$$\lbrack \hat{y_{1}},\hat{y_{2}}]=i.$$The operators made up of $\hat{y_{1}}$ and $\hat{y_{2}}$ $$\hat{O}=\sum_{m,n}C_{mn}\hat{y}_{1}^{m}\hat{y}_{2}^{n}$$form the noncommutative plane $R^{2}$. All operators on $R^{2}$ which commute with $U_{1}$ and $U_{2}$ $$U_{1}=e^{-il\hat{y_{2}}},~~~~~~~~~~U_{2}=e^{il(\tau _{2}\hat{y_{1}}-\tau _{1}%
\hat{y_{2}})},$$where $l,\tau _{1},\tau _{2}$ are all real numbers and $l,\tau
_{2}>0,\tau =\tau _{1}+i\tau _{2}$, constitute the noncommutative torus $T^{2}$. We have $$\begin{aligned}
U_{1}^{-1}\hat{y_{1}}U_{1} &=&\hat{y_{1}}+l,~~~~~~~U_{2}^{-1}\hat{y_{1}}%
U_{2}=\hat{y_{1}}+l\tau _{1},  \nonumber \\
U_{1}^{-1}\hat{y_{2}}U_{1} &=&\hat{y_{2}},~~~~~~~~~~~U_{2}^{-1}\hat{y_{2}}%
U_{2}=\hat{y_{2}}+l\tau _{2}.\end{aligned}$$The operators $U_{1}$ and $U_{2}$ are two different wrapping operators around the noncommutative torus and their commutation relation is $%
U_{1}U_{2}=U_{2}U_{1}e^{-2\pi i\frac{l^{2}\tau _{2}}{2\pi }}$. When $A=\frac{%
l^{2}\tau _{2}}{2\pi }$ is an integer, we call the noncommutative torus integral.Introduce two operators $u_{1}$ and $u_{2}$: $$\begin{aligned}
\label{eq:40}
u_{1} &=&e^{-il\hat{y_{2}}/A},~~~~~~~~~~u_{2}=e^{-il(\tau _{2}\hat{y_{1}}%
-\tau _{1}\hat{y_{2}})/A},  \nonumber  \label{eq:14} \\
u_{1}u_{2} &=&u_{2}u_{1}e^{2\pi
i/A},~~~~~~u_{1}^{A}=U_{1},~~u_{2}^{A}=U_{2}^{-1}\end{aligned}$$The operators on the noncommutative torus are composed of the Laurant series of $u_{1}$ and $u_{2}$, $$\hat{O}_{T^{2}}=\sum_{m,n}C_{mn}^{\prime }u_{1}^{m}u_{2}^{n}
\label{eq:5}$$where $m,n\in Z$ and $C_{00}^{\prime }$is called the trace of the operators, Eq.(\[eq:5\]) includes all operators on the noncommutative torus $T^{2}$, satisfying the relation $U_{i}^{-1}\hat{O}_{T^{2}}U_{i}=\hat{O}_{T^{2}}$. From (\[eq:40\]) we can rewrite the equation(\[eq:5\]) as $$\label{eq:41}
\hat{O}_{T^{2}}=\sum_{s,t=0}^{A-1}u_{1}^{s}u_{2}^{t}\Psi
_{st}(u_{1}^{A},u_{2}^{A})  \label{eq:6}$$where $\Psi _{st}$ is Laurant series of the operators $u_{1}^{A}$and $%
u_{2}^{A}$. we call this formula the standard expression for the operator on the noncommutative torus $T^{2}$. The trace for the operator is 