For each unital $C^*$-algebra $A$, we denote $\operatorname{cel}_{\operatorname{CU}}(A)=\sup\{\operatorname{cel}(u):u\in\operatorname{CU}(A)\}$, where $\operatorname{cel}(u)$ is the exponential length of $u$ and $\operatorname{CU}(A)$ is the closure of the commutator subgroup of $U_0(A)$. In this paper, we prove that $\operatorname{cel}_{\operatorname{CU}}(A)\geq2\pi$ provided that $A$ is an AH-algebra with slow dimension growth whose real rank is not zero. On the other hand, we prove that $\operatorname{cel}_{\operatorname{CU}}(A)\leq 2\pi$ when $A$ is an AH-algebra with ideal property and of no dimension growth (if we further assume that $A$ is not of real rank zero, we have $\operatorname{cel}_{\operatorname{CU}}(A)=2\pi$).

中文翻译：

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$C^*$ AH 代数中换向器幺正的指数长度

对于每个单位 $C^*$-代数 $A$，我们表示 $\operatorname{cel}_{\operatorname{CU}}(A)=\sup\{\operatorname{cel}(u):u\in \operatorname{CU}(A)\}$，其中 $\operatorname{cel}(u)$ 是 $u$ 的指数长度，$\operatorname{CU}(A)$ 是$U_0(A)$。在本文中，我们证明 $\operatorname{cel}_{\operatorname{CU}}(A)\geq2\pi$ 条件是 $A$ 是一个维度增长缓慢的 AH-代数，其实际秩不为零。另一方面，我们证明了 $\operatorname{cel}_{\operatorname{CU}}(A)\leq 2\pi$ 当 $A$ 是一个具有理想性质且没有维数增长的 AH-代数（如果我们进一步假设 $A$ 不是真正的零阶，我们有 $\operatorname{cel}_{\operatorname{CU}}(A)=2\pi$)。