Exponentials in simple Z-stable C⁎-algebras

Let A be a unital separable simple Z-stable C⁎-algebra which has rational tracial rank at most one and let u∈U0(A), where U0(A) is the connected component of the unitary group of A containing the identity. We show that, for any ϵ>0, there exists a self-adjoint element h∈A such that(0.1)‖u−exp(ih)‖<ϵ. But there is no control of ‖h‖ in general. Let CU(A) be the closure of the commutator subgroup of the unitary group of A and let u∈CU(A). We prove that there exists a self-adjoint element h∈A such that(0.2)‖u−exp(ih)‖<ϵand‖h‖⩽2π. Examples are given that the bound 2π for ‖h‖ is optimal in general. For the Jiang-Su algebra Z, we show that, if u∈U0(Z) and ϵ>0, there exists a real number −π