Proof. Suppose that the lemma is false. Then we obtain an $\eta>0$ and a sequence of unital $C^*$ -algebras $\{A_k\}$ and a sequence of n-tuples $\{a_{j,k}\}\subset \{(A_k)_{s.a.}\}$ with $\|a_{j,k}\|\le 1$ ( $1\le j\le n$ ), $k\in \mathbb {N}$ such that

(e2.23) $$ \begin{align} \lim_{k\to\infty}\|a_{i,k}a_{j,k}-a_{j,k}a_{i,k}\|=0,\,\,\, 1\le j\le n, \end{align} $$

and, for case (1), $s\mathrm {Sp}^\eta ((a_{1, k}, a_{2,k},\ldots ,a_{n,k}))=\emptyset , k\in \mathbb {N},$ (or, for case (2), $nSp^\eta ((a_{1,k},a_{2,k},\ldots ,a_{n,k}))=\emptyset , k\in \mathbb {N}.$ )

Consider $B=\prod _kA_k$ and quotient map $\Pi : B\to B/\bigoplus _kA_k.$ Let $s_j=\Pi (\{a_{j,k}\}), j=1,2,\ldots ,n.$ Then $s_is_j=s_js_i, 1\le i,j\le n.$ Let C be the $C^*$ -subalgebra of $B/\bigoplus _k A_k$ generated by $s_0=1, s_1,s_2,\ldots ,s_n.$ Then $C=C(X_0)$ for some non-empty compact subset $X_0$ of ${\mathbb {I}}^n.$ Let $\varphi : C(X_0)\to B/\bigoplus _kA_k$ be the monomorphism given by C with $\varphi (e_j|_{X_0})=s_j, j=0,1,2,\ldots ,n.$

To show (1), consider $D^\eta =\{x_1,x_2,\ldots , x_m\}$ (see Definition 2.8). Fix $x\in X_0.$ Choose $x_j\in D^\eta $ such that $\mathrm {dist}(x_j, x)<\eta /2,$ then $\|\Theta _{x_j, \epsilon }|_{X_0}\|=1.$ Hence, since $\varphi $ is a monomorphism,

(e2.24) $$ \begin{align} \|\varphi(\Theta_{x_j,\eta}|_{X_0})\|=1. \end{align} $$

Recall that $\Pi (\{\Theta _{x_j,\eta }(a_{1,k}, a_{2,k},\ldots ,a_{n,k})\}_{k\in \mathbb {N}})=\varphi (\Theta _{x_j,\eta }|_{X_0}).$ Hence, there is an infinite subset $S\subset \mathbb {N}$ such that, for all $k\in S,$

(e2.25) $$ \begin{align} \|\Theta_{x_j,\eta}(a_{1,k}, a_{2,k},\ldots,a_{n,k})\|\ge 1-\eta. \end{align} $$

It follows that $X_0\subset \bigcup _{\|\Theta _{x_j,\epsilon }(a_{1,k},a_{2,k},\ldots ,a_{n,k})\|\ge 1-\eta }B(x_j, \eta )$ for all $k\in S.$ In other words, $(a_{1,k}, a_{2,k},\ldots ,a_{n,k})$ has $\eta $ -synthetic-spectrum containing $X_0\not =\emptyset $ for all $k\in S.$ This contradicts “ $s\mathrm {Sp}^\eta ((a_{1, k}, a_{2,k},\ldots ,a_{n,k}))=\emptyset , k\in \mathbb {N}$ .” Thus (1) follows.

For (2), we obtain, by the Choi–Effros Theorem [Reference Choi and Effros4], a c.p.c. map $L: C(X_0)\to B$ such that $\Pi \circ L=\varphi .$ Write $L=\{L_k\},$ where each $L_k: C(X_0)\to A_k$ is a c.p.c map, $k\in \mathbb {N}.$ Then

(e2.26) $$ \begin{align} \lim_{k\to\infty}\|L_k(e_j|_{X_0})-a_{j,k}\|=0. \end{align} $$

Let $\{x_1, x_2,\ldots ,x_m\}\subset X_0$ be an $\eta /4$ -dense subset, and $f_1, f_2,\ldots ,f_m\in C(X_0)_+$ with $0\le f_j\le 1, f_j(x)=1$ if $\mathrm {dist}(x, x_j)\le \eta /4$ and $f_j(x)=0$ if $\mathrm {dist}(x, x_j)\ge \eta /2, j=1,2,\ldots ,m.$

Since $\Pi \circ L=\varphi $ is injective, there exists a subsequence $\{k(l)\}\subset \mathbb {N}$ such that

(e2.27) $$ \begin{align} \|L_{k(l)}(f_j)\|\ge (1-\eta/4),\,\,\,j=1,2,\ldots,m. \end{align} $$

Let $f\in C(X_0)_+$ and $x\in X_0$ be such that $f(y)\ge 1$ for all $y\in \overline {B(x, \eta )}.$ Since $\{x_1,x_2,\ldots ,x_l\}$ is $\eta /4$ -dense in $X_0,$ there is $x_j$ such that $\mathrm {dist}(x, x_j)<\eta /4.$ Then $f\ge f_j.$ It follows that

(e2.28) $$ \begin{align} \|L_{k(l)}(f)\|\ge \|L_{k(l)}(f_j)\|\ge (1-\eta/4),\,\,\, j=1,2,\ldots,m. \end{align} $$

Thus, since $\{e_j|_{X_0}: 0\le j\le n\}$ generates $C(X_0),$ by (e 2.26) and (e 2.28), for all large $k(l), X_0$ is an $\eta $ -near-spectrum of $(a_{1,k},a_{2,k},\ldots ,a_{n,k}).$ A contradiction.

Proof. Suppose that the lemma is false. Then there exist $\epsilon _0>0,$ a finite subset ${\cal F}\subset C(X)$ and a sequence of unital $C^*$ -algebras $\{A_n\},$ and two sequences of unital c.p.c. maps $\varphi _{1,n}, \varphi _{2, n}: C(X)\to A_n$ such that

(e2.32) $$ \begin{align} &\lim_{n\to\infty}\|\varphi_{1,n}(g)-\varphi_{2, n}(g)\|=0\,\,\,\mathrm{for\,\,\,all}\,\,\, g\in {\cal G}, \end{align} $$

(e2.33) $$ \begin{align} &\lim_{n\to\infty}\|\varphi_{i,n}(gh)-\varphi_{i,n}(g)\varphi_{i,n}(h)\|=0\,\,\,\mathrm{for\,\,\,all}\,\,\, g,h\in {\cal G},\,\,i=1,2,\,\,\,\mathrm{and}\end{align} $$

(e2.34) $$ \begin{align} &\max\{\|\varphi_{1,n}(f)-\varphi_{2,n}(f)\|: f\in {\cal F}\}\ge \epsilon_0\,\,\,\mathrm{for\,\,\,all}\,\,\, n\in \mathbb{N}. \end{align} $$

Moreover, in the case that $X\subset {\mathbb {I}}^n$ and ${\cal G}=\{e_j|_X: 0\le j\le n\},$ there exists $\epsilon _1>0$ and a finite subset ${\cal H}_0$ such that

(e2.35) $$ \begin{align} \max\{\|\varphi_{1,n}(h(e_j))-h(\varphi_{1,n}(e_j))\|: h\in {\cal H}_0,1\le j\le n\}\ge \epsilon_1 \end{align} $$

for all $n\in \mathbb {N}.$

Put $B=\prod _nA_n.$ Let $\Pi : B\to B/\bigoplus _nA_n$ be the quotient map. Then, since ${\cal G}$ is a generating set of $C(X),$ by (e 2.33), $\psi _i:=\Pi \circ (\{\varphi _{i,n}\}): C(X)\to B/\bigoplus _nA_n$ is a unital homomorphism, $i=1,2,$ and by (e 2.32), $\psi _1=\psi _2.$ Hence,

(e2.36) $$ \begin{align} \psi_1(f)=\psi_2(f)\,\,\,\mathrm{for\,\,\,all}\,\,\, f\in C(X). \end{align} $$

But this implies that, for all sufficiently large $n,$

(e2.37) $$ \begin{align} \|\varphi_{1,n}(f)-\varphi_{2,n}(f)\|<\eta_0/2\,\,\,\mathrm{for\,\,\,all}\,\,\, f\in {\cal F}. \end{align} $$

This leads a contradiction to (e 2.34). The first part of the lemma then follows.

For the second part, since $\psi _1$ is a unital homomorphism, for any $h\in {\cal H}_0,$

(e2.38) $$ \begin{align} \psi_1(h(e_j|_X))=h(\psi_1(e_j|_X)),\,\,\, 1\le j\le n. \end{align} $$

It follows that, for all large n and $h\in {\cal H}_0,$

(e2.39) $$ \begin{align} \|\varphi_{1,n}(h(e_j|_X))-h(\varphi_{1,n}(e_j|_X)\|<\epsilon_1,\,\,\, 1\le j\le n. \end{align} $$

Another contradiction. So the lemma follows.

Proof. Let $D^\eta =\{x_1,x_2,\ldots ,x_m\}\subset {\mathbb {I}}^n$ be as defined in Definition 2.8, where $x_j=(x_{j,1}, x_{j,2},\ldots ,x_{j,n}\}$ ( $1\le j\le m$ ). There is $\delta _1(\eta )>0$ satisfying the following: for any unital $C^*$ -algebra A and any pair of self-adjoint elements $a, b\in A_{s.a.}$ with ${\|a\|, \|b\|\le 1,}$ if $\|a-b\|<\delta _1,$ then

(e2.42) $$ \begin{align} \|\theta_{j,i, \eta}(a)-\theta_{j,i,\eta}(b)\|<\eta/2(n+1),\,\,\, 1\le j\le m, \end{align} $$

where $\theta _{j,i,\eta }$ is defined in Definition 2.8.

