The algebraic and geometric approaches to $\cal PT$ -symmetric non-Hermitian relativistic quantum mechanics with maximal mass

A non-Hermitian fermion model containing a $\gamma_5$-matrix in the mass term $m \rightarrow m_1 + \gamma_5 m_2$ is considered. A new refined condition of the unbroken $\cal PT$ - symmetry of the theory is proposed that comprises an indication that the initial domain of unbroken $\cal PT$ - symmetry is divided into subdomains corresponding to the description of ordinary and exotic particles. A relationship is established between the theory under study and the quantum field theory with maximal mass $M$ developed on the basis of the geometric approach. The operator {ie223-1} is calculated explicitly, which is required for the construction of a new scalar product in the theory with a non-Hermitian Hamiltonian.