Higher dimensional Bott classes and the stability of rotation relations

Let $\mathrm{\Theta }=({\theta }_{jk}{)}_{n\times n}$ be a real skew-symmetric $n\times n$ matrix for $n\ge 2$. Under some mild non-integrality conditions on $\mathrm{\Theta },$ we construct Rieffel-type projections as higher dimensional Bott classes in the $n$-dimensional noncommutative torus ${\mathcal{A}}_{\mathrm{\Theta }}\mathrm{.}$ These projections generate ${\mathrm{K}}_0({\mathcal{A}}_{\mathrm{\Theta }})$ when $\mathrm{\Theta }$ is strongly totally irrational. As an application, when $\mathrm{\Theta }$ is strongly totally irrational, we show that: For any $\epsilon >0,$ there exists $\delta >0$ (depending only on $\epsilon$ and $\mathrm{\Theta }$) satisfying the following: For any unital simple separable $C^{\ast }$-algebra $\mathcal{A}$ with tracial rank at most one, and for any $n$-tuple of unitaries $u_1,u_2,\dots ,u_n$ in $\mathcal{A}$, if $u_1,u_2,\dots ,u_n$ satisfy certain trace conditions and \begin{eqnarray*}|u_ku_j-e^{2\pi i\theta_{jk}}u_ju_k|<\delta,,j,k=1,2,\dots,n, \end{eqnarray*} then there exists an $n$-tuple of unitaries ${\tilde{u}}_1,{\tilde{u}}_2,\dots ,{\tilde{u}}_n$ in $\mathcal{A}$ such that \begin{eqnarray*}\tilde{u}_k\tilde{u}j=e^{2\pi i\theta{jk}}\tilde{u}_j\tilde{u}_k, {\rm and}, |\tilde{u}_j-u_j|<\varepsilon,, j,k=1,2,\dots,n. \end{eqnarray*} We also show that these trace conditions are also necessary in the above application.