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

TL;DR

This paper investigates the approximation of unitaries in Z-stable simple C*-algebras by exponentials of self-adjoint elements, establishing bounds on the length of these elements and providing optimal bounds in general.

Contribution

It introduces bounds on the exponential length for unitaries in Z-stable simple C*-algebras, including optimal bounds and approximation results for the Jiang-Su algebra.

Findings

Any unitary in the connected component can be approximated by an exponential of a self-adjoint element.

The bound on the self-adjoint element's norm can be arbitrarily large, but is at most 2π for unitaries in the commutator subgroup.

In the Jiang-Su algebra, unitaries can be approximated up to a phase shift with a self-adjoint element of norm at most 2π.

Abstract

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 $\epsilon>0,$ 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 ${\cal Z},$ we show that, if $u\in U_0({\cal Z})$ and $\epsilon>0,$ there exists a real number $-\pi<t\le \pi$ and a self-adjoint element $h\in {\cal Z}$ with $|h|\le 2\pi$ such that $$ |e^{it}u-\exp(ih)|<\epsilon. $$