Proof. A norm-convergent sequence is bounded, so (3) defines a diagonal operator on \(H\), with norm \(\sup_n\left\|x_n\right\|\). Algebra operations and adjoints preserve the limit condition, and the constant identity belongs to the algebra. To check norm closedness, suppose \(x^{(k)}\to x\) in the supremum norm, with coordinate limits \(p_k\in P\). Then \[\left\|p_k-p_j\right\|\le\sup_n\left\|x_n^{(k)}-x_n^{(j)}\right\|.\] Thus \(p_k\to p\in P\). The estimate \[\left\|x_n-p\right\|\le\left\|x_n-x_n^{(k)}\right\| +\left\|x_n^{(k)}-p_k\right\|+\left\|p_k-p\right\|\] shows that \(x_n\to p\). Moreover, \[ \mathcal E_D(P)=c_0(\mathbb N,D)+\{(p,p,\ldots):p\in P\}. \tag{6}\] The two summands are norm-separable: \(P\) is a subspace of the separable metric space \(D\), and finite-support sequences from a countable dense subset of \(D\) are dense in \(c_0(\mathbb N,D)\). Their sum is norm-separable.

For the metric assertion, write \(\alpha(P,Q)=\sup_{p\in P_1}\mathop{\mathrm{dist}}(p,Q_1)\). Given a contraction \(x\in\mathcal E_D(P)\) with limit \(p\) and \(\eta>0\), choose \(q\in Q_1\) with \(\left\|p-q\right\|<\alpha(P,Q)+\eta\), and choose \(N\) so that \(\sup_{n>N}\left\|x_n-p\right\|<\eta\). Keeping the first \(N\) entries of \(x\) and putting \(q\) thereafter gives a contraction \(y\in\mathcal E_D(Q)\) with \(\left\|x-y\right\|<\alpha(P,Q)+2\eta\). This proves the upper bound for the directed distance. Conversely, for a constant contraction \((p,p,\ldots)\) and any contraction \(y\in\mathcal E_D(Q)\) with limit \(q\in Q_1\), \[\left\|(p,p,\ldots)-y\right\|=\sup_n\left\|p-y_n\right\|\ge\left\|p-q\right\| \ge\mathop{\mathrm{dist}}(p,Q_1).\] Taking infima and suprema proves equality of the directed distances. Interchanging \(P,Q\) gives (4).

Let \(e_n\) be the projection onto the \(n\)th copy of \(L\). Every \(e_n\) is a central element of \(\mathcal E_D(P)\). Conversely, a central element \((x_n)\) commutes with all finite-support \(D\)-valued sequences, so each \(x_n\) lies in \(Z(D)=\mathbb CI_L\). Thus the center consists exactly of convergent scalar sequences. Every nonzero central projection contains an \(e_n\), and hence the minimal nonzero central projections are precisely the \(e_n\). The same holds for \(\mathcal E_D(Q)\).

If \(S\mathcal E_D(P)S^*=\mathcal E_D(Q)\), conjugation by \(S\) therefore permutes the projections \(e_n\). Its restriction from the \(n\)th coordinate to its image is a unitary \(U_n:L\to L\). The corresponding coordinate corners of both algebras are exactly \(D\), since every finitely supported \(D\)-valued sequence is allowed. Consequently \(U_nDU_n^*=D\). For \(b\in Q\), pull back the constant sequence \((b,b,\ldots)\) under \(S\). Its \(n\)th entry is \(U_n^*bU_n\), regardless of the coordinate permutation. It belongs to \(\mathcal E_D(P)\), so its entries converge in norm to some \(p_b\in P\). This is equivalent to (5) and proves (ii).

Finally, an operator commuting with \(\mathcal E_D(P)\) commutes with all \(e_n\), so it is block diagonal. Commutation with the finite-support \(D\) corners means that each block lies in \(D'\). These conditions are also sufficient, and do not depend on \(P\). Thus both commutants equal \[\{(z_n):z_n\in D',\ \sup_n\left\|z_n\right\|<\infty\}.\] This proves (iii). ◻

Proof. If \(UDU^*=D\), then \(UD'U^*=D'\). For \(q_i=Up_iU^*\) and \(z_i\in D'\), apply (7) to \(p_i\) and \(U^*z_iU\). Conjugate its product sum by \(U\) and its tensor sum by \(U\otimes U\). This gives (7) for \(q_i,z_i\).

