$C^{\ast }$ exponential length of commutators unitaries in AH-algebras

Abstract

For each unital $C^{\ast }$-algebra $A$, we denote ${\mathrm{cel}}_{\mathrm{CU}}(A)=\sup \{\mathrm{cel}(u):u\in \mathrm{CU}(A)\}$, where $\mathrm{cel}(u)$ is the exponential length of $u$ and $\mathrm{CU}(A)$ is the closure of the commutator subgroup of $U_0(A)$. In this paper, we prove that ${\mathrm{cel}}_{\mathrm{CU}}(A)\ge 2\pi$ provided that $A$ is an AH-algebra with slow dimension growth whose real rank is not zero. On the other hand, we prove that ${\mathrm{cel}}_{\mathrm{CU}}(A)\le 2\pi$ 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 ${\mathrm{cel}}_{\mathrm{CU}}(A)=2\pi$).