On A Generalized Non-commutative Weyl-von Neumann Theorem In Operator Algebras

In recent years,Weyl-von Neumann theorem was extensively studied.In 1976,Voicules-cu proved a non-commutative version of the Weyl-von Neumann theorem.In this paper,we generalize that the Voiculescu’s theorem(short for the Voiculescu’s noncommutative Weyl-von Neumann theorem)in operator algebras,and we extend a theorem of Halmos to arbitrary factor von Neumann algebras with separable predual.The main contents are as follows:In chapter 3,we extend a result of the D.Hadwin for approximate summands of represen-tations into a finite von Neumann algebra.Let A be a unital AF algebra and R be a type Ⅱ1 factor with a faithful normal tracial state τ.If P is a projection in R,π:A→R is a unital*-homomorphism and ρ:A→PRP is a unital*-homomorphism such thatτ(R(ρ(X)))≤τ(R(π(X))),(?)X∈A.we prove that there exists a unital*-homomorphism γ:A→P-RP such that γ⊕ρ～aπ(R).In chapter 4,we extend the "compact operator" part of D.Voiculescu’s theorem on approx-imate equivalence of unital*-homomorphisms of an AF algebra when the range is in a semifinite von Neumann algebra.Let R be a countably decomposable,properly infinite.semifinite factor with a faithful normal semifinite tracial weight τ.Suppose that A is an AF-subalgebra of R with an identity IA(.If φ and ψ are unital*-homomorphisms of into we prove that the following are equivalent:(1)φ～aψ,namely,φ and ψ are approximately unitarily equivalent in R;(2)φ～Aψ mod K(R,τ),namely,φ and ψ are strongly approximately unitarily equivalent over A.In chapter 5,let M be a factor von Neumann algebra with separable predual and let T ∈ M.We call T an irreducible operator(relative to M)if W*(T)is an irreducible subfactor of M,i.e.,W*(T)’∩ M=CI.We show that the set of irreducible operators in M is a dense Gδ subset of M in the operator norm.This result generalizes a theorem of Halmos.