Exponential rank and exponential length for Z-stable simple C*-algebras

Let $A$ be a unital separable simple ${\cal Z}$-stable C*-algebra which has rational tracial rank at most one and let $u\in U_0(A),$ the connected component of the unitary group of $A.$ We show that, for any $ε>0,$ there exists a self-adjoint element $h\in A$ such that $|u-\exp(ih)|<ε.$ The lower bound of $|h|$ could be as large as one wants. If $u\in CU(A),$ the closure of the commutator subgroup of the unitary group, we prove that there exists a self-adjoint element $h\in A$ such that $|u-\exp(ih)| <εand |h|\le 2π.$ Examples are given that the bound $2π$ for $|h|$ is the optimal in general. For the Jiang-Su algebra ${\cal Z},$ we show that, if $u\in U_0({\cal Z})$ and $ε>0,$ there exists a real number $-π<t\le π$ and a self-adjoint element $h\in {\cal Z}$ with $|h|\le 2π$ such that $|e^{it}u-\exp(ih)|<ε.$

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