We know that amenability of countable discrete group $\Gamma$ has many equivalent characterizations. In particular, there are two: a) there is a sequence of finitely supported positive definite functions $\phi_i$ defined on $\Gamma$ such that $\lim_{i\to\infty}\phi(\gamma)=1$ for any $\gamma\in \Gamma$; b) $\Gamma$ satisies the Følner condition, i.e., there exists a sequence of $F_k\subset\Gamma$ such that $\lim_{k\to\infty}\frac{|F_k\cap sF_k|}{|F_k|}=1$ for any $s\in \Gamma$. Can somebody provide a proof of $a) \Rightarrow b)$? I have checked N. P. Brown and N. Ozawa's classical book "$C^\ast$-algebras and Finite-Dimensional Approximations", but no hint. It would be nice if somebody gives a hint or a reference. Thanks in advance.
Let $\varphi$ be a positive definite function on $\Gamma$. Then $m_{\varphi}: \sum c_t\lambda_t \rightarrow \sum\varphi(t)c_t\lambda(t)$ extends to a completely positive map $\phi$ from $C^\ast_\lambda (\Gamma)$ into itself. My question is if it is possible to factor $m_{\varphi}$ through finite-dimensional matrix algebra $M_n(\mathbb{C})$ as Brown and Ozawa show in their Theorem 2.6.8.
For convenience, I'd like to recall the construction there w.r.t. F{\o}lner condition. Given a sequence of F{\o}lner sets $F_k$, for each $k$, let $P_k\in B(l^2(\Gamma))$ be the orthogonal projection onto the finite-dimensional subspace spanned by $\{\delta_g:g\in F_k\}$. Identity $P_kB(l^2(\Gamma))P_k$ with the matrix algebra $M_{F_k}(\mathbb{C})$ and let $\{e_{p,q}\}_{p,q\in F_k}$ be the canonical matrix units of $M_{F_k}(\mathbb{C})$. One can chechk that for each $s\in \Gamma$ we have $e_{p,p}\lambda_se_{q,q}=0$ unless $sq=p$, and $e_{p,p}\lambda_se_{q,q}=e_{p,q}$ if $sq=p$. Since $P_k=\sum_{p\in F_k}e_{p,p}$, we have $P_k\lambda_sP_k=\sum_{p,q\in F_k}e_{p,p}\lambda_se_{q,q}=\sum_{p\in F_k\cap sF_k}e_{p,s^{-1}p}$. Let $\varphi_k:C^\ast_\lambda (\Gamma)\rightarrow M_{F_k}(\mathbb{C})$ be the u.c.p. map defined by $x\mapsto P_kxP_k$. Now define a map $\psi:M_{F_k}(\mathbb{C})\rightarrow C^\ast_\lambda (\Gamma)$ by sending $e_{p,q}\mapsto\frac{1}{|F_k|}\lambda_p\lambda_{q^{-1}}$. Evidently this map is unital; it is also completely positive. Since the linear span of $\{\lambda_s:s\in\Gamma\}$ is norm dense in $C^\ast_\lambda (\Gamma)$, it suffices to check that $\|\lambda_s-\psi_k\circ \varphi_k(\lambda_s)\|\rightarrow 0$ for all $s\in \Gamma$. This follows from the definition of F{\o}lner sets together with the following computation $\psi_k\circ \varphi_k(\lambda_s)=\psi_k(\sum_{p\in F_k\cap sF_k}e_{p,s^{-1}p})=\sum_{p\in F_k\cap sF_k}\frac{1}{|F_k|}\lambda_s=\frac{|F_k\cap sF_k|}{|F_k|}\lambda_s$.
