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 $A\mathrm{.}$ Suppose that $u\in U_0(A),$ the connected component of $U(A)$ containing the identity. We show that, for any $\text{ep}>0,$ there exists a selfadjoint element $h\in A_{s.a}$suchthat$$\mathrm{\parallel }u-\exp (ih)\mathrm{\parallel }<\text{ep}\mathrm{.}$$Wealsostudytheproblemwhen$u$canbeapproximatedbyunitariesin$A$ 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 $CU(A)\subset U_0(A)\mathrm{.}$ Denote by $\widehat{a}$ the affine function on $T(A)$ defined by $\widehat{a}(\tau )=\tau (a)\mathrm{.}$ We show that $u$ can be approximated by unitaries in $A$ with finite spectrum if and only if $u\in CU(A)$ and $\widehat{u^n+(u^n{)}^{\ast }},i(\widehat{u^n-(u^n{)}^{\ast }})\in \overline{{\rho }_A(K_0(A)}$ 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.