Approximate unitary equivalence in simple $C^*$-algebras of tracial rank one

Abstract:

Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple $C^*$-algebra with tracial rank no more than one. Suppose that $\phi , \psi : C\to A$ are two unital monomorphisms. With some restriction on $C,$ we show that $\phi$ and $\psi$ are approximately unitarily equivalent if and only if \begin{eqnarray}\nonumber [\phi ]&=&[\psi ]\,\,\,\text {in}\,\,\, KL(C,A),\\\nonumber \tau \circ \phi &=&\tau \circ \psi \ \text {for all tracial states of}\,\,\, A\ \text {and}\\\nonumber \phi ^{\ddagger }&=&\psi ^{\ddagger }, \end{eqnarray} where $\phi ^{\ddagger }$ and $\psi ^{\ddagger }$ are homomorphisms from $U(C)/CU(C)\to U(A)/CU(A)$ induced by $\phi$ and $\psi ,$ respectively, and where $CU(C)$ and $CU(A)$ are closures of the subgroup generated by commutators of the unitary groups of $C$ and $B.$

A more practical but approximate version of the above is also presented.