_{1}>  \nonumber \\
&=&\sum_{j=0}^{N-1}\int_{\sigma _{j}}dk_{j}dq_{0j}\sum_{n}|k_{j},q_{0j}+\frac{ln}{A}%
>a_{n}(k_{j},q_{0j})  \nonumber \\
&=&\sum_{j=0}^{N-1}\int_{\sigma _{j}}dkdq_{0}\sum_{nn_{1}}|k,q_{0}+\frac{ln}{A}%
>A_{nn_{1}(k,q_{0})}^{j}a_{n_{1}}(k,q_{0})  \nonumber \\
&=&\sum_{j=0}^{N-1}R_{N}^{j}\int_{\sigma _{0}}dkdq_{0}\sum_{n}|k,q_{0}+\frac{ln}{A}%
>a_{n}(k,q_{0}).\end{aligned}$$Thus we have $$\label{eq:25}
R_{N}|\phi _{1}>=|\phi _{1>}.$$In the same way, we get $$\label{eq:26}
<\phi _{2}|R_{N}=<\phi _{2}|.$$That is to say that the state vectors $|\phi _{1}>$ and $<\phi
_{2}|$ are invariant under the rotation $R_{N}$.

More generally, we can take any state vectors $|\phi _{1}>$ and $<\phi _{2}|$ satisfying $$\label{eq:50}
R_{N}|\phi _{1}>=e^{i\alpha _{1}}|\phi _{1}>,~~~~~~~<\phi
_{2}|R_{N}^{-1}=e^{-i\alpha _{2}}<\phi _{2}|$$to construct a projection operator on noncommutative orbifold $T^{2}/Z_{N}$. Let $M(k,q_{0})$ be given by (\[eq:8\]) with $$\begin{aligned}
a_{n}(k,q_{0}) &=&\frac{<k,q_{0}+\frac{ln}{A}|\phi _{1}>}{\sqrt{%
\sum_{n^{\prime }}<k,q_{0}+\frac{ln^{\prime }}{A}|\phi _{1}><\phi
_{2}|k,q_{0}+\frac{ln^{\prime }}{A}>}},  \label{eq:4} \\
b_{n}(k,q_{0}) &=&\frac{<\phi _{2}|k,q_{0}+\frac{ln}{A}>}{\sqrt{%
\sum_{n^{\prime }}<k,q_{0}+\frac{ln^{\prime }}{A}|\phi _{1}><\phi
_{2}|k,q_{0}+\frac{ln^{\prime }}{A}>}}.\end{aligned}$$The projector of minimal trace and with continuous coefficient functions is surely of this form. It can be verified that $M^{2}=M$. And it is also covariant under $R_{N}$. From (\[eq:18\]) we have

$$\begin{aligned}
&&<\phi _{2}|k^{^{\prime }},q_{0}^{^{\prime }}+\frac{n^{^{\prime
}}l}{A}>
\\
&=&<\phi _{2}|R_{N}\sum_{n^{"}}A^{-1}(k,q_{0})_{n"n^{\prime
}}|k,q_{0}+\frac{n^{"}l}{A}> \\
&=&e^{i\alpha _{2}}\sum_{n^{"}}A^{-1}(k,q_{0})_{n^{"}n^{^{\prime
}}}<\phi _{2}|k,q_{0}+\frac{n^{"}l}{A}>\end{aligned}$$

and similarly

$$<k^{^{\prime }},q_{0}^{^{\prime }}+\frac{n^{^{\prime }}l}{A}|\phi
_{1}>=e^{-i\alpha _{1}}\sum_{n^{"}}<k,q_{0}+\frac{n^{"}l}{A}|\phi
_{1}>A(k,q_{0})_{n^{^{\prime }}n^{"}},$$ giving

$$\begin{aligned}
&&\sum_{n}<k^{^{\prime }},q_{0}^{^{\prime }}+\frac{ln}{A}|\phi
_{1}><\phi
_{2}|k^{^{\prime }},q_{0}^{^{\prime }}+\frac{ln}{A}>  \nonumber \\
&=&\sum_{n}<k,q_{0}+\frac{ln}{A}|\phi _{1}><\phi _{2}|k,q_{0}+\frac{ln}{A}%
>e^{-i(\alpha _{1}-\alpha _{2})}.\end{aligned}$$

Thus $$M(k^{\prime },q_{0}^{\prime })_{nn^{\prime }}=a_{n}(k^{\prime
},q_{0}^{\prime })b_{n^{\prime }}(k^{\prime },q_{0}^{\prime
})=[AMA^{-1}](k,q_{0})_{nn^{\prime }},$$$P$ is invariant under rotation $R_{N}$ due to (\[eq:24\]) and 