  Study of certain angular correlations in the three-photon annihilations of
the triplet state of positronium, the electron-positron bound state, may be
used as a probe of potential CP and CPT-violating effects in the leptonic
sector. We present the perspectives of CP and CPT tests using this process
recorded with a novel detection system for photons in the positron annihilation
energy range, the Jagiellonian PET (J-PET). We demonstrate the capability of
this system to register three-photon annihilations with an unprecedented range
of kinematical configurations and to measure the CPT-odd correlation between
positronium spin and annihilation plane orientation with a precision improved
by at least an order of magnitude with respect to present results. We also
discuss the the means to control and reduce detector asymmetries in order to
allow J-PET to set the first measurement of the correlation between positronium
spin and momentum of the most energetic annihilation photon which has never
been studied to date.