Let ${\cal G}=\{e_i: 1\le i\le n\}\cup \{1\}$ and ${\cal H}=\{\theta _{j,i,\eta }: 1\le i\le m, \,\, 1\le j\le n\}.$ For $X={\mathbb {I}}^n,$ choose $\delta _2(n, \eta /4)>0$ such that (the second part of) Lemma 2.13 holds for ${\cal H}, {\cal G}$ above and $\eta /4(n+1)$ (in place of $\eta $ ). Choose $\delta _3>0$ such that, for any unital c.p.c. map $\Phi : C(Y)\to A$ (any unital $C^*$ -algebra A), that

(e2.43) $$ \begin{align} \|\Phi(e_ie_j|_Y)-\Phi(e_i|_Y)\Phi(e_j|_Y)\|<\delta_3\,\,\, (1\le i, j\le n) \end{align} $$

implies, for $1\le j\le m,$

(e2.44) $$ \begin{align} \|\Phi(\Theta_{x_j,\eta}|_Y)-\Phi(\theta_{j,1, \eta}|_Y)\Phi(\theta_{j,2, \eta}|_Y)\ldots \Phi(\theta_{j,n, \eta}|_Y)\| <\eta/4. \end{align} $$

Choose $\delta =\min \{\delta _1, \delta _2, \delta _3\}.$

Suppose that L and $\{a_i: 1\le i\le n\}$ satisfy the assumption of this lemma for $\delta .$ Since $\Theta _{x_j,\eta }=\theta _{j,1,\eta }\theta _{j,2,\eta }\ldots \theta _{j,n,\eta }$ and

(e2.45) $$ \begin{align} \Theta_{x_j,\eta}(b_1, b_2,\ldots,b_n)=\theta_{j,1,\eta}(b_1)\theta_{j,2,\eta}(b_2)\ldots \theta_{j,n,\eta}(b_n) \end{align} $$

for any n-tuple $(b_1,b_2,\ldots ,b_n)$ of self-adjoint elements in A with $\|b_i\|\le 1$ ( $1\le i\le n$ ), by the choice of $\delta $ (and $\delta _1$ ) and by the second part of (e 2.40) and (e 2.42), we have, for each $1\le j\le m,$

(e2.46) $$ \begin{align} \|\Theta_{x_j,\eta}(L(e_1|_X), L(e_2|_X),\ldots, L(e_n|_X))-\Theta_{x_j,\eta}(a_1, a_2,\ldots,a_n)\|<\eta/2. \end{align} $$

By the choice of $\delta _3,$ we also have

(e2.47) $$ \begin{align} \|L(\Theta_{x_j,\eta}|_X)-L(\theta_{j,1, \eta}(e_1|_X))L(\theta_{j,2, \eta}(e_2|_X))\ldots L(\theta_{j,n, \eta}(e_n|_X))\| <\eta/4. \end{align} $$

Define $\tilde L: C({\mathbb {I}}^n)\to A$ by $\tilde L(f)=L(f|_Y)$ for all $f\in C(Y).$ By the choice $\delta $ and applying (the second part of) Lemma 2.13 to $\tilde L_i=\tilde L, i=1,2,$ we obtain

(e2.48) $$ \begin{align} \|\tilde L(\theta_{j,i,\eta}(e_i))-\theta_{j,i,\eta}({\tilde L}(e_i))\|<\eta/4(n+1),\,\,\,1\le i\le n, \,\,\, 1\le j\le m. \end{align} $$

In other words,

(e2.49) $$ \begin{align} \|L(\theta_{j,i,\eta}(e_i)|_Y)-\theta_{j,i,\eta}(L(e_i|_Y))\|<\eta/4(n+1),\,\,\,1\le i\le n, \,\,\, 1\le j\le m. \end{align} $$

Note that $\theta _{j,i,\eta }(e_i)|_Y=\theta _{j,i,\eta }(e_i|_Y).$ It follows that (keeping in mind of (e 2.45))

(e2.50) $$ \begin{align}\nonumber &L(\theta_{j,1, \eta}(e_1|_Y))L(\theta_{j,2, \eta}(e_2|_Y))\ldots L(\theta_{j,n, \eta}(e_n|_Y))\\ &\quad\approx_{n\eta\over{4(n+1)}}\Theta_{x_j,\eta}(L(e_1|_Y), L(e_2|_Y),\dots, L(e_n|_Y)). \end{align} $$

Combining (e 2.50) with (e 2.47), we obtain

(e2.51) $$ \begin{align} \|L(\Theta_{x_j, \eta}|_Y)-\Theta_{x_j,\eta}(L(e_1|_Y), L(e_2|_Y),\dots, L(e_n|_Y))\|<\eta/2,\,\,\, 1\le j\le n. \end{align} $$

We conclude that, by (e 2.51) and (e 2.46),

(e2.52) $$ \begin{align} &\|L(\Theta_{x_j, \eta}|_Y)-\Theta_{x_j,\eta}(a_1, a_2,\ldots,a_n)\|\nonumber \\&\quad\le \|L(\Theta_{x_j, \eta}|_Y)-\Theta_{x_j,\eta}(L(e_1|_Y), L(e_2|_Y), L(e_n|_Y))\| \end{align} $$

(e2.53) $$ \begin{align} &+ \|\Theta_{j,\eta}(L(e_1|_Y), L(e_2|_Y), L(e_n|_Y))-\Theta_{x_j,\eta}(a_1, a_2,\ldots,a_n)\| \end{align} $$

(e2.54) $$ \begin{align} &<\eta/2+\eta/2<\eta.\\[-33pt] \nonumber \end{align} $$

Proof. We may assume that $0<\eta <1.$ We first prove (1).

Let $\{x_1, x_2,\ldots , x_m\}$ be an $\eta /8$ -dense subset of ${\mathbb {I}}^n.$ Choose $f_j\in C({\mathbb {I}}^n)_+$ with $0\le f_j\le 1, f_j(x)=1$ if $\mathrm {dist}(x, x_j)\le \eta /4,$ and $f_j(x)=0,$ if $\mathrm {dist}(x, x_j)\ge \eta /2, j=1,2,\ldots ,m.$ Put ${\cal F}=\{f_j: 1\le j\le m\}.$

Note that $\{e_j: 0\le j\le n\}$ is a generating set of $C({\mathbb {I}}^n).$ Let $\delta _1:=\delta ({\cal F}, \eta /32)>0$ be given by Lemma 2.13 (for $\eta /32$ and ${\cal F}$ as well as ${\mathbb {I}}^n$ ) and $\delta _2:=\delta (n,\eta /2)>0$ be given by Corollary 2.14. Choose $\delta =\min \{\delta _1/2, \delta _2/2, \eta /33\}.$

Now suppose that X and Y are $\delta $ -near-spectra of the n-tuple $(a_1,a_2,\ldots ,a_n).$ Let $L_1,\, L_2: C(X)\to A$ be unital c.p.c maps such that

(e2.58) $$ \begin{align} &\|L_1(e_ie_j|_X)-L_1(e_i|_X)L_1(e_j|_X)\|<\delta,\,\,\,1\le i,\, j\le n, \,\,\, \end{align} $$

(e2.59) $$ \begin{align} &\|L_2(e_ie_j|_Y)-L_2(e_i|_Y)L_2(e_j|_Y)\|<\delta,\,\,\,1\le i,\, j\le n, \,\,\, \end{align} $$

(e2.60) $$ \begin{align} &\|L_1(e_j|_X)-a_j\|<\delta, \,\,\, \|L_2(e_j|_Y)-a_j\|<\delta, \,\,\, 1\le j\le n,\,\,\,\mathrm{and}\,\,\, \end{align} $$

(e2.61) $$ \begin{align} &\|L_1(f|_X)\|\ge (1-\delta) \,\,\,\mathrm{and}\,\,\, \|L_2(f|_Y)\|\ge (1-\delta) \end{align} $$

for any $f\in C({\mathbb {I}}^n)_+$ with $f(x)=1$ for all $x\in B(\xi , \delta ),$ for some $\xi \in X$ (for $L_1$ ), or $\xi \in Y$ (for $L_2$ ).

