Local Maps And W～*-Probability Spaces

This paper is composed of two chapters.In chapter 1, we introduce the notions of the approximately 2-local (*-) automorphisms, the approximately local Hilbert space (*-) representations and the approximately 2- local Hilbert space (*-) representations;We prove that every surjective and multiplicative approximately local *-automorphisms of a C*- algebra with a unique faithful tracial state is *-automorphisms;surjective approximately 2- local *- automorphisms of C*-algebra with unique faithful tracial states are Jordan *- automorphisms;Surjective approximately 2- local *- automorphisms of a finit factor are *-automorphisms;Approximately 2- local *- automorphisms of M_n(C) are *-automorphisms. Also prove that every bounded linear approximately local Hilbert space (*-) representation of a abelian von Neumann algebra is a (*-) representation,every bounded linear approximately 2- local Hilbert space (*-) representation of a von Neumann algebra is a (*-) representation. If let H be a complex separable Hilbert space ,dimH ≥ 3, we prove that every approximately 2- local automorphism of P(H) (B_s(H),E(H)) is an automorphism. Every real linear approximately local automorphism of B_s(H) is an automorphism.In chapter 3, we study the expectation, variance and the *-freeness of normal operators in a W*- probability space, and show the Techebycheff's inequality and law of large numbers of the normal random variables. And in this chapter, we prove that, for n = 2,3,every pair of orthogonal MASA's in M_n(C) is standard.