One can define the noncommutative torus $A_{\theta}$ (assume that $\theta$ is irrational) as a universal algebra generated by two unitaries $u,v$ satisfying the relation $vu=e^{2 \pi i \theta} uv$. This is an abstract defnition: however one can show that this algebra is simple and can be concretely represented as a $C^*$-subalgebra of $B(L^2(\mathbb{T}))$ generated by $U$ and $V$ where $Uf(x)=e^{2\pi i x}f(x)$ and $Vf(x)=f(x+\theta)$. Denote this concrete algebra as $\mathfrak{A}$ and consider $\mathfrak{A}''$ which is von Neumann algebra.

How to prove that $\mathfrak{A}''$ jest a type $II_1$ factor (correct me if it isn't true)?

Le $\theta$ be irrational. One can define the noncommutative torus $A_{\theta}$ as a universal algebra generated by two unitaries $u,v$ satisfying the relation $vu=e^{2 \pi i \theta} uv$. This is an abstract defnition: however one can show that this algebra is simple and can be concretely represented as a $C^*$-subalgebra of $B(L^2(\mathbb{T}))$ generated by $U$ and $V$ where $Uf(x)=e^{2\pi i x}f(x)$ and $Vf(x)=f(x+\theta)$. Denote this concrete algebra as $\mathfrak{A}$ and consider $\mathfrak{A}''$ which is von Neumann algebra.

How to prove that $\mathfrak{A}''$ is a type $II_1$ factor (correct me if it isn't true)?