Let $\xi \in X.$ We may assume that $\xi \in B(x_j, \eta /8).$ Then $B(x_j, \delta )\subset B(\xi , \eta /4).$ On the other hand, by (e 2.61), one must have, as $\eta /4>\delta ,$

(e2.62) $$ \begin{align} \|L_1(f_j|_X)\|\ge (1-\delta). \end{align} $$

Define $L^{\prime }_l: C({\mathbb {I}}^n)\to A$ ( $l=1,2$ ) by $L^{\prime }_1(f)=L_1(f|_X)$ and $L_2'(f)=L_2(f|_Y)$ for all $f\in C({\mathbb {I}}^n).$ By (e 2.59) and applying Lemma 2.13, we have, for $1\le j\le m,$

(e2.63) $$ \begin{align} \|L_1'(f_j)-L_2'(f_j)\|<\eta/32,\,\,\, \mathrm{or}\,\,\, \|L_1(f_j|_X)-L_2(f_j|_Y)\|<\eta/32. \end{align} $$

Then, by (e 2.62), we obtain, for $1\le j\le m,$

(e2.64) $$ \begin{align} \|L_2(f_j|_Y)\|\ge \|L_1(f_j|_X)\|-\eta/32\ge 1-\delta-\eta/32>3/4. \end{align} $$

Since $f_j(x)=0,$ if $\mathrm {dist}(x, x_j)\ge \eta /2,$ we must have $\mathrm {dist}(x_j, Y)<\eta /2.$ Therefore,

(e2.65) $$ \begin{align} \mathrm{dist}(\xi, Y)\le \mathrm{dist}(\xi, x_j)+\mathrm{dist}(x_j, Y)<\eta/8+\eta/2=5\eta/8. \end{align} $$

This holds for all $\xi \in X.$ Exchanging the role of X and $Y,$ we obtain

(e2.66) $$ \begin{align} d_H(X, Y)<\eta. \end{align} $$

Let $D^\eta =\{\xi _1,\xi _2,\ldots ,\xi _N\}$ and $Z=\bigcup _{\|\Theta _{\xi _j, \eta }(a_1,a_2,\ldots ,a_k)\|\ge 1-\eta } \overline {B(\xi _j, \eta )}.$

By the choice of $\delta $ and applying Corollary 2.14, we have that

(e2.67) $$ \begin{align} \|L_1(\Theta_{\xi_j,\eta}|_X)-\Theta_{\xi_j,\eta}(a_1,a_2,\ldots,a_k)\|<\eta/2,\,\,\, j=1,2,\ldots,N. \end{align} $$

Pick $\xi \in X.$ There is $\xi _j\in D^\eta $ such that $\mathrm {dist}(\xi , \xi _j)<\eta /2.$ Recall that $\Theta _{\xi _j,\eta }(y)=1$ if $\mathrm {dist}(y, \xi _j)\le 3\eta /4.$ As $\delta <\eta /32,$ we have $\eta /2+\delta <3\eta /4.$ Hence, $\Theta _{\xi _j, \eta }(y)=1$ for all $y\in B(\xi , \delta )\subset B(\xi _j, 3\eta /4).$ It follows that

(e2.68) $$ \begin{align} \|L_1(\Theta_{\xi_j,\eta}|_X)\|\ge 1-\delta. \end{align} $$

Thus, by (e 2.67),

(e2.69) $$ \begin{align} \|\Theta_{\xi_j,\eta}(a_1,a_2,\ldots,a_k)\|\ge 1-\delta-\eta/2\ge 1-\eta. \end{align} $$

This implies that

(e2.70) $$ \begin{align} X\subset \bigcup_{\|\Theta_{\xi_j,\eta}(a_1,a_2,\ldots,a_k)\|\ge 1-\eta}\overline{B(\xi_j, \eta)}=s\mathrm{Sp}^\eta((a_1,a_2,\ldots,a_k)). \end{align} $$

Since $X\not =\emptyset ,$ we conclude that $Z\not =\emptyset .$ The same argument shows that $Y\subset s\mathrm {Sp}^\eta ((a_1,a_2,\ldots ,a_k)).$

Next, let $\|\Theta _{\xi _j,\eta }(a_1,a_2,\ldots ,a_k)\|\ge 1-\eta .$ By (e 2.67),

(e2.71) $$ \begin{align} \|L_1(\Theta_{\xi_j, \eta}|_X)\|\ge 1-\eta-\eta/2>1/4. \end{align} $$

Hence, $\Theta _{\xi _j, \eta }|_X\not =0.$ Thus,

(e2.72) $$ \begin{align} X\cap B(\xi_j, \eta)\not=\emptyset. \end{align} $$

It follows that

(e2.73) $$ \begin{align} B(\xi_j, \eta)\subset X_{2\eta}. \end{align} $$

Hence,

(e2.74) $$ \begin{align} Z\subset X_{2\eta}. \end{align} $$

This completes the proof of part (1).

For (2), let us keep notation above. Then $\Omega _1\subset Z$ is immediate.

Suppose that $\Omega _2\not =\emptyset .$ Then that $\Omega _2\subset \Omega _1 \subset (\Omega _2)_{2\eta }$ follow from part (1). It remains to show that $\Omega _2\subset \overline {X_\eta }.$

There is a unital c.p.c. map $\Phi : C(\Omega _2)\to A/J$ such that

(e2.75) $$ \begin{align} &\|\Phi(e_ie_j|_{\Omega_2})-\Phi(e_i|_{\Omega_2})\Phi(e_j|_{\Omega_2})\|<\delta,\,\,\, 1\le i,j\le k; \end{align} $$

(e2.76) $$ \begin{align} &\|\Phi(e_j|_{\Omega_2})-\pi(a_j)\|<\delta,\,\,\, 1\le j\le k,\,\,\,\mathrm{and}\,\,\, \end{align} $$

(e2.77) $$ \begin{align} &\|\Phi(f)\|\ge 1-\delta \end{align} $$

for any $f\in C(\Omega _2)_+$ with $f(x)=1$ if $x\in B(\zeta , \delta )$ for some $\zeta \in \Omega _2.$

Choose $y\in \Omega _2.$ We may assume that $\mathrm {dist}(y, x_j)<\eta /8.$ Hence, $B(y, \delta )\subset \mathrm {dist}(x_j,\eta /4).$ It follows from (e 2.77) that

(e2.78) $$ \begin{align} \|\Phi(f_j)\|\ge 1-\delta. \end{align} $$

Let $\tilde L_1: C({\mathbb {I}}^n)\to A/J$ be defined $\tilde L_1(f)=\pi \circ L_1(f|_X)$ and $\tilde \Phi : C({\mathbb {I}}^n)\to A/J$ by $\tilde \Phi (f)=\Theta (f|_{\Omega _2})$ for all $f\in C({\mathbb {I}}^n).$ By the choice of $\delta ,$ (e 2.59), (e 2.76), and (e 2.60), and applying Lemma 2.13 to $\pi \circ \tilde L_1$ and $\tilde \Phi ,$ we have

(e2.79) $$ \begin{align} \|\pi\circ \tilde L_1(f_j)-\tilde \Phi(f_j)\|<\eta/32,\,\,\, 1\le j\le m. \end{align} $$

In other words,

(e2.80) $$ \begin{align} \|\pi\circ L_1(f_j|_X)-\Phi(f_j|_{\Omega_2})\|<\eta/32, \,\,\, 1\le j\le m. \end{align} $$

Thus,

(e2.81) $$ \begin{align} \|L_1(f_j|_X)\|\ge \|\pi\circ L_1(f_j|_X)\|\ge 1-\delta-\eta/32\ge 3/4. \end{align} $$

Hence, $f_j|_X\not =0.$ It follows that $\mathrm {dist}(x_j, X)<\eta /2.$

(e2.82) $$ \begin{align} \mathrm{dist}(y, X)<\eta/2+\eta/8<\eta. \end{align} $$

Hence, $\Omega _2\subset \overline {X_\eta }.$

Proof. By Proposition 2.16, it suffices to show that ${\mathrm sp}(a_1+a_2)\subset n{\mathrm Sp}^\eta((a_1,a_2))$ . Otherwise, one obtains $\eta _0>0,$ a sequence of unital $C^*$ -algebras $\{A_n\},$ a sequence of elements $a_n, b_n\in (A_n)_{s.a.}$ with $\|a_n\|, \|b_n\|\le 1$ such that

(e2.84) $$ \begin{align} \lim_{n\to\infty}\|a_nb_n-b_na_n\|=0, \end{align} $$

and a sequence of numbers $\lambda _n\in \mathrm {sp}(a_n+ib_n) \setminus s\mathrm {Sp}^{\eta _0}((a_n, b_n)).$ Put $c_n=a_n+ib_n$ and $Z_n=n\mathrm {Sp}^{\eta _0/4}((a_n, b_n)), n\in \mathbb {N}.$ Note that $\lambda _n\in {\mathbb {I}}^2\subset \mathbb {C}$ and $Z_n\subset {\mathbb {I}}^2, n\in \mathbb {N}.$ Since $(F({\mathbb {I}}^2), d_H)$ is compact, without loss of generality, by passing to a subsequence, we may assume that $Z_n\to Z$ for some compact subset $Z\subset {\mathbb {I}}^n$ in $(F({\mathbb {I}}^2), d_H)$ as $n\to \infty ,$ and $\lambda _n\to \lambda \in {\mathbb {I}}^2.$

By (4) of Remark 2.11, $(n\mathrm {Sp}^{\eta _0/4}((a_n, b_n)))_{\eta _0/4}\subset n\mathrm {Sp}^{\eta _0}((a_n, b_n)), n\in \mathbb {N}.$ It follows that

(e2.85) $$ \begin{align} \mathrm{dist}(\lambda_n, Z_n)\ge \eta_0/4,\,\,\,n\in \mathbb{N}. \end{align} $$

Hence, there exists $n_0\in \mathbb {N}$ such that

(e2.86) $$ \begin{align} \mathrm{dist}(\lambda, Z_{n})\ge \eta_0/5 \,\,\,\mathrm{for\,\,\,all}\,\,\, n\ge n_0. \end{align} $$

Let $B=\prod _nA_n$ and $\Pi : B\to B/\bigoplus _nA_n$ be the quotient map. Put $x=\Pi (\{c_n\}).$ Then, by (e 2.84), x is normal. Let $\mathrm {sp}(x)=X.$ Then $\varphi : C(X)\to B/\bigoplus _nA_n$ defined by $\varphi (f)=f(x)$ for $f\in C(X)$ is a unital monomorphism. By the Choi–Effros Lifting Theorem [Reference Choi and Effros4], there is a sequence of unital c.p.c. maps $L_n: C(X)\to A_n$ such that $\Pi \circ \{L_n\}=\varphi .$ Let $\delta :=\delta (2, \eta _0/4)$ be as in Proposition 2.16. Then, since $\varphi $ is a homomorphism, for all large $n, X$ is a (non-empty) $\delta $ -near-spectrum for $(a_n, b_n).$ Applying Proposition 2.16, we may assume that, for all $n,$

(e2.87) $$ \begin{align} X\subset Z_n. \end{align} $$

Note that, by (e 2.86) and (e 2.87), $\lambda \not \in X.$ Let $y=(\lambda -x)^{-1}.$ Note that $\Pi (\{\lambda _n-(a_n+ib_n)\})=\lambda -x.$ Hence, there are $y_n\in A_n$ such that ( $\|y_n\|\le \|y\|+1$ )

(e2.88) $$ \begin{align} \Pi(\{y_n\})=y\,\,\,\mathrm{and}\,\,\, \lim_{n\to\infty}\|y_n(\lambda_n-(a_n+ib_n))-1\|=0. \end{align} $$

Hence, for all large $n,$

(e2.89) $$ \begin{align} \lambda_n\not\in \mathrm{sp}(a_n+ib_n). \end{align} $$

This is a contradiction. So the proposition follows.

