  In this work, we study the localized $CP$ violation in $B^-\rightarrow
K^-\pi^+\pi^-$ and $B^-\rightarrow K^- \sigma(600)$ decays by employing the
quasi two-body QCD factorization approach. Both the resonance and the
nonresonance contributions are studied for the $B^-\rightarrow K^-\pi^+\pi^-$
decay. The resonance contributions include those not only from $[\pi\pi]$
channels including $\sigma(600)$, $\rho^0(770)$ and $\omega(782)$ but also from
$[K\pi]$ channels including $K^*(892)$, $K_0^*(1430)$, $K^*(1410)$, $K^*(1680)$
and $K_2^*(1430)$. By fitting the experimental data
$\mathcal{A_{CP}}(K^-\pi^+\pi^-)=0.678\pm0.078\pm0.0323\pm0.007$ for
$m_{K^-\pi^+}^2<15$ $\mathrm{GeV}^2$ and $0.08<m_{\pi^+\pi^-}^2<0.66$
$\mathrm{GeV}^2$, we get the end-point divergence parameters in our model,
$\phi_S \in [4.75, 5.95]$ and $\rho_S\in[4.2, 8]$. Using these results for
$\rho_S$ and $\phi_S$, we predict that the $CP$ asymmetry parameter
$\mathcal{A_{CP}} \in [-0.094, -0.034]$ and the branching fraction $\mathcal{B}
\in [1.82, 20.0]\times10^{-5}$ for the $B^-\rightarrow K^-\sigma(600)$ decay.
In addition, we also analyse contributions to the localized $CP$ asymmetry
$\mathcal{A_{CP}}(B^-\rightarrow K^-\pi^+\pi^-)$ from $[\pi\pi]$, $[K\pi]$
channel resonances and nonresonance individually, which are found to be
$\mathcal{A_{CP}}(B^-\rightarrow K^-[\pi^+\pi^-] \rightarrow
K^-\pi^+\pi^-)=0.585\pm0.045$, $\mathcal{A_{CP}}(B^-\rightarrow [K^-\pi^+] \pi
\rightarrow K^-\pi^+\pi^-)=0.086\pm0.021$ and
$\mathcal{A_{CP}}^{NR}(B^-\rightarrow K^-\pi^+\pi^-)=0.061\pm0.0042$,
respectively. Comparing these results, we can see that the localized $CP$
asymmetry in the $B^-\rightarrow K^-\pi^+\pi^-$ decay is mainly induced by the
$[\pi\pi]$ channel resonances while contributions from the $[K\pi]$ channel
resonances and nonresonance are both very small.
