About the bound of the C* exponential length

TL;DR

This paper demonstrates that in certain C*-algebras, the exponential length of specific unitaries cannot be bounded by pi, highlighting limitations in controlling this metric.

Contribution

It provides explicit examples showing the unboundedness of the C* exponential length in certain contexts, including simple inductive limit C*-algebras.

Findings

Existence of unitaries with unbounded exponential length

Counterexamples in C(X)⊗M_n with determinant 1

Implications for the structure of simple inductive limit C*-algebras

Abstract

In this paper, we give an example to show that, if $u\in C(X)\otimes M_n$ with $\det (u)=1$ then the C* exponential length of $u$ (denoted by $cel(u)$) can not be controlled by $\pi$. Moreover, in the simple inductive limit C*-algebras, similar examples exist.