Approximation by Unitaries with Finite Spectrum in Purely Infinite C*-Algebras

We prove that a unital purely infinite simple C*-algebra has C*-exponential rank at most 1 + ϵ, and therefore has the property weak (FU): Every element of the identity component of the unitary group is norm limit of unitaries with finite spectrum. We also prove that the appropriate modification of this statement for nonunital algebras.