Now fix finite families \(b_i\in Q\) and \(z_i\in D'\). Put \(q_{i,n}=U_np_{b_i}U_n^*\). For each \(n\), the first part gives the identity for the entire list \((q_{i,n})_i\), since its entries all lie in \(U_nPU_n^*\). Both errors \[\left\|\sum_i(q_{i,n}-b_i)z_i\right\|,\qquad \left\|\sum_i(q_{i,n}-b_i)\otimes z_i\right\|\] are at most \(\sum_i\left\|q_{i,n}-b_i\right\|\left\|z_i\right\|\), which tends to zero. Taking the limit proves the identity for the prescribed families in \(Q\) and \(D'\). ◻

Proof. A spatial conjugacy would give the unitaries in Proposition 2(ii), contradicting Lemma 4. For the distance bound, \(p\mapsto WpW^*\) bijects the closed unit balls and \[\left\|WpW^*-p\right\|\le2\left\|p\right\|\left\|W-I_L\right\|.\] Apply Proposition 2(i). ◻

Proof. Use the tree of reduced words with edges given by right multiplication by a generator or its inverse, rooted at \(e\). Its distance is \(d(x,y)=|x^{-1}y|\), so left translations preserve distance. For \(x\ne e\), let \(p(x)\) be the neighbor of \(x\) toward \(e\), obtained by deleting the last letter of its reduced word. Recursively define \[ \xi_e=\delta_e,\qquad \xi_x=t\xi_{p(x)}+r\delta_x\quad(x\ne e). \tag{11}\] These are unit vectors supported on their root paths. If \(z\) is the last common vertex of the root paths to \(x\) and \(y\), the common-support components are \(t^{d(x,z)}\xi_z\) and \(t^{d(y,z)}\xi_z\), respectively. The remaining supports are disjoint. Hence \[ \left\langle \xi_x,\xi_y\right\rangle=t^{d(x,y)}. \tag{12}\] Also \(\delta_e=\xi_e\) and \(\delta_x=(\xi_x-t\xi_{p(x)})/r\) for \(x\ne e\), so their span is dense. The assignment \(\xi_x\mapsto\xi_{gx}\) preserves every finite Gram matrix by (12). It therefore defines an isometry on the span, and the inverse assignment for \(g^{-1}\) makes the extension unitary. Denote this unitary by \(\pi_t(g)\). The representation law holds on the dense span and hence on \(K\). Since \(\pi_t(s_i)\delta_e=\xi_{s_i}=t\delta_e+r\delta_{s_i}\), (9) follows.

It remains to prove that the perturbation is uniformly small over all group elements, rather than just over the generators. Fix \(g\ne e\), write \(h_0=e,h_1,\ldots,h_l=g\) for its root path, and put \(T=\pi_t(g)\lambda(g)^*\). For \(y\ne g\), (11) gives \[ T\delta_y=\frac{\xi_y-t\xi_{g p(g^{-1}y)}}{r}. \tag{13}\] The vertex \(g p(g^{-1}y)\) is the neighbor of \(y\) toward \(g\). If \(y\) is off the path \([e,g]\), this neighbor is also \(p(y)\), the neighbor toward \(e\); see Figure 1. Thus \(T\delta_y=\delta_y\) off that path. On the path, the neighbor toward \(g\) of \(h_k\), for \(k<l\), is \(h_{k+1}\), so \[ T\delta_{h_k}=r\xi_{h_k}-t\delta_{h_{k+1}}\quad(0\le k<l), \qquad T\delta_{h_l}=\xi_{h_l}. \tag{14}\]

