Weakly Approximate Diagonalization of Homomorphisms into Finite von Neumann Algebras.

Let A be a unital C*-algebra and B a unital C*-algebra with a faithful trace τ. Let n be a positive integer. We give the definition of weakly approximate diagonalization (with respect to τ) of a unital homomorphism ϕ : A → M n (B) . We give an equivalent characterization of McDuff II₁ factors. We show that, if A is a unital nuclear C*-algebra and B is a type II₁ factor with faithful trace τ, then every unital *-homomorphism ϕ : A → M n (B) is weakly approximately diagonalizable. If B is a unital simple infinite dimensional separable nuclear C*-algebra, then any finitely many elements in M n (B) can be simultaneously weakly approximately diagonalized while the elements in the diagonals can be required to be the same. [ABSTRACT FROM AUTHOR]