Symmetry violation under the combined transformation of charge (C), parity (P), and time reversal (T) in the charged leptonic sector can be sought through the non-vanishing expectation value of certain angular correlation operators that are odd under the CPT transformation. Here, we discuss the experimental approach for a CPT symmetry test by measuring angular correlations between the spin and momenta of photons originating from ortho-positronium (o-Ps$\rightarrow$3${\gamma}$) decays. This experiment is performed with the J-PET detector which measures a broad range of kinematical configurations of ortho-positronium annihilation into three photons and is the first experiment to determine the full range of the CPT-odd angular correlation.
A novel technique to estimate the spin of ortho-positronium and momenta of annihilation photons for a single recorded ortho-positronium event allowed J-PET to measure the expectation value of CPT symmetry odd angular correlation operator at the precision level of 10$^{-4}$, a factor of three better than the previous best result. There are new measures and perspectives for J-PET in improving the sensitivity to CPT violating effects beyond the level of 10$^{-4}$ by increasing the efficiency for detection of photons from ortho-positronium decays by means of using a new modular J-PET detector and spherical annihilation chamber.