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

We propose a global Jacquet-Langlands correspondence for the modules over the von Neumann algebras of S-arithmetic subgroups of GL(2) and of a quaternion algebra D, which are both defined over a totally real number field F. If a representation π'=⊗π'ᵥ of D×(AF) corresponds to a representation π=⊗ πᵥ of GL(2,AF), we have L(SL(2,OS))πSCπ'S=|ζD(0)ζF(0)|, where ζF,ζD are the zeta functions of $F,D$ respectively.