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

Abstract

For each unital C * -algebra A, we denote cel CU (A) = sup{cel(u) : u ∈ 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) = 2π 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) ≤ 2π 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) = 2π).