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

TL;DR

This paper determines the exponential length of commutators in certain $AH$ algebras, showing it equals $2\pi$ under specific conditions related to dimension growth and real rank.

Contribution

It establishes the exact value of $cel_{CU}(A)$ for $AH$ algebras with slow dimension growth and non-zero real rank, and provides bounds for other classes.

Findings

$cel_{CU}(A)=2\pi$ for $AH$ algebras with slow dimension growth and non-zero real rank

$cel_{CU}(A)\leq 2\pi$ for $AH$ algebras with ideal property and no dimension growth

Exact value of exponential length for specific classes of $AH$ 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 $cel_{CU}(A)= 2\pi$).