  For the famous Camassa-Holm equation, the well-posedness in
$B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $ p\in [1,\infty)$ and the
ill-posedness in $B^{1+\frac{1}{p}}_{p,r}(\mathbb{R})$ with $ p\in [1,\infty],\
r\in (1,\infty]$ had been studied in \cite{d1,d2,glmy,yyg}, that is to say, it
only left an open problem in the critical case $B^{1}_{\infty,1}(\mathbb{R})$
proposed by Danchin in \cite{d1,d2}. In this paper, we solve this problem by
proving the norm inflation and hence the ill-posedness for the Camassa-Holm
equation in $B^{1}_{\infty,1}(\mathbb{R})$. Therefore, the well-posedness and
ill-posedness for the Camassa-Holm equation in all critial Besov spaces
$B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $ p\in [1,\infty]$ have been
completed. Finally, since the norm inflation occurs by choosing an special
initial data $u_0\in B^{1}_{\infty,1}(\mathbb{R})$ but $u^2_{0x}\notin
B^{0}_{\infty,1}(\mathbb{R})$ (an example implies
$B^{0}_{\infty,1}(\mathbb{R})$ is not a Banach algebra), we then prove that
this condition is necessary. That is, if $u^2_{0x}\in
B^{0}_{\infty,1}(\mathbb{R})$ holds, then the Camassa-Holm equation has a
unique solution $u(t,x)\in \mathcal{C}_T(B^{1}_{\infty,1}(\mathbb{R}))\cap
\mathcal{C}^{1}_T(B^{0}_{\infty,1}(\mathbb{R}))$ and the norm inflation will
not occur.
