The stability of AF-relations

For given \(\ell,s\in \mathbb{N}, \Lambda=\{\rho_j\}_{j=1,\cdots,s},\rho_j\in\mathbb{T}\) $$\ell ,s\in \mathbb{N},\mathrm{\Lambda }=\{{\rho }_j{\}}_{j=1,\cdots ,s},{\rho }_j\in \mathbb{T}$$ , the C*-algebra \(\mathcal{B}:=\mathcal{E}(\{r_j\}_{j=1,\cdots,s},\Lambda,\ \ell)\) $$\mathcal{B}:=\mathcal{E}(\{r_j{\}}_{j=1,\cdots ,s},\mathrm{\Lambda },\ell )$$ is defined to be the universal C*-algebra generated by ℓ unitaries \(\mathfrak{u}_1,\cdots,\mathfrak{u}_{\ell}\) $${\mathfrak{u}}_1,\cdots ,{\mathfrak{u}}_{\ell }$$ subject to the relations \(r_{j}(\mathfrak{u}_1,\cdots,\mathfrak{u}_{\ell})-\rho_j=0\) $$r_j({\mathfrak{u}}_1,\cdots ,{\mathfrak{u}}_{\ell })-{\rho }_j=0$$ for all j = 1, ⋯, s, where the r_j is monomial in \(\mathfrak{u}_1,\cdots,\mathfrak{u}_{\ell}\) $${\mathfrak{u}}_1,\cdots ,{\mathfrak{u}}_{\ell }$$ and their inverses for j = 1, 2, ⋯, s. If \(\mathcal{B}\) $$\mathcal{B}$$ is a unital AF-algebra with a unique tracial state, and \(K_0(\mathcal{B})\) $$K_0(\mathcal{B})$$ is a finitely generated group, we say that the relations \((\{r_j\}_{j=1,\cdots,s},\Lambda,\ell)\) $$(\{r_j{\}}_{j=1,\cdots ,s},\mathrm{\Lambda },\ell )$$ are AF-relations. If the relations \((\{r_j\}_{j=1,\cdots,s},\Lambda,\ell)\) $$(\{r_j{\}}_{j=1,\cdots ,s},\mathrm{\Lambda },\ell )$$ are AF-relations, we prove that, for any ε > 0, there exists a δ > 0 satisfying the following: for any unital C*-algebra \(\mathcal{A}\) $$\mathcal{A}$$ with the cancellation property, strict comparison, nonempty tracial state space, and any ℓ unitaries \(u_1,u_2,\cdots,u_\ell\in\mathcal{A}\) $$u_1,u_2,\cdots ,u_{\ell }\in \mathcal{A}$$ satisfying \(\|r_j(u_1,u_2,\cdots,u_\ell)-\rho_j\|<\delta,\,\,j=1,2,\cdots,s,\) $$\parallel r_j(u_1,u_2,\cdots ,u_{\ell })-{\rho }_j\parallel <\delta ,j=1,2,\cdots ,s,$$ and certain trace conditions, there exist ℓ unitaries \(\tilde{u}_1,\tilde{u}_2,\cdots,\tilde{u}_{\ell}\in\mathcal{A}\) $${\stackrel{\sim }{u}}_1,{\stackrel{\sim }{u}}_2,\cdots ,{\stackrel{\sim }{u}}_{\ell }\in \mathcal{A}$$ such that \(r_j(\tilde{u}_1,\tilde{u}_2,\cdots,\tilde{u}_\ell)=\rho_j\,\,{\rm for}\,\,j=1,2,\cdots,s, \,\,{\rm and}\,\,\|u_i-\tilde{u}_i\|<\varepsilon\,\,{\rm for}\,\,i=1,2,\cdots,\ell.\) $$r_j({\stackrel{\sim }{u}}_1,{\stackrel{\sim }{u}}_2,\cdots ,{\stackrel{\sim }{u}}_{\ell })={\rho }_j\mathrm{f}\mathrm{o}\mathrm{r}j=1,2,\cdots ,s,\mathrm{a}\mathrm{n}\mathrm{d}\parallel u_i-{\stackrel{\sim }{u}}_i\parallel <\epsilon \mathrm{f}\mathrm{o}\mathrm{r}i=1,2,\cdots ,\ell .$$

Finally, we give several applications of the above result.