Unitaries in a Simple C-algebra of Tracial Rank One

Let A be a unital separable simple infinite dimensional C∗-algebra with tracial rank no more than one and with the tracial state space T (A) and let U(A) be the unitary group of A. Suppose that u ∈ U0(A), the connected component of U(A) containing the identity. We show that, for any ǫ > 0, there exists a selfadjoint element h ∈ As.a such that ‖u− exp(ih)‖ < ǫ. We also study the problem when u can be approximated by unitaries in A with finite spectrum. Denote by CU(A) the closure of the subgroup of unitary group of U(A) generated by its commutators. It is known that CU(A) ⊂ U0(A). Denote by â the affine function on T (A) defined by â(τ) = τ(a). We show that u can be approximated by unitaries in A with finite spectrum if and only if u ∈ CU(A) and ̂ u + (u), i( ̂ u − (u)) ∈ ρA(K0(A) for all n ≥ 1. Examples are given that there are unitaries in CU(A) which can not be approximated by unitaries with finite spectrum. Significantly these results are obtained in the absence of amenability.