Let \(M_g=\mathop{\mathrm{span}}\{\delta_{h_k}:0\le k\le l\}\). On this space define the truncated shift \(S\delta_{h_k}=\delta_{h_{k+1}}\) for \(k<l\) and \(S\delta_{h_l}=0\), and put \(X\delta_{h_k}=\xi_{h_k}\). For \(R=\operatorname{diag}(1,r,\ldots,r)\), the recursion gives \[ X=R(I-tS^*)^{-1}. \tag{15}\] Indeed, its \(k\)th column is \(t^k\delta_e+r\sum_{j=1}^k t^{k-j}\delta_{h_j}\). Since \(\left\|S\right\|\le1\) and \(\left\|R\right\|=1\), \[ \left\|X-I\right\|\le\left\|R-I\right\| +\left\|R\right\|\sum_{j=1}^{\infty}t^j \le1-r+\frac{t}{1-t}. \tag{16}\] Writing \(R'=\operatorname{diag}(r,\ldots,r,1)\), (14) becomes \(T|_{M_g}=XR'-tS\). Therefore \[\begin{align*} \left\|T|_{M_g}-I\right\| &\le\left\|(X-I)R'\right\|+\left\|R'-I\right\|+t\left\|S\right\|\\ &\le2(1-r)+\frac{t}{1-t}+t=c(t). \end{align*}\] The same bound holds on \(K\), since \(T\) is the identity on \(M_g^\perp\). Multiplication by \(\lambda(g)\) proves (10). For \(g=e\) the difference is zero, so the supremum includes this case as well. ◻

Proof. The operator is the orthogonal direct sum of the unitaries \(\sigma(x)\). On \(\delta_x\otimes v\), the left side of (17) equals \[\delta_{gx}\otimes\sigma(gx)\sigma(x)^*v =\delta_{gx}\otimes\sigma(g)v. \qedhere\] ◻

Proof. Commutation with all \(I_K\otimes E_{xy}\) forces a bounded operator to be \(Y\otimes I_K\): commutation with the diagonal matrix units removes off-diagonal blocks, and commutation with the remaining units makes the diagonal blocks identical. The first-factor generators then impose exactly \(Y\in\lambda(G)'\). Such operators commute with all the second-factor generators, proving (20). In particular, \(\rho(g)\otimes I_K\in D'\).

If \(Y\otimes I_K\in Z(D)\), then \(Y\) commutes with \(\lambda(G)\) by (20). It also commutes with \(\rho(G)\), since \(Y\otimes I_K\) belongs to \(D\) and the right translations belong to \(D'\). Thus \(Y\delta_e\) is invariant under \(\lambda(g)\rho(g)\), which acts on the basis by conjugation \(x\mapsto gxg^{-1}\). Every nonidentity conjugacy class in \(G\) is infinite: conjugate a fixed nonidentity reduced word by distinct generators absent from that word. A square-summable vector invariant under conjugation must therefore be supported at \(e\). Write \(Y\delta_e=\alpha\delta_e\). Commuting with left translations gives \(Y\delta_x=\alpha\delta_x\) for every \(x\), proving \(Z(D)=\mathbb CI_L\).

For the final assertion, Lemma 8 shows that \(W=W_{\pi_t}W_{\pi_0}^*\) sends every \(Q_0(g)\) to \(Q_t(g)\) by conjugation. Moreover, the direct-sum description of these unitaries gives \[\left\|W-I_L\right\|=\left\|W_{\pi_t}-W_{\pi_0}\right\| =\sup_{x\in G}\left\|\pi_t(x)-\lambda(x)\right\|\le c(t). \qedhere\] ◻

Proof. Define a unitary on \(L\) by \[V(v\otimes\delta_x)=\lambda(x)^*v\otimes\delta_x.\] It commutes with every element of \(D'\) by (20). On elementary vectors one computes \[VQ_0(g)V^*(v\otimes\delta_x)=v\otimes\delta_{gx}.\] Thus \(VP_0V^*=I_K\otimes C^*(\lambda(G))\). Given finite families \(p_i\in P_0\) and \(z_i\in D'\), write \[Vp_iV^*=I_K\otimes X_i,\qquad z_i=Y_i\otimes I_K.\] Conjugation by \(V\) gives \[\left\|\sum_i p_i z_i\right\|=\left\|\sum_i Y_i\otimes X_i\right\|.\] On \(L\otimes L=K^{\otimes4}\), conjugation by \(V\otimes I_L\) gives \[\left\|\sum_i p_i\otimes z_i\right\| =\left\|\sum_iI_K\otimes X_i\otimes Y_i\otimes I_K\right\| =\left\|\sum_iY_i\otimes X_i\right\|.\] The last equality permutes the four tensor factors and uses the fact that tensoring with a nonzero identity operator does not change norm. This proves (7). ◻

