| This paper is mainly discuss the phase isometry in complex L~∞(Γ)-type spaces,and demonstrate the Wigner theorem in the above spaces.At the same time,the Wigner theorem combined with the Tingley’s problem,further proved on the unit spheres of complex-type spaces.The first chapter mainly introduced the development process and present situation of the Wigner theorem,Mazur-Ulam theorem and Tingley’s problem.Also,the basic definitions,such as linear isometry and phase isometry,are introduced one by one.In second chapter,we discussed the phase isometry problems in complex L~∞(Γ)-type spaces.The main research method is based on some conclusions of phase isometry in real L~∞(Γ)-type space,also obtain different forms of phase isometry in complex-L~∞(Γ)type space,and got the following conclusions of the surjective phase isometry mapping in two complex L~∞(Γ)-type spaces is phase equivalent to a real linear isometry mapping.In third chapter,combining with the Wigner theorem and Tingley’s problem,we research the problem of the extension of phase isometries between the unit spheres of L~∞(Γ)spaces.The main research method is based on the conclusion that the mapping phase isometries on the unit sphere of real L~∞(Γ)space extended to the whole space,we explored the different situations in complex spaces,and obtained the following conclusions of the phase isometry mapping in two unit spheres of complex L~∞(Γ)-type spaces can be extended to the whole spaces,and the extending mapping is phase equivalent to a real linear isometry mapping. |
