About the bound of the C* exponential length

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 $π$. Moreover, in the simple inductive limit C*-algebras, similar examples exist.

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