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- W3196895890 abstract "Let $Theta=(theta_{jk})_{ntimes n}$ be a real skew-symmetric $ntimes n$ matrix for $ngeq 2$. Under some mild non-integrality conditions on $Theta,$ we construct Rieffel-type projections as higher dimensional Bott classes in the $n$-dimensional noncommutative torus $mathcal{A}_{Theta}.$ These projections generate $operatorname{K}_0(mathcal{A}_Theta)$ when $Theta$ is strongly totally irrational. As an application, when $Theta$ is strongly totally irrational, we show that: For any $varepsilon>0,$ there exists $delta>0$ (depending only on $varepsilon$ and $Theta$) satisfying the following: For any unital simple separable $C^*$-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^{2pi itheta_{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}_ktilde{u}_j=e^{2pi itheta_{jk}}tilde{u}_jtilde{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." @default.
- W3196895890 created "2021-09-13" @default.
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- W3196895890 date "2021-09-02" @default.
- W3196895890 modified "2023-09-25" @default.
- W3196895890 title "Higher dimensional Bott classes and the stability of rotation relations" @default.
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- W3196895890 doi "https://doi.org/10.48550/arxiv.2109.00739" @default.
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