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

Abstract

Let $A$ be a unital separable simple $\mathcal{Z}$-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 $\epsilon >0,$ there exists a self-adjoint element $h\in A$ such that $$|u-\exp (ih)|<\epsilon .$$ 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 $$|u-\exp (ih)|<\epsilon and|h|\le 2\pi .$$ Examples are given that the bound $2\pi$ for $|h|$ is the optimal in general. For the Jiang-Su algebra $\mathcal{Z},$ we show that, if $u\in U_0(\mathcal{Z})$ and $\epsilon >0,$ there exists a real number $-\pi <t\le \pi$ and a self-adjoint element $h\in \mathcal{Z}$ with $|h|\le 2\pi$ such that $$|e^{it}u-\exp (ih)|<\epsilon .$$

Cited by in corpus (1)

• Kishimoto's Conjugacy Theorems in simple $C^{\ast }$-algebras of tracial rank one