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THE STABILITY OF AF-RELATIONS
Jiajie HUA
Acta mathematica scientia,Series B. 2024, 44 (6):
2443-2464.
DOI: 10.1007/s10473-024-0621-1
For given $\ell,s\in \mathbb{N},$ $\Lambda=\{\rho_j\}_{j=1,\cdots,s},\rho_j\in\mathbb{T}$, the $C^*$-algebra $\mathcal{B}:=\mathcal{E}(\{r_j\}_{j=1,\cdots,s},\Lambda,\\ \ell)$ is defined to be the universal $C^*$-algebra generated by $\ell$ unitaries $\mathfrak{u}_1,\cdots,\mathfrak{u}_{\ell}$ subject to the relations $r_{j}(\mathfrak{u}_1,\cdots,\mathfrak{u}_{\ell})-\rho_j=0$ for all $j=1,\cdots,s,$ where the $r_j$ is monomial in $\mathfrak{u}_1,\cdots,\mathfrak{u}_{\ell}$ and their inverses for $j=1,2,\cdots,s$. If $\mathcal{B}$ is a unital $AF$-algebra with a unique tracial state, and $K_0(\mathcal{B})$ is a finitely generated group, we say that the relations $(\{r_j\}_{j=1,\cdots,s},\Lambda,\ell)$ are $AF$-relations. If the relations $(\{r_j\}_{j=1,\cdots,s},\Lambda,\ell)$ are $AF$-relations, we prove that, for any $\varepsilon>0,$ there exists a $\delta>0$ satisfying the following: for any unital $C^*$-algebra $\mathcal{A}$ with the cancellation property, strict comparison, nonempty tracial state space, and any $\ell$ unitaries $u_1,u_2,\cdots,u_\ell\in\mathcal{A}$ satisfying $$\|r_j(u_1,u_2,\cdots,u_\ell)-\rho_j\|<\delta,\,\,j=1,2,\cdots,s,$$ and certain trace conditions, there exist $\ell$ unitaries $\tilde{u}_1,\tilde{u}_2,\cdots,\tilde{u}_{\ell}\in\mathcal{A}$ such that $$r_j(\tilde{u}_1,\tilde{u}_2,\cdots,\tilde{u}_\ell)=\rho_j\,\,{\rm for}\,\,j=1,2,\cdots,s, \,\,{\rm and}\,\,\|u_i-\tilde{u}_i\|<\varepsilon\,\,{\rm for}\,\,i=1,2,\cdots,\ell.$$ Finally, we give several applications of the above result.
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