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.
It is clear to see that γ is a continuous path from 0 to w̅.
We will show that (tin[0,1]mapsto C_{varphi_{t}}) is a continuous path in (mathcal{C}(A_{alpha}^{2}(Pi^))).
We are going to prove that T t) is a continuous path which connects uC φ and vC ψ, and here we simply denote by C t = C φ t.
Here, we obtain that {T t)} is a continuous path which connects any different uC φ and vC ψ in C w ( A ). Next, for any given u C φ ∈ C w ( A ), we need to find out continuous pathes from uC φ to either θuC φ or uC κφ where θ < 0 and κ < 0 with κφ ∈ S B N ) ∩ A. It is obvious that -uC φ and θuC φ can be connected.
(c) (Homotopy invariance) Suppose that (H colon[0,1] timesoverline{G} to X) is an admissible affine homotopy with a common continuous essential inner map and (h: [0,1] to X) is a continuous path in X such that (h(t) notin H t,partial G)) for all (t in[0,1]).
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 thatv(0)="v,v(1)="1and‖uv(t)−v(t)u‖<ϵfor allt∈[0,1].
If (h:[0,1]rightarrow H) is a continuous path in H such that h(t)notinbigl(L+N t,cdot bigr) bigl(partial{G}cap D(L bigr)quad textit{for all } tin[0,1], then (d (L+N t,cdot), G, h(t))) is constant for all (tin[0,1]).
(c) (Homotopy invariance) If (H colon[0,1] times overline{G} to X) is a bounded admissible affine homotopy with a common continuous essential inner map and (h: [0,1] to X) is a continuous path in X such that (h(t) notin H t,partial G)) for all (t in[0,1]), then the value of (d_{B}(H t, cdot), G, h(t))) is constant for all (t in[0,1]).
(Homotopy invariance) If (H colon[0,1] times overline{G} to X) is a bounded admissible affine homotopy with a common continuous essential inner map and (h: [0,1] to X) is a continuous path in X such that (h(t) notin H t,partial G)) for all (t in[0,1]), then the value of (d_{B}(H t, cdot), G, h(t))) is constant for all (t in[0,1]).
