Von Neumann Dimensions and Trace Formulas II: A Jacquet-Langlands correspondence for Arithmetic Group Algebras in $\mathrm{G}\mathrm{L}(2)$

We propose a global Jacquet-Langlands correspondence for the modules over the von Neumann algebras of $S$-arithmetic subgroups of $\mathrm{G}\mathrm{L}(2)$ and of a quaternion algebra $D$, which are both defined over a totally real number field $F$. If a representation ${\pi }^{\mathrm{\prime }}=\otimes {\pi }_v^{\mathrm{\prime }}$ of $D^{\times }({\mathbb{A}}_F)$ corresponds to a representation $\pi =\otimes {\pi }_v$ of $\mathrm{G}\mathrm{L}(2,{\mathbb{A}}_F)$, we have $\frac{{\dim }_{\mathcal{L}(\mathrm{S}\mathrm{L}(2,{\mathcal{O}}_S))}{\pi }_S}{{\dim }_{\mathbb{C}}{\pi }_S^{\mathrm{\prime }}}=\mathrm{\mid }\frac{{\zeta }_D(0)}{{\zeta }_F(0)}\mathrm{\mid }$, where ${\zeta }_F,{\zeta }_D$ are the zeta functions of $F,D$ respectively.