Huaxin Lin: Homotopy of unitaries in simple C*-algebras with tarcial rank one. (arXiv:0805.0583v2 [math.OA] UPDATED)

[0805.0583] Homotopy of unitaries in simple C*-algebras with tracial rank one

Let $\epsilon>0$ . Is there $\delta>0$ satisfying the following? Given any pair of unitaries $u$ and $v$ in a unital separable simple C*-algebra $A$ for which $\|uv-vu\|<\delta$ , there is a continuous path of unitaries $\{v(t): t\in [0,1]\}\subset A$ such that $v(0)=v$ , $v(1)=1$ and $\|uv(t)-v(t)u\|<\epsilon$ for all $t\in [0,1]$ . We give an answer to this question when $A$ is assumed to be a unital simple C*-algebra with tracial rank no more than one.
Let $C$ be a unital separable simple C*-algebra, with tracial rank no more than one. Suppose that $\phi: C\to A$ is a unital monomorphism and $u\in A$ is a unitary such that $\phi$ almost commutes with $u$ . We give an answer to the question when there is a continuous path of unitaries $\{u(t): t\in [0,1]\}$ with $u(0)=u$ and $u(1)=1$ such that entire path $u(t)$ almost commute with $\phi$ . Other versions of so-called basic homotopy lemma are also presented.