Normal elements with finite spectrum in C*-algebras of real rank zero

grantor: University of TorontoWe give a short and elementary proof of Huaxin Lin's theorem that almost commuting pairs of self-adjoint matrices are near commuting pairs of self-adjoint matrices in a uniform way. The theorem is a consequence of a result that normal elements of a certain C*-algebra can be approximated by ones with finite spectrum. Our methods yield a complete characterization of C*-algebras with the latter property, generalizing earlier results of Lin. Also, we generalize Lin's theorem to a class of C*-algebras that includes not only matrix algebras, but all simple purely infinite C*-algebras and all of stable rank one. Using similar methods, we give a fairly elementary proof of the Brown-Douglas-Fillmore theorem on essentially normal operators. This dissertation is based on the author's joint work with Mikael Rørdam.Ph.D