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

Let ǫ > 0 be a positive number. Is there a number δ > 0 satisfying the following? Given any pair of unitaries u and v in a unital simple C∗-algebra A with [v] = 0 in K1(A) for which ‖uv − vu‖ < δ, there is a continuous path of unitaries {v(t) : t ∈ [0, 1]} ⊂ A such that v(0) = v, v(1) = 1 and ‖uv(t)− v(t)u‖ < ǫ for all t ∈ [0, 1]. An answer is given 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 amenable simple 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 ∈ A is a unitary with [v] = 0 in 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) = v and v(1) = 1 such that the entire path 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.