Homotopy of unitaries in simple C∗C∗-algebras with tracial rank one

Let ϵ>0ϵ>0 be a positive number. Is there a number δ>0δ>0 satisfying the following? Given any pair of unitaries u and v   in a unital simple C∗C∗-algebra A   with [v]=0[v]=0 in K1(A)K1(A) for which‖uv−vu‖<δ,‖uv−vu‖<δ, there is a continuous path of unitaries {v(t):t∈[0,1]}⊂A such thatv(0)=v,v(1)=1and‖uv(t)−v(t)u‖<ϵfor allt∈[0,1]. An answer is given to this question when A   is assumed to be a unital simple C∗C∗-algebra with tracial rank no more than one. Let C   be a unital separable amenable simple C∗C∗-algebra with tracial rank no more than one which also satisfies the UCT. Suppose that ϕ:C→A is a unital monomorphism and suppose that v∈Av∈A is a unitary with [v]=0[v]=0 in K1(A)K1(A) such that v almost commutes with ϕ  . It is shown that there is a continuous path of unitaries {v(t):t∈[0,1]} in A   with v(0)=vv(0)=v and v(1)=1v(1)=1 such that the entire path v(t)v(t) almost commutes with ϕ, provided that an induced Bott map vanishes. Other versions of the so-called Basic Homotopy Lemma are also presented.