Here is a quotation:
Corollary 3.7.12 If $\Gamma$ is a non-amenable residually finite group, then $C^{*}(\Gamma)$ is not exact.
It follows from this corollary that $B(l^{2})$ is not exact either (since exactness passes to subalgebras).
My question is: How to use the corollary to judge $B(l^{2})$ is not exact?
A discrete group $\Gamma$ is called residually finite if there exist subgroups $\Gamma\supset\Gamma_{1}\supset\Gamma_{2}\supset...$ such that each $\Gamma_{i}$ is a finite-index, normal subgroup of $\Gamma$ and $\bigcap_{n}\Gamma_{n}=\{e\}$(the neutral element of $\Gamma$).
From earlier in the book you know that $\mathbb F_2$ is countable, non amenable, and residually finite. By the Corollary, $C^*(\mathbb F_2)$ is not exact. It is also separable, so it embeds in $B(\ell^2(\mathbb N))$. If $B(\ell^2(\mathbb N))$ were exact, then so would be all of its subalgebras, contradicting the fact that $C^*(\mathbb F_2)$ is not exact.