Proof. We prove this by induction on the number of convex sets.

If Y is convex, then Y is contractive. Hence, $K_0(C(Y))=\mathbb {Z}$ and $K_1(C(Y))=\{0\}.$

Suppose that the proposition is proved for m many closed convex sets, $m\ge 1.$

Now suppose that Y is a union of $m+1$ many compact convex sets in $\mathbb {R}^n.$ Write $Y=\cup _{i=1}^{m+1} Y_i,$ where each $Y_i$ is a compact convex subset of $\mathbb {R}^n.$

Put $X_1=\cup _{i=1}^m Y_i.$ Then $Y_{m+1}\cap X_1=\cup _{i=1}^m (Y_i\cap Y_{m+1})$ which is a union of m many compact convex subsets. By the induction assumption, $K_i(C(X_1\cap Y_{m+1}))$ is finitely generated ( $i=0,1$ ).

Then

(e3.91) $$ \begin{align} C(Y)=\{(f, g): (f, g)\in C(X_1)\oplus C(Y_{m+1}): f|_{X_1\cap Y_{m+1}}=g|_{X_1\cap Y_{m+1}}\} \end{align} $$

is a pull back. By the Mayer–Vietoris sequence in K-theory for $C^*$ -algebras (see [Reference Hilgert15]), we obtain the following commutative diagram:

(e3.92) $$ \begin{align} \begin{array}{ccccc} {\small{K_0(C(Y))}} & {\longrightarrow}& K_0(C(X_1))\oplus K_0(C(Y_{m+1})) & {\rightarrow} & K_0(C(X_1\cap Y_{m+1}))\\ \uparrow & && & \downarrow\\ K_1(C(X_1\cap Y_{m+1})) & {\leftarrow}& K_1(C(X_1))\oplus K_1(C(Y_{m+1})) & {\leftarrow} & K_1(C(Y))\,. \end{array} \end{align} $$

Since we have established that $K_i(C(X_1)), K_i(C(Y_{m+1})),$ and $K_i(C(X_1\cap Y_{m+1}))$ are all finitely generated abelian groups, we conclude that $K_i(C(Y))$ is also finitely generated. This ends the induction.

Proof. Since X has property (F), we may write $C(X)=C(\sqcup _{i=1}^m X_i)=\bigoplus _{i=1}^m C(X_i),$ where each $X_i$ is path connected and $K_j(C(X_i))$ is finitely generated ( $j=0,1$ ), $ 1\le i\le m.$ Let $d_i\in C(X)$ be such that ${d_i}|_{X_i}=1$ and $d_i(x)=0$ if $x\not \in X_i, 1\le i\le m.$ Put $E_i=\varphi (d_i)$ ( $1\le i\le m$ ). By considering $\varphi |_{C(X_i)}\to E_iBE_i$ for each $i,$ we reduce the general case to the case that X has only one path connected component. Note that, if originally, X is not connected, then we do not need to consider the “Moreover” part of the theorem. If X is connected, then we need to point out that $\varphi (1_{C(X)})=1_A.$

So now we assume that X is connected. Put $\Omega :=\Omega (X)=X\setminus \{x\}$ for some $x\in X.$ Then we have a splitting short exact sequence

$$ \begin{align*}0\to C_0(\Omega)\to C(X)\to \mathbb{C}\to 0. \end{align*} $$

Recall that $K_i(C(X))$ is finitely generated ( $i=0,1$ ). It follows that, for any unital $C^*$ -algebra $B,$

(e3.96) $$ \begin{align} KK(C(X), B)=KK(C_0(\Omega), B)\oplus KK(\mathbb{C}, B). \end{align} $$

Define $\varphi _1: C(X)\to B$ by $\varphi _1(f)=f(x)\cdot 1_B$ for all $f\in C(X).$ Then one computes that, by (e 3.94),

(e3.97) $$ \begin{align} [\varphi_1]=[\varphi]\,\,\,\mathrm{in}\,\, \, KK(C(X), B). \end{align} $$

Fix $\epsilon>0$ and finite subset ${\cal F}\subset C(X).$ Applying Theorem A of [Reference Dadarlat6], we obtain an integer $k,$ a unitary $u\in M_{k+1}(B),$ and $x_1, x_2,\ldots ,x_k\in X$ such that, for all $f\in {\cal F},$

(e3.98) $$ \begin{align}& \|\mathrm{diag}(\varphi(f), f(x_1), f(x_2),\ldots,f(x_k))\nonumber\\&\qquad -u^*\mathrm{diag}(\varphi_1(f), f(x_1),f(x_2),\ldots,f(x_k))u\|<\epsilon/3. \end{align} $$

Define $\varphi _0: C(X)\to M_k(B)$ by $\varphi _0(f)=\mathrm {diag}(f(x_1), f(x_2),\ldots ,f(x_k))$ for all $f\in C(X).$ Then we may rewrite $\varphi _0(f)=\sum _{i=1}^k f(x_i)p_i$ for all $f\in C(X),$ where $\{p_1, p_2,\ldots ,p_k\}$ is a set of mutually orthogonal projections such that $\sum _{i=1}^k p_i=1_{M_k}.$ Choose mutually orthogonal projections $p_{0,1}, p_{0,2},\ldots ,p_{0,k}\in M_{k+1}(B)$ such that $p_01_{M_k}=1_{M_k} p_0=0,$ where $p_0=\sum _{i=1}^k p_{0,i},$ and $[p_{0,i}+p_i]=0$ in $K_0(B)$ (recall that B is purely infinite simple). Define $\varphi _{00}: C(X)\to (1_{M_k}+p_0)M_{k+1}(B)(1_{M_k}+p_0)\subset M_{k+1}(B)$ by $\varphi _{00}(f)=\sum _{i=1}^k f(x_i)(p_i+p_{0,i})$ for all $f\in C(X).$ By replacing $\varphi _0$ by $\varphi _{00}$ and u by $u\oplus 1_B,$ we may assume, without loss of generality, that $[p_i]=0, i=1,2,\ldots ,k,$ and

(e3.99) $$ \begin{align} \|\mathrm{diag}(\varphi(f), \varphi_0(f))-u^*\mathrm{diag}(\varphi_1(f), \varphi_0(f))u\|<\epsilon/3\,\,\,\mathrm{for\,\,\,all}\,\,\, f\in {\cal F}. \end{align} $$

(But then u is a unitary in $M_{k+2}(B)$ .)

Recall that, by Zhang’s theorem [Reference Zhang37], B has real rank zero. Since $\varphi $ is injective, by applying Lemma 4.1 of [Reference Elliott, Gong, Lin and Pasnicu8], we obtain a set of mutually orthogonal nonzero projections $q_1, q_2,\ldots ,q_k\in B$ with $q=\sum _{i=1}^kq_i<1_B$ such that

(e3.100) $$ \begin{align} \left\|\varphi(f)-\left((1-q)\varphi(f)(1-q)+\sum_{i=1}^k f(x_i) q_i\right)\right\|<\epsilon/3\,\,\,\mathrm{for\,\,\,all}\,\,\, f\in {\cal F}. \end{align} $$

There are nonzero subprojections $q_i'\le q_i$ such that $[q_i']=0$ in $K_0(B), i=1,2,\ldots ,k$ (recall again that B is purely infinite simple). We may then write

(e3.101) $$ \begin{align} \left\|\varphi(f)-\left(\gamma(f)+\sum_{i=1}^kf(x_i)q_i'\right)\right\|<\epsilon/3\,\,\,\mathrm{for\,\,\,all}\,\,\, f\in {\cal F}, \end{align} $$

where $\gamma (f)=(1-q)\varphi (f)(1-q)+\sum _{i=1}^k f(x_i)(q_i-q_i')$ for all $f\in C(X).$ There are $w_i\in M_{2k+3}(B)$ such that

(e3.102) $$ \begin{align} w_i^*w_i=q_i'\,\,\,\mathrm{and}\,\,\, w_iw_i^*=q_i'\oplus p_i,\,\,\, i=1,2,\ldots,k. \end{align} $$

Let $w=(1-q)+\sum _{i=1}^k (q_i-q_i')+\sum _{i=1}^k w_i.$ Then $w^*w=1_B$ and $ww^*=1_{M_{k+1}}+\sum _{i=1}^k p_i.$ Moreover,

(e3.103) $$ \begin{align} \ \ w^*(\mathrm{diag}\left(\gamma(f)+\sum_{i=1}^k f(x_i)q_i', \varphi_0(f)\right)w=\gamma(f)+\sum_{i=1}^kf(x_i)q_i' \end{align} $$

