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The time irreversible evolution of quantum systems with infinite number ofparticles (QSINP) was studied within a newly constructed algebraic framework.The QSINP could be described by the quantum infinite lattice field. The*-Algebra R is the set of all dynamic variables of the QSINP, in which aspecial addition and multiplication operations were defined. The full purestate vector (FPSV) \rho, the pure state vector {\rho_f} and the equivalentrelations within them are also well defined. The set of all pure state vectors,which are equivalent to the FPSV \rho, is a Hilbert Space {H_\rho}, if theaddition, the inner product and the norm were defined in it. The set of alllinear transformation {N_\rho} on {H_\rho} is isomorphic to R, and{{N_\rho},{H_\rho}} is the representation of R, associated with \rho. The*-Algebra R has infinitely many non-equivalent irreducible GNS constructions(representations), associated with different nonequivalent FPSVs respectively.  It is proved that, the dynamical motions of the QSINP is totally within theGNS construction associated with the initial pure state vector; However, thetime reversal transformation makes the initial FPSV and its correspondingdynamical evolution into another non-equivalent GNS construction. Therefore,within the GNS construction associated with the FPSV, the dynamics of the QSINPis time irreversible.  Finally, due to the Liouville operator has real continuous spectrum withinthe GNS construction, the longtime asymptotic solution of Liouville equation inthe GNS construction could be treated by the formal scattering theory. TheMaster equation with dissipative term was obtained, which is formallyirrelevant of the initial state and the corresponding GNS construction. Thismaster equation could be regarded as the evolution equation of the QSINP on R.

Time irreversibility in the quantum systems with infinite number of particles