| In this dissertation, we explore the approximate diagonalization of unital homomorphisms between C*-algebras. In particular, we prove that unital homomorphisms from commutative C*-algebras into simple separable unital C*-algebras with tracial rank at most one are approximately diagonalizable. This is equivalent to the approximate diagonalization of commuting sets of normal matrices.;We also prove limited generalizations of this theorem. Namely, certain injective unital homomorphisms from commutative C*-algebras into simple separable unital C*-algebras with rational tracial rank at most one are shown to be approximately diagonalizable. Also unital injective homomorphisms from AH-algebras with unique tracial state into separable simple unital C*-algebras of tracial rank at most one are proved to be approximately diagonalizable. Counterexamples are provided showing that these results cannot be extended in general.;Finally, we prove that for unital homomorphisms between AF-algebras, approximate diagonalization is equivalent to a combinatorial problem involving sections of lattice points in cones. |
