The Wigner's Theorem Of CL-spaces And Its Unit Sphere

Wigner's theorem says that any symmetric operator in a quantum mechanical system is induced by a unitary operator or an inverse unitary operator.Wigner's theorem is the basis of quantum mechanics theory and is closely related to projection space theory.A symmetric transformation form written by Wigner's theorem has profound significance and wide application in quantum mechanics,such as quantum state space,density operator space.Since the Wigner's theorem was put forward,many scholars have proved the theorem strictly along with the idea of the theorem.They have proved the Wigner's theorem in Hilbert space in different ways.With the continuous development,Wigner's theorem is popularized in many general Banach spaces.This paper first introduces the definition of Wigner's theorem and Tingley's problem,reviews the proof of Wigner's theorem in CL-spaces,and finally proves the Wigner's theorem on the unit sphere of C(K).In the first chapter,we describes the development of Mazur-Ulam theorem,Tingley's problem and Wigner's theorem,and introduces some basic concepts of CL-spaces and C(K)spaces,which lays a foundation for the study of CL-spaces and C(K)spaces in the following chapters.In the second chapter,we first introduce some basic knowledge of CL-spaces.Then the properties of phase-isometry in Banach spaces are proved.Finally,an important conclusion is obtained:all CL-spaces have Wigner properties.Chapter 3 introduces the development of Wigner's theorem and introduces the C(K)spaces.Then,it is proved that the surjective phase-isometry fromS_C(K)onto S_C(?)is phase-equivalent to a surjective linear isometry which can is the restriction of a linear isometry from C(K)onto C(?).Finally,it is proved that every surjective phase-isometry between the unit spheres of C(K)and an arbitrary Banach space Y is also phase-equivalent to an isometry which can be extended to a linear isometry from C(K)onto Y.