Weakly Approximate Diagonalization of Homomorphisms into Finite von Neumann Algebras

Let \(\cal{A}\) be a unital C*-algebra and \(\cal{B}\) 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 \(\phi:\cal{A}\rightarrow M_{n}(\cal{B})\). We give an equivalent characterization of McDuff II₁ factors. We show that, if \(\cal{A}\) is a unital nuclear C*-algebra and \(\cal{B}\) is a type II₁ factor with faithful trace τ, then every unital *-homomorphism \(\phi:\cal{A}\rightarrow M_{n}(\cal{B})\) is weakly approximately diagonalizable. If \(\cal{B}\) is a unital simple infinite dimensional separable nuclear C*-algebra, then any finitely many elements in \(M_{n}(\cal{B})\) can be simultaneously weakly approximately diagonalized while the elements in the diagonals can be required to be the same.