Hence,

(e3.104) $$ \begin{align} \|\varphi(f)-w^*(\mathrm{diag}\left(\gamma(f)+\sum_{i=1}^k f(x_i)q_i', \varphi_0(f)\right)w\|<\epsilon/3\,\,\,\mathrm{for\,\,\,all}\,\,\, f\in {\cal F}. \end{align} $$

Define, for all $f\in C(X),$

(e3.105) $$ \begin{align} \psi(f)=w^*u^*(\mathrm{diag}(\varphi_1(f), \varphi_0(f))uw= f(x) w^*u^*1_Au^*w+\sum_{i=1}^k f(x_i)w^*u^*p_iuw. \end{align} $$

Then $\psi $ is a unital homomorphism from $C(X)$ to B with finite-dimensional range and, by (e 3.99), (e 3.101), and (e 3.104), for all $f\in {\cal F},$

$$ \begin{align*}\nonumber &\|\varphi(f)-\psi(f)\| \le \|\varphi(f)-w^*(\mathrm{diag}(\gamma(f)+\sum_{i=1}^k f(x_i)q_i', \varphi_0(f)))w\|\\\nonumber &\quad +\|w^*(\mathrm{diag}(\gamma(f)+\sum_{i=1}^k f(x_i)q_i', \varphi_0(f)))w-w^*u^*\mathrm{diag}(\varphi_1(f), \varphi_0(f))uw\|\\\nonumber &\quad<\epsilon/3+\|\mathrm{diag}(\gamma(f)+\sum_{i=1}^k f(x_i)q_i', \varphi_0(f))-u^*\mathrm{diag}(\varphi_1(f), \varphi_0(f))u\|\\\nonumber &\quad<\epsilon/3+\epsilon/3+\|\mathrm{diag}(\varphi(f), \varphi_0(f))-u^*\mathrm{diag}(\varphi_1(f), \varphi_0(f))u\|<2\epsilon/3+\epsilon/3=\epsilon. \end{align*} $$

Finally, since $[p_i]=0$ in $K_0(B)$ as we arranged earlier, by (e 3.105), the “Moreover” part of the statement also holds.

Proof. Let $\epsilon>0$ and finite subset ${\cal F}\subset C({\mathbb {I}}^n)$ be given. Choose $0<\delta <\min \{\epsilon /2, 1/8\}$ such that, for all $g\in {\cal F},$

(e3.108) $$ \begin{align} |g(\xi)-g(\xi')|<\epsilon/4\,\,\,\mathrm{for\,\,\,all}\,\,\, \xi, \xi'\in {\mathbb{I}}^n\,\,\,\mathrm{with}\,\,\, \mathrm{dist}(\xi, \xi')<2\delta. \end{align} $$

Suppose that Y is a compact subset of ${\mathbb {I}}^n$ such that $X\subset Y\subset X_\delta $ which has property (F). Since Y is compact, there is $\delta _1>0$ with $0<\delta _1<\delta $ such that

(e3.109) $$ \begin{align} Y\subset X_{\delta_1}. \end{align} $$

Put $\eta _0={\delta -\delta _1\over {4}}>0$ and $r=\delta _1+\eta _0.$ Then $0<r=\delta _1+\eta _0<\delta _1+2\eta _0<\delta .$ There is a finite $\eta _0$ -net $\{\xi _1, \xi _2,\ldots , \xi _m\}\subset X$ such that $\cup _{i=1}^m B(\xi _i,\eta _0)\supset X,$ where $B(\xi _i, \eta _0)=\{\xi \in \mathbb {R}^n: \mathrm {dist}(\xi , \xi _i)< \eta _0\}.$ Then $\cup _{i=1}^m B(\xi _i, r)\supset Y.$ Note that we assume that $\xi _i\not =\xi _j,$ if $i\not =j.$

Choose a nonzero projection $p\in B$ with $[p]=0$ in $K_0(B)$ and $1-p\not =0.$ Since $pBp$ is a purely infinite simple $C^*$ -algebra, there is a unital embedding $\Phi : O_2\to pBp.$ Applying Theorem 2.8 of [Reference Kirchberg and Phillips18], one obtains a unital embedding $\psi _0: C(Y)\to \Phi (O_2)\subset pBp.$ Note $[\psi _0]=0$ in $KK(C(Y),pBp).$ There is a partial isometry $w\in B$ such that $w^*w=1-p$ and $ww^*=1.$ Define $\psi _1: C(Y)\to B$ by $\psi _1(f)=w^*\varphi (f|_X)w\oplus \psi _0(f)$ for all $f\in C(Y).$ Then $\psi _1$ is a unital injective homomorphism.

Choose mutually orthogonal nonzero projections $p_1, p_2,\ldots ,p_m\in \Phi (O_2)\subset pBp$ such that $\sum _{i=1}^m p_i=p.$ Define $\varphi _0: C(X)\to \Phi (O_2)\subset pBp$ by $\varphi _0(f)=\sum _{i=1}^mf(\xi _i)p_i.$ Then $[\varphi _0]=0$ in $KK(C(X),B).$ Define $\varphi _1: C(X)\to B$ by $\varphi _1(f)=w^*\varphi (f)w+\varphi _0(f)$ for all $f\in C(X).$ Then $\varphi _1$ is injective. One computes that

(e3.110) $$ \begin{align} [\varphi]=[\varphi_1]\,\,\, \mathrm{in}\,\,\, KK(C(X), B). \end{align} $$

It follows from Theorem 1.7 of [Reference Dadarlat6] that there is a unitary $u\in B$ such that

(e3.111) $$ \begin{align} \|\varphi(f|_X)-u^*\varphi_1(f|_X)u\|<\epsilon/4\,\,\,\mathrm{for\,\,\,all}\,\,\, f\in {\cal F}. \end{align} $$

Applying Theorem 3.5, there is a unital homomorphism $\psi _{00}: C(Y)\to \Phi (O_2)\subset pBp$ with finite-dimensional range such that

(e3.112) $$ \begin{align} \|\psi_0(g|_Y)-\psi_{00}(g|_Y)\|<\epsilon/4\,\,\,\mathrm{for\,\,\,all}\,\,\, g\in {\cal F}. \end{align} $$

We may write $\psi _{00}(g)=\sum _{j=1}^K g(y_j) d_i$ for all $g\in C(Y),$ where $y_j\in Y$ and $d_1, d_2,\ldots ,d_K$ are mutually orthogonal nonzero projections in $\Phi (O_2)$ with $p=\sum _{j=1}^K d_j.$ By choosing a better approximation if necessary, without loss of generality, we may assume that $\{y_1, y_2,\ldots ,y_K\}$ is $\eta _0$ -dense in $Y, K\ge m.$ Recall that $\{\xi _1, \xi _2,\ldots ,\xi _n\}$ is $\eta _0$ -dense in X and r-dense in $Y.$ For each $i,$ choose exactly one $y_{(i,1)}\subset \{y_1, y_2,\ldots ,y_K\}$ such that $\mathrm {dist}(y_{(i,1)}, \xi _i)<\eta _0.$ Then, we may assume $\cup _{i=1}^m\{y_{(i, j)}: 1\le j\le k(i)\}=\{y_1, y_2,\ldots ,y_K\},$ where $\{y_{(i,j)}: 1\le j\le k(i), 1\le i\le m\}$ is a set of distinct points such that

(e3.113) $$ \begin{align} \mathrm{dist}(y_{(i,j)}, \xi_i)<\delta_1+2\eta_0<\delta,\,\,\,1\le j\le k(i),\,\,\,1\le i\le n. \end{align} $$

Note that we view $(i,j)\in \{1,2,\ldots ,K\}.$ Then set $q_i'=\sum _{j=1}^{k(i)}d_{(i,j)}.$ Put $\psi _{00}': C(Y)\to \Phi (O_2)\subset pBp,$ where $\psi _{00}'(g)=\sum _{i=1}^m g(\xi _i)q_i'$ for all $g\in C(Y).$

By the choice of $\delta ,$ we have, for all $g\in {\cal F},$

(e3.114) $$ \begin{align} \|\psi_{00}(g|_Y)-\psi_{00}'(g|_Y)\|<\epsilon/4. \end{align} $$

Recall that $p_i, q_i'\in \Phi (O_2).$ Hence, $[p_i]=[q_i']=0$ in $K_0(B).$ There are partial isometries $w_i\in pB(u^*pu)$ such that $w_i^*w_i=u^*p_iu$ and $w_iw_i^*=q_i', i=1,2,\ldots ,m.$ Define $v=(1-p)u\oplus \sum _{i=1}^m w_i.$ Then

(e3.115) $$ \begin{align} v^*v&=u^*(1-p)u\oplus \sum_{i=1}^m u^*p_iu=u^*(1-p)u+u^*pu=1\,\,\,\mathrm{and}\,\,\, \end{align} $$

(e3.116) $$ \begin{align} vv^*&=(1-p)uu^*(1-p)\oplus \sum_{i=1}^m q_i'=(1-p)+\sum_{i=1}^Kd_i=1. \end{align} $$

So v is a unitary in $B.$ Note that

(e3.117) $$ \begin{align} v^*\psi_{00}'(g)v=v^*\left(\sum_{i=1}^n g(\xi_i)q_i'\right)v= \sum_{i=1}^n g(\xi_i) u^*p_iu=u^*\varphi_0(g|_X)u \,\,\,\mathrm{for\,\,\,all}\,\,\, g\in C(Y). \end{align} $$