Proof. Let \(F_i\) be the orthogonal projection onto the closed span of words beginning with \(s_i^{-1}\). The \(F_i\) are pairwise orthogonal. Write \(\lambda(s_i)=A_i+B_i\), where \[A_i=\lambda(s_i)(I_K-F_i),\qquad B_i=\lambda(s_i)F_i.\] There is no cancellation when \(s_i\) is prepended to a word not beginning with \(s_i^{-1}\). Thus the range of \(A_i\) consists of vectors supported on words beginning with \(s_i\), and these ranges are mutually orthogonal. For every \(v\in K\), \[\left\|\sum_i A_iv\right\|^2=\sum_i\left\|(I_K-F_i)v\right\|^2 \le m\left\|v\right\|^2,\] while \[\left\|\sum_i B_iv\right\|\le\sum_i\left\|F_iv\right\| \le\sqrt m\left(\sum_i\left\|F_iv\right\|^2\right)^{1/2} \le\sqrt m\left\|v\right\|.\] Adding the two bounds proves (22). ◻

Proof. Choose an integer \(m>4/t^2\), so \(mt>2\sqrt m\), and take \[p_i=Q_t(s_i),\qquad z_i=\rho(s_i)\otimes I_K, \qquad 1\le i\le m.\] These belong to \(P_t\) and \(D'\), respectively. Since \(\lambda(s_i)\rho(s_i)\delta_e=\delta_e\), (9) implies \[ \left\|\sum_i p_i z_i\right\| \ge\left|\left\langle \left(\sum_i p_i z_i\right) (\delta_e\otimes\delta_e),\delta_e\otimes\delta_e\right\rangle\right| =mt. \tag{23}\] On the other hand, the tensor sum on \(L\otimes L\) is \[\sum_{i=1}^m\lambda(s_i)\otimes\pi_t(s_i) \otimes\rho(s_i)\otimes I_K.\] Apply Lemma 8 to the representation \(\sigma=\pi_t\otimes\rho\otimes1\) on \(K^{\otimes3}\), where \(1\) is the trivial representation on the last copy of \(K\). Together with Lemma 11, it gives \[ \left\|\sum_i p_i\otimes z_i\right\| =\left\|\sum_{i=1}^m\lambda(s_i)\right\|\le2\sqrt m<mt. \tag{24}\] Equations (23) and (24) contradict the identity for this finite list. ◻

Proof of Theorem 1. Given \(\varepsilon>0\), choose \(0<t<1\) so small that \(2c(t)<\varepsilon\). Use the algebras \(P_0,P_t\subseteq D\subseteq\mathcal B(L)\) from Section 4, and put \[H=\bigoplus_{n\ge1}L,\qquad A=\mathcal E_D(P_0),\qquad B=\mathcal E_D(P_t).\] The space \(H\) is separable. Lemma 9 supplies the scalar center of \(D\), so Proposition 2 shows that \(A,B\) are unital norm-separable \(C^*\)-algebras with common identity and that \(A''=B''\). The same lemma supplies a unitary \(W\) with \(WP_0W^*=P_t\) and \(\left\|W-I_L\right\|\le c(t)\). By Corollary 5, \[d_{\mathrm{KK}}(A,B)\le2c(t)<\varepsilon.\] Finally, Lemmas 10 and 12 show that the relative tensor-norm identity holds for \(P_0\) but not for \(P_t\). Corollary 5 therefore excludes every unitary conjugating \(A\) onto \(B\). ◻