# From positive definite function to Følner sequence --— a question on amenability and nuclearity

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 $\psi_k\circ \varphi_k(\lambda_s)=\psi_k(\sum_{p\in F_k\cap sF_k}e_{p,s^{-1}p})=\sum_{p\in F_k\cap sF_k}\frac{1}{|F_k|}\lambda_s=\frac{|F_k\cap sF_k|}{|F_k|}\lambda_s$.

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You will find these proved in appendices G and C in the recent monograph by Bekka, de la Harpe and Valette (see perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf). –  user2412 Apr 15 '13 at 12:08
You might want to look at Brown and Ozawa again, in particular Theorem 2.6.8. –  Mike Jury Apr 15 '13 at 12:11
Thank you. I clarify my question further above. –  Chao You Apr 15 '13 at 14:07
It took me a while to realise that your initial question is not about operator algebras as such but about harmonic analysis: (a) is a variant on Leptin's condition for the Fourier algebra, and (b) is Folner's condition as you say. Are you just looking for any proof that (a) implies (b), or are you looking for a proof that follows the Brown-Ozawa ideas? –  Yemon Choi Apr 15 '13 at 20:39
Yes, I want to mimic Brown-Ozawa's proof. What I am most interested in is how to factor $m_{\varphi}$ through $M_n(\mathbb{C})$. Brown-Ozawa's proof gives a model for that. But I don't know what to do with finitely supported positive definite functions. –  Chao You Apr 16 '13 at 1:17