Then, combining with (e 3.112) and (e 3.114), for all $g\in {\cal F},$

(e3.118) $$ \begin{align}\nonumber \|v^*\psi_0(g|_Y)v-u^*\varphi_0(g|_X)u\|&=\|v^*\psi_0(g|_Y)v-v^*\psi_{00}(g|_Y)v\|\\&\quad+ \|v^*\psi_{00}(g|_Y)v-v^*\psi_{00}'(g|_Y)v\|\nonumber\\ &\quad<\epsilon/4+\epsilon/4=\epsilon/2. \end{align} $$

Define $\psi : C(Y)\to B$ by

(e3.119) $$ \begin{align} \psi(g)=v^*(w^*\varphi(g|_X)w+\psi_0(g|_Y))v \,\,\,\mathrm{for\,\,\,all}\,\,\, g\in C(Y). \end{align} $$

Then $\psi $ is a unital injective homomorphism. Moreover, for all $g\in {\cal F},$ by (e 3.111), the definition of $\varphi _1$ and v above and by (e 3.118),

(e3.120) $$ \begin{align}\nonumber \|\varphi(g|_X)&-\psi(g|_Y)\|=\|\varphi(g|_X)-u^*\varphi_1(g|_X)u\|+\|u^*\varphi_1(g|_X)u-\psi(g|_Y)\|\\\nonumber &<\epsilon/4+\|u^*w^*\varphi(g|_X)wu\oplus u^*\varphi_0(g|_X)u-v^*(w^*\varphi(g|_X)w\oplus \psi_0(g|_Y))v\|\\\nonumber &=\epsilon/4+\|u^*w^*\varphi(g|_X)wu\oplus u^*\varphi_0(g|_X)u-v^*\nonumber\\&\times((1-p)w^*\varphi(g|_X)w(1-p)\oplus \psi_0(g|_Y))v\|\nonumber\\\nonumber &=\epsilon/4+\|u^*w^*\varphi(g|_X)wu\oplus u^*\varphi_0(g|_X)\nonumber\\&\quad u-u^*w^*\varphi(g|_X)wu\oplus v^*\psi_0(g|_Y))v\|\nonumber\\ &=\epsilon/4+\|u^*\varphi_0(g|_X)u-v^*\psi_0(g|_Y)v\|<\epsilon/4+\epsilon/2<\epsilon. \end{align} $$

To see the second formula in (e 3.107), we may write, from (e 3.119), since $\psi _0$ factors through $\Phi (O_2),$

(e3.121) $$ \begin{align} [\psi]=[\varphi\circ \pi_X]+[\psi_0]=[\varphi\circ \pi_X] \,\,\, \mathrm{in}\,\,\, KK(C(Y), B).\\[-33pt] \nonumber\end{align} $$

Proof. At the beginning of the proof of Theorem 3.6, choose ${\cal F}=\{e_j: 1\le j\le n\}.$ Then, with $\delta :=\epsilon /8,$ when $\mathrm {dist}(\xi , \xi ')<2\delta ,$

(e3.123) $$ \begin{align} |e_j(\xi)-e_j(\xi')|<\epsilon/4. \end{align} $$

Then the rest of the proof of Theorem 3.6 applies (with $\delta =\epsilon /8$ ).

Proof. Let $\epsilon>0.$ It follows from Corollary 1.20 of [Reference Lin22] that there exists $\delta _1>0$ such that when $\|s_is_j-s_js_i\|<\delta _1,$ there are $a_1, a_2,\ldots ,a_n \in A_{s.a.}$ such that

(e3.127) $$ \begin{align} a_ia_j=a_ja_i\,\,\,\mathrm{and}\,\,\, \|s_i-a_i\|<\epsilon/2\,\,\,\mathrm{for\,\,\,all}\,\,\, 1\le i,\,j\le n. \end{align} $$

Let C be the commutative $C^*$ -subalgebra generated by $a_1,a_2,\ldots ,a_n.$ Then there exists a unital homomorphism $\varphi _0: C({\mathbb {I}}^n)\to C$ such that $\varphi _0(e_j)=a_j, j=1,2,\ldots ,n.$ Hence, $C\cong C(X_0)$ for some compact subset of ${\mathbb {I}}^n.$ Denote $\varphi _1: C(X_0)\to C$ the isomorphism given above. We may assume that $\varphi _1(e_j|{X_0})=a_j, j=1,2,\ldots ,n.$

Put $\delta =\min \{\epsilon /2, \delta _1/2\}.$ Choose any $k\in \mathbb {N}$ with $2\sqrt {n}/k<\delta $ and $1/k\le \eta <\delta .$ By Corollary 3.8, we obtain a $1/k$ -brick combination $X\subset {\mathbb {I}}^n$ and a unital injective homomorphism $\varphi : C(X)\to B$ such that

(e3.128) $$ \begin{align} \|\varphi(e_j|_X)-a_j\|<\epsilon/2, \,\,\, j=1,2,\ldots,n. \end{align} $$

Hence,

(e3.129) $$ \begin{align} \|\varphi(e_j|_X)-s_j\|<\epsilon,\,\,\, j=1,2,\ldots,n.\\[-33pt] \nonumber \end{align} $$

Proof. Let $\epsilon>0.$ Choose $\delta =\epsilon /128.$ Since X is a $\delta $ -near-spectrum of $(T_1,T_2,\ldots ,T_n),$ by Definition 2.10, there is a unital c.p.c. map $\Phi : C(X)\to A$ such that

(e4.132) $$ \begin{align} \|\Phi(e_j|_X)-T_j\|<\delta \,\,\,\mathrm{and}\,\,\, \|\Phi(e_ie_j|_X)-\Phi(e_i|_X)\Phi(e_j|_X)\|<\delta,\,\,\, i,j\in \{1,2,\ldots,n\}. \end{align} $$

Since Y is a $\delta $ -spectrum of $(\pi (T_1),\pi (T_2),\ldots , \pi (T_n)),$ there exists, by Definition 2.10 (near (e 2.21)), a compact subset $Z\subset {\mathbb {I}}^n$ such that $Z\subset Y\subset Z_\delta $ and a unital monomorphism $\varphi : C(Z)\to A/J$ such that

(e4.133) $$ \begin{align} \|\varphi(e_j|_Z)-\pi(T_j)\|<\delta,\,\,\, j=1,2,\ldots,n. \end{align} $$

Let $e'\in A/J$ be a nonzero projection such that $1-e'\not =0.$ Choose a partial isometry $w\in A/J$ such that

(e4.134) $$ \begin{align} w^*w=1\,\,\,\mathrm{and}\,\,\, ww^*=e\le e'. \end{align} $$

It follows that $[1-e]=0$ in $K_0(A/J).$ Let $\{\lambda _1, \lambda _2,\dots ,\lambda _m\}$ be a $\delta /2$ -dense subset of Z which is also $\delta $ -dense in $Y.$ Let $q_1, q_2,\ldots ,q_m$ be mutually orthogonal nonzero projections in $(1-e)(A/J)(1-e)$ such that $[q_i]=0$ in $K_0(A/J).$ Define $\psi : C(Z)\to A/J$ by

(e4.135) $$ \begin{align} \psi(f)=\sum_{k=1}^m f(\lambda_k)q_k+w\varphi(f)w^*\,\,\,\mathrm{for\,\,\,all}\,\,\, f\in C(Z). \end{align} $$

Note that $[\varphi ]=[\psi ]$ in $KK(C(Z), A/J).$ Then, since $A/J$ is purely infinite and simple, by Theorem 1.7 of [Reference Dadarlat6], there exists a unitary $u\in A/J$ such that

(e4.136) $$ \begin{align} \|u\psi(e_j|_Z)u^*-\varphi(e_j|_Z)\|<\delta/16\,\,\,\mathrm{for\,\,\,all}\,\,\, j=1,2,\ldots,n. \end{align} $$

Let $q_k'=uq_ku^*, k=1,2,\ldots ,n,$ and

$$ \begin{align*}\varphi_0: C(Z)\to ueu^*(A/J)ueu^* \,\,\,\mathrm{by}\,\,\, \varphi_0(f)=uw\varphi(f)w^*u^*\,\,\,\mathrm{for\,\,\,all}\,\,\, f\in C(Z). \end{align*} $$

Then we may write

(e4.137) $$ \begin{align} u\psi(f)u^*=\sum_{k=1}^m f(\lambda_k)q_k'+\varphi_0(f)\,\,\,\mathrm{for\,\,\,all}\,\,\, f\in C(Z). \end{align} $$

Since $[q_k']=0$ in $K_0(A/J),$ by applying Lemma 4.1, we obtain projections $P_k\in A$ such that

(e4.138) $$ \begin{align} &\pi(P_k)=q_k',\,\,\, k=1,2,\ldots,m, \,\,\, \pi(1-\sum_{k=1}^m P_k)=ueu^*\,\,\,\mathrm{and}\,\,\, \end{align} $$

(e4.139) $$ \begin{align} & P_kP_{k'}=P_{k'}P_k=0,\,\,\,\mathrm{if}\,\,\, k\not=k'. \end{align} $$

Let $P=\sum _{k=1}^m P_k.$ Then $P\in A\setminus J$ is a projection. By the Choi–Effros lifting Theorem [Reference Choi and Effros4], there is a c.p.c. map $\Psi : C(Y)\to (1-P)A(1-P)$ such that $\pi \circ \Psi =\varphi _0\circ \pi _Z,$ where $\pi _Z: C(Y)\to C(Z)$ is the quotient map.

