Unitaries in a Simple C*-algebra of Tracial Rank One

Let $A$ be a unital separable simple infinite dimensional \CA 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 $u\in U_0(A),$ the connected component of $U(A)$ containing the identity. We show that, for any $\ep>0,$ there exists a selfadjoint element $h\in A_{s.a}$ such that $$\|u-\exp(ih)\|<\ep.$$ 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 $CU(A)\subset U_0(A).$ Denote by $\widehat{a}$ the affine function on $T(A)$ defined by $\widehat{a}(\tau)=\tau(a).$ We show that $u$ can be approximated by unitaries in $A$ with finite spectrum if and only if $u\in CU(A)$ and $\widehat{u^n+(u^n)^*},i(\widehat{u^n-(u^n)^*})\in \overline{\rho_A(K_0(A)}$ for all $n\ge 1.$ 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.