  Observations are reported of different sources of $CP$ violation from an
amplitude analysis of $B^+ \to \pi^+ \pi^+ \pi^-$ decays, based on a data
sample corresponding to an integrated luminosity of $3 \; {\rm fb}^{-1}$ of
$pp$ collisions recorded with the LHCb detector. A large $CP$ asymmetry is
observed in the decay amplitude involving the tensor $f_2(1270)$ resonance, and
in addition significant $CP$ violation is found in the $\pi^+ \pi^-$ S-wave at
low invariant mass. The presence of $CP$ violation related to interference
between the $\pi^+ \pi^-$ S-wave and the P-wave $B^+ \to \rho(770)^0 \pi^+$
amplitude is also established; this causes large local asymmetries but cancels
when integrated over the phase space of the decay. The results provide both
qualitative and quantitative new insights into $CP$-violation effects in
hadronic $B$ decays.