Then, by (e 4.132), (e 4.133), and (e 4.136), there are $h_j\in J, j=1,2,\ldots ,n,$ such that

(e4.140) $$ \begin{align} \|\Phi(e_j|_X)-\left(\sum_{k=1}^m e_j(\lambda_k)P_k+\Psi(e_j|_Y)\right)-h_j\|<2\delta+\delta/16,\,\,\, j=1,2,\ldots,n. \end{align} $$

Let $\{d_{k, l}\}$ be an approximate identity for $P_kJP_k, k=1,2,\ldots ,m,$ and $\{d_{0, l}\}$ be an approximate identity for $(1-P)J(1-P)$ consisting of projections. Put

$$ \begin{align*}d_l=d_{0,l}+\sum_{k=1}^m d_{k,l}\,\,\,\mathrm{and}\,\,\, d_l'=\sum_{k=1}^m d_{k,l},\,\,\,l\in \mathbb{N}. \end{align*} $$

Then $\{d_l\}$ is an approximate identity for J, and $\{d_l'\}$ is an approximate identity for $PJP.$ Choose l such that

(e4.141) $$ \begin{align} \|(1-d_l)h_j\|<\delta/64\,\,\,\mathrm{and}\,\,\, \|h_j(1-d_l)\|<\delta/64,\,\,\, j=1,2,\ldots,n. \end{align} $$

Hence, since $P-d_l'\le 1-d_l,$

(e4.142) $$ \begin{align} \|(P-d_l')h_j\|<\delta/64\,\,\,\mathrm{and}\,\,\, \|h_j(P-d_l')\|<\delta/64,\,\,\, j=1,2,\ldots,n. \end{align} $$

Put $p_k=P_k-d_{k,l}, k=1,2,\ldots ,m.$ Then, by (e 4.140), for $j=1,2,\ldots ,n,$

(e4.143) $$ \begin{align} (P-d_l')\Phi(e_j|_X)&\approx_{33\delta/16} (P-d_l')\left(\sum_{k=1}^m e_j(\lambda_k)P_k+\Psi(e_j|_Y)+h_j\right) \end{align} $$

(e4.144) $$ \begin{align} &\approx_{\epsilon/64}\sum_{k=1}^m e_j(\lambda_k)p_k. \end{align} $$

Similarly, for $j=1,2,\ldots ,n,$

(e4.145) $$ \begin{align} \Phi(e_j|_X)(P-d_l')&\approx_{33\delta/16} \left(\sum_{k=1}^m e_j(\lambda_k)P_k+\Psi(e_j|_Y)+h_j\right)(P-d_l') \end{align} $$

(e4.146) $$ \begin{align} &\approx_{\delta/64}\sum_{k=1}^m e_j(\lambda_k)p_k. \end{align} $$

Put $p=P-d_l'.$ Then

(e4.147) $$ \begin{align} &\|p\Phi(e_j|_X)-\sum_{k=1}^m e_j(\lambda_k)p_k\|<33\delta/16+\delta/64, \end{align} $$

(e4.148) $$ \begin{align} &\|p\Phi(e_j|_X)p-\sum_{k=1}^m e_j(\lambda_k)p_k\|<33\delta/16+\delta/64\,\,\,\mathrm{and}\,\,\, \end{align} $$

(e4.149) $$ \begin{align} &\|p\Phi(e_j|_X)-\Phi(e_j|_X)p\|<33\delta/8+\delta/32,\,\,\,\,\,\, j=1,2,\ldots,n. \end{align} $$

Put $L(f)=(1-p)\Phi (f)(1-p)$ for $f\in C(X).$ Then it is a c.p.c. map. Moreover,

(e4.150) $$ \begin{align} \Phi(e_j|_X)&=p\Phi(e_j|_X)+(1-p)\Phi(e_j|_X) \end{align} $$

(e4.151) $$ \begin{align} &\approx_{33\delta/8+3\delta/64} \sum_{k=1}^m e_j(\lambda_k)p_k +(1-p)\Phi(e_j|X)(1-p). \end{align} $$

Hence, using the inequality above and (e 4.132),

(e4.152) $$ \begin{align}\nonumber &\|\sum_{k=1}^m e_j(\lambda_k)p_k+L(e_j|_X)-T_j\| \\&\quad \le \|\sum_{k=1}^m e_j(\lambda_k)p_k+L(e_j|_X)-\Phi(e_j|_X)\|+ \|\Phi(e_j|_X)-T_j\|\nonumber\\ &\quad<33\delta/8+3\delta/64+\delta<\epsilon,\,\,\,\,\,\, j=1,2,\ldots,n. \end{align} $$

To arrange $(1-p)\not \in J,$ we may split $p_1.$ We write $p_1=p_1'+p_1",$ where $p_1', p_1"\in p_1Ap_1$ are two mutually orthogonal projections and both are not in J as $p_1\not \in J.$ Then define $L': C(X)\to (1-p+p_1")A(1-p+p_1")$ by $L'(f)=f(\lambda _1)p_1"+L(f)$ for all $f\in C(X).$ We then replace L with $L'$ and replace $p_1$ with $p_1'.$

Finally, by (e 4.149) and (e 4.132),

$$ \begin{align*}\nonumber &L(e_ie_j|_X)-L(e_i|_X)L(e_j|_X)=(1-p)\Phi(e_ie_j|_X)(1-p)\nonumber\\&\quad -(1-p)\Phi(e_i|_X)(1-p)\Phi(e_j|_X)(1-p)\\\nonumber &\quad\approx_{33\delta/8+\delta/32}(1-p)(\Phi(e_i|_X)-\Phi(e_i|_X)\Phi(e_j|_X))(1-p)\approx_\delta 0. \end{align*} $$

Thus (e 4.131) also holds.

Proof. First, by [Reference Cuntz and Higson5], $K_i(M(A\otimes {\cal K}))=\{0\}, i=0,1.$ We also have, for any $k\in \mathbb {N}, M_k(M(A\otimes {\cal K}))\cong M(A\otimes {\cal K}).$ For any projection $p\in M(A\otimes {\cal K}),$ we have $p\oplus 1_{M(A\otimes {\cal K})}\sim 1_{M(A\otimes {\cal K})}.$ This immediately implies (1), (4), and (7).

It follows from Theorem 1.1 of [Reference Zhang38] that $\mathrm {cel}(M(A\otimes {\cal K}))\le 3\pi $ and $\mathrm {cer}(M(A\otimes {\cal K}))\le 4.$ Therefore, (2), (3), (5), (6), (8), and (9) hold.

Proof. Let us first fix a compact subset $X.$ Let $\epsilon>0$ and ${\cal F}=\{e_j|_X: 0\le j\le n\}.$

Since we assume that $K_i(A\otimes C(Y))=0$ for any compact metric space $Y, i=0,1,$ for any finite subset ${\cal P}$ as in Theorem 3.1 of [Reference Gong and Lin11], we have that $[L]|_{\cal P}=0$ for any sufficiently multiplicative c.p.c. maps $L: C(X)\to A.$ Then, by Proposition 4.3 and applying Theorem 3.1 of [Reference Gong and Lin11], we obtain $\delta _0$ and a finite subset ${\cal G}_0\subset C(X)$ such that, if

(e4.155) $$ \begin{align} \|L_i(fg)-L_i(f)L_i(g)\|<\delta_0\,\,\,\mathrm{for\,\,\,all}\,\,\, f,g\in {\cal G}_0,\,\,\, i=1,2, \end{align} $$

there exists an integer $l>0,$ a unital homomorphism $\sigma : C(X)\to M_l(A)$ with finite-dimensional range, and a unitary $u\in M_{l+1}(A)$ such that

(e4.156) $$ \begin{align} \|u^*\mathrm{diag}(L_1(e_j|_X), \sigma(e_j|_X))u - \mathrm{diag}(L_2(e_j|_X), \sigma(e_j|_X))\|<\epsilon/2,\,\,\, 1\le j\le n. \end{align} $$

(Note as mentioned above, that $[L_i]|_{\cal P}=0,$ whenever it makes sense, since $K_i(A\otimes C(Y))=0$ for any compact metric space $Y, i=0, 1.$ )

However, since $\{e_j: 0\le j\le n\}$ generates $C(X)$ as $C^*$ -algebra, one obtains $\delta (n, \epsilon )>0$ such that when (e 4.153) holds, then

(e4.157) $$ \begin{align} \|L_i(fg)-L_i(f)L_i(g)\|<\delta_0\,\,\,\mathrm{for\,\,\,all}\,\,\, f,g\in {\cal G}_0,\,\,\, i=1,2. \end{align} $$

So Theorem 3.1 of [Reference Gong and Lin11] applies and the theorem would follow if we fix X first.

