  Let A be a unital separable simple C*-algebra such that either (1) A has real
rank zero, strict comparison and cancellation or (2) A is TAI. We study the
kernel of the de la Harpe--Skandalis determinant on GL^0(A), proving that the
determinant vanishes exactly on elements which are products of 8 multiplicative
commutators. We also have results for the unitary case.
