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

Abstract: We propose a global Jacquet-Langlands correspondence for the modules over the von Neumann algebras of $S$-arithmetic subgroups of $\rm{GL}(2)$ and of a quaternion algebra $D$, which are both defined over a totally real number field $F$. If a representation $\pi'=\otimes\pi'v$ of $D^{\times}(\mathbb{A}_F)$ corresponds to a representation $\pi=\otimes \pi_v$ of $\rm{GL}(2,\mathbb{A}_F)$, we have $\frac{\dim{\mathcal{L}(\rm{SL}(2,\mathcal{O}S))}\pi_S}{\dim{\mathbb{C}}\pi'S}=|\frac{\zeta{D}(0)}{\zeta_{F}(0)}|$, where $\zeta_F,\zeta_{D}$ are the zeta functions of $F,D$ respectively.