To find a common $\delta $ and $l,$ independent of $X\subset {\mathbb {I}}^n,$ we note that there are finitely many compact subsets $X_1, X_2,\ldots ,X_m$ of ${\mathbb {I}}^n$ such that, for any non-empty compact subset $X,$ there is $s\in \{1,2,\ldots ,m\}$ such that

(e4.158) $$ \begin{align} d_H(X, X_s)<\epsilon/6 \end{align} $$

(see Definition 2.15). Let $\Omega _s=\{x\in {\mathbb {I}}^n: \mathrm {dist}(x, X_s)\le \epsilon /6\}, s=1,2,\ldots ,m.$ Then, for any non-empty compact subset X of ${\mathbb {I}}^n,$ there is $s\in \{1,2,\ldots ,m\}$ such that

(e4.159) $$ \begin{align} d_H(X, \Omega_s)\le \epsilon/3 \,\,\,\mathrm{and}\,\,\, X\subset \Omega_s. \end{align} $$

When $\epsilon>0$ is given, by the first part of the proof, we obtain $\delta _s>0$ and $l_s\in \mathbb {N}$ such that the conclusion of the first part holds for $\Omega _s, s=1,2,\ldots , m.$ Choose $\delta =\min \{\delta _1, \delta _2,\ldots ,\delta _m\}$ and $l=\max \{l_1,l_2,\ldots ,l_m\}.$

Now suppose $X\subset {\mathbb {I}}^n$ and $L_1, L_2: C(X)\to A$ are unital c.p.c. maps satisfying the assumption (e 4.153). Suppose that

(e4.160) $$ \begin{align} d_H(X,\Omega_s)\le \epsilon/3\,\,\,\mathrm{and}\,\,\, X\subset \Omega_s. \end{align} $$

Define $\tilde L_k: C(\Omega _s)\to A$ by $\tilde L_k(e_j|_{\Omega _s})=L_k(e_j|_X)$ (this makes sense as $X\subset \Omega _s$ ), $j=1,2,\ldots ,n, k=1,2.$ Therefore,

(e4.161) $$ \begin{align} \|\tilde L_k(e_ie_j|_{\Omega_s})-\tilde L_k(e_i|_{\Omega_s})\tilde L_k(e_j|_{\Omega_s})\|<\delta,\,\,i,j\in \{1,2,\ldots,n\} \end{align} $$

and $k=1,2.$ Hence, with the choice of $\delta ,$ we obtain homomorphism $\sigma ': C(\Omega _s)\to M_l(A)$ with finite-dimensional range and a unitary $u\in M_{l+1}(A)$ such that, for $j=1,2,\ldots ,n,$

(e4.162) $$ \begin{align} \|u^*\mathrm{diag}(L_1(e_j|_{\Omega_s}), \sigma'(e_j|_{\Omega_s}))u - \mathrm{diag}(L_2(e_j|_{\Omega_s}), \sigma'(e_j|_{\Omega_s}))\|<\epsilon/2. \end{align} $$

We may write $\sigma '(e_j|_{\Omega _s})=\sum _{k=1}^K e_j(\xi _k) p_k,$ where $\xi _k\in \Omega _s$ ( $k=1,2,\ldots ,K$ ) and $\{p_1,p_2,\ldots ,p_K\}$ is a set of mutually orthogonal projections in $M_l(A).$ Since $d_H(X, \Omega _s)\le \epsilon /4,$ for each $\xi _k,$ we may choose $x(\xi _k)\in X$ such that

(e4.163) $$ \begin{align} \|\xi_k-x(\xi_k)\|_2=\mathrm{dist}(\xi_k, x(\xi_k))\le \epsilon/3,\,\,1\le k\le K. \end{align} $$

Define $\sigma : C(X)\to M_{l}(A)$ by

(e4.164) $$ \begin{align} \sigma(f)=\sum_{k=1}^K f(x(\xi_k))p_k \,\,\,\mathrm{for\,\,\,all}\,\,\, f\in C(X). \end{align} $$

Note that

(e4.165) $$ \begin{align} \|\sigma(e_j|_X)-\sigma'(e_j|_{\Omega_s})\|\le \epsilon/3,\,\,\, 1\le j\le n. \end{align} $$

Hence, by (e 4.162), for $1\le j\le n,$

(e4.166) $$ \begin{align} \|u^*\mathrm{diag}(L_1(e_j|_{X}), \sigma(e_j|_{X}))u - \mathrm{diag}(L_2(e_j|_{X}), \sigma(e_j|_{X}))\|\le \epsilon/2 +\epsilon/3<\epsilon. \end{align} $$

Proof. Fix $n\in \mathbb {N}.$ Let $\epsilon>0.$ We will apply Theorem 4.5. Let $\delta _1>0$ (in place of $\delta $ ) be given by Theorem 4.5 for $\epsilon /16.$

Put $\epsilon _0=\min \{\epsilon /128,\delta _1/4\}.$ Let $\delta _2'>0$ (in place of $\delta $ ) be given by Lemma 4.2 for $\epsilon _0$ (in place of $\epsilon $ ). Choose $\delta _2=\min \{\delta _2', \epsilon _0/2\}.$ Put $\epsilon _1=\min \{\epsilon _0/2, \delta _2/4\}.$

Let $\delta _3>0$ be given by Proposition 2.16 for $\epsilon _1$ (in place of $\eta $ ). Choose $\epsilon _2=\min \{\epsilon _1/2, \delta _3/2\}.$ Let $\delta _4>0$ be given by Proposition 2.12 for $\epsilon _2$ (in place of $\eta $ ) and let $\delta _5>0$ be given by Theorem 3.9 for $\epsilon _2.$

Let $\delta :=\min \{\delta _1/2, \delta _2/2, \delta _4, \delta _5/2, \epsilon _2\}.$

Now suppose that $T_1, T_2,\ldots ,T_n\in A_{s.a.}$ with $\|T_i\|\le 1$ ( $1\le i\le n$ ) such that

(e4.169) $$ \begin{align} \|T_iT_j-T_jT_i\|<\delta\,\,\,\mathrm{for\,\,\,all}\,\,\, 1\le j\le n\,\,\,\mathrm{and}\,\,\, d_H(X, Y)<\epsilon/8. \end{align} $$

By the choice of $\delta $ (and $\delta _4, \delta _5$ ) and, applying Proposition 2.12, we obtain non-empty sets $X=s\mathrm {Sp}^{\epsilon /8}((T_1, T_2,\ldots ,T_n))$ and $Y=s\mathrm {Sp}^{\epsilon /8}((\pi (T_1), \pi (T_2),\ldots ,\pi (T_n)),$ and the n-tuple $(T_1, T_2,\ldots , T_n)$ has an $\epsilon _2$ -near-spectrum $X_1$ and, by Theorem 3.9, $(\pi (T_1), \pi (T_2),\ldots ,\pi (T_n))$ has an $\epsilon _2$ -spectrum $Y_1.$ Moreover (by the choice of $\delta _3$ ), since $\epsilon _2=\min \{\epsilon _1/2, \delta _3/2\},$ we have (by applying Proposition 2.16)

(e4.170) $$ \begin{align} Y_1\subset \overline{(X_1)_{\epsilon_1}},\,\,\, X_1\subset X\subset \overline{(X_1)_{2\epsilon_1}}, \,\,\,\mathrm{and}\,\,\, Y_1\subset Y\subset \overline{(Y_1)_{2\epsilon_1}}. \end{align} $$

By the choice of $\delta _2$ (and $\epsilon _2<\delta _2'$ ), applying Lemma 4.2, we obtain mutually orthogonal nonzero projections $p_1,p_2,\ldots ,p_m\in A\setminus J,$ a $\delta _2$ -dense subset $\{\lambda _1, \lambda _2,\ldots ,\lambda _m\}\subset Y_1,$ and a c.p.c. map $L: C(X_1)\to (1-p)A(1-p)$ such that (with $p=\sum _{i=1}^m p_i$ )

(e4.171) $$ \begin{align} &\|\sum_{i=1}^m e_j(\lambda_i)p_i+L(e_j|_{X_1})-T_j\|<\epsilon_0<\delta_1\,\,\,\mathrm{and}\,\,\, \end{align} $$

(e4.172) $$ \begin{align} &\|L(e_ie_j|_X)-L(e_i|_X)L(e_j|_X)\|<\epsilon_0, \end{align} $$

$j=1,2,\ldots ,n,$ and $1-p\not \in J.$ Put

$$ \begin{align*}\Phi_1(f)=\sum_{i=1}^m f(\lambda_i)p_i+L(f|_{X_1})\,\,\,\mathrm{for\,\,\,all}\,\,\, f\in C({\mathbb{I}}^n). \end{align*} $$

Note that, since $(1-p)\sim 1, (1-p)A(1-p)\cong A.$ Choose $\xi _0\in X_1$ and define $\Phi _0: C(X_1)\to (1-p)A(1-p)$ by $\Phi _0(f)=f(\xi _0)(1-p).$

By the choice of $\delta _1$ and (e 4.172), as $(1-p)A(1-p)\cong A,$ and applying Theorem 4.5, we obtain a unital homomorphism $\sigma : C(X_1)\to M_l((1-p)A(1-p))$ (for some integer $l\in \mathbb {N}$ ) with finite-dimensional range and a unitary $u_1\in M_{l+1}((1-p)A(1-p))$ such that

(e4.173) $$ \begin{align} \|u_1^*\mathrm{diag}(\Phi_0(e_j|_{X_1}), \sigma(e_j|_{X_1}))u_1-\mathrm{diag}(L(e_j|_{X_1}), \sigma(e_j|_{X_1}))\|<\epsilon/16, \,\,\, 1\le j\le n. \end{align} $$

Since $\sigma $ has a finite-dimensional range, there are mutually orthogonal projections $q_1,q_2,\ldots ,q_L\in M_{l}((1-p)A(1-p))$ (for some integer $L\ge 1$ ) such that

(e4.174) $$ \begin{align} \sigma(e_j|_{X_1})= \sum_{k=1}^L e_j(\xi_k)q_k,\,\,\, 1\le j\le n, \end{align} $$

where $\xi _k\in X_1, k=1,2,\ldots ,L.$ Put $q=\sum _{i=1}^L q_i.$ Note that, without loss of generality, by replacing l by $l+1$ and adding a projection $q_i"\in M_l((1-p)A(1-p))\setminus M_l((1-p)J(1-p))$ to each $q_i,$ if necessarily, we may assume that $q_i\not \in M_l((1-p)A(1-p))\setminus M_l((1-p)J(1-p)), i=1,2,\ldots ,L.$