$C^*$ exponential length of commutators unitaries in $AH$ algebras

Abstract

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