop2.12 in [5]. This is a contradiction.

In the following, we consider $2\le q<p<+\infty$. Suppose that the canonical map from $C_{l_p}^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$ onto $C_{l_q}^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$ is injective from some $q<p$. Take a real number $\alpha$ with

$$3^{-\frac{1}{q}}<\alpha <3^{-\frac{1}{p}}.$$

For ${\Phi }_{\alpha }(s_1,s_2)={\alpha }^{|s_1|+|s_2|}$, by Corollary 3 we have

$$\left|{\omega }_{{\Phi }_{\alpha }}\left(f\right)\right|\le {∥f∥}_{{\ell }_p}={∥f∥}_{{\ell }_q},forf\in c_c\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right).$$

Therefore, it follows again that ${\Phi }_{\alpha }$ can be extended to the state on $C_{l_q}^{*}\left({\mathbb{F}}_2\times {\mathbb{F}}_2\right)$, but it contradicts to the choice of $\alpha$ and Corollary 3. □