On perturbation

We assume $ε < \frac{1}{2}$ . As in the textbook, let $F_{i, j}^{(s)}$ be a matrix unit in $A_{0}$ . Bythe lemma we can take a unitary operator $U$ in $C^{*} (A, B)$ such that ${UAU}^{*} \subset B$ and $∥ U - I ∥ \leq ε$ . Put $G_{i, j}^{(s)} = {UF}_{i, j}^{(s)} U^{*}$ . Then we have $∥ G_{i, j}^{(s)} - F_{i, j}^{(s)} ∥ \leq 2 ε$ . In particular, $G_{i, 1}^{(s)}$ is bounded belowby $1 - 2 ε$ on the range of $F_{1, i}^{(s)}$ and vice versa.Thus $Y = \sum_{s, i} G_{i, 1}^{(s)} F_{1, i}^{(s)}$ is an invertibleelement of $B$ . By the polar decomposition $Y = W_{0} | Y |$ , we get anunitary operator $W_{0}$ in $B$ .

Note that $W_{0}$ maps the range of $F_{i, i}^{(s)}$ onto that of $G_{i, i}^{(s)}$ and also it is sufficiently close to $I$ : $∥ W_{0} - I ∥ \leq ∥ Y - | Y | ∥ \cdot ∥ Y ∥^{- 1}$ . From $Y^{*} Y = \sum_{s, i} F_{i, 1}^{(s)} G_{1, 1}^{(s)} F_{1, i}^{(s)}$ , we get $∥ | Y |^{2} - I ∥ \leq 2 dim A_{0} * ε$ , consequently $∥ | Y | - I ∥ \leq 2 dim A_{0} * ε$ . We also have $∥ Y - I ∥ \leq 2 {dimA}_{0} * ε$ , hence $∥ Y - | Y | ∥ \leq 4 dim A_{0} * ε$ and $∥ W_{0} - I ∥ \leq \frac{4 dim A_{0} * ε}{1 - 2 ε} .$

Put $W_{j}^{(s)} = G_{j, 1}^{(s)} W_{0} F_{1, j}^{(s)}$ . Then $W = \sum_{s, j} W_{j}^{(s)}$ is unitary and satisfies ${WF}_{1, j}^{(s)} W^{*} = G_{1, j}^{(s)}$ . Finally, the unitary $V = W^{*} U$ commutes with the $F_{i, j}^{(s)}$ (hence with $A_{0}$ ) and we have ${VAV}^{*} \subset B$ . On the other hand we have $∥ W_{j}^{(s)} - F_{j, j}^{(s)} ∥ \leq ∥ G_{j, 1}^{(s)} W_{0} - F_{j, 1}^{(s)} ∥ \leq ∥ G_{j, 1}^{(s)} - F_{j, 1}^{(s)} ∥ + ∥ W_{0} - I ∥ \leq (2 + \frac{4 dim A_{0} * ε}{1 - 2 ε}) ε .$ This leads to $∥ W - I ∥ \leq (2 + \frac{4 dim A_{0} * ε}{1 - 2 ε}) dim A_{0} \cdot ε \Rightarrow ∥ V - I ∥ \leq (2 + 2 dim A_{0} + \frac{4 (dim A_{0})^{2} ε}{1 - 2 ε}) ε .$