Given a number field F and a reductive group G over F, the unitary dual \(\widehat{G({\mathbb {A}}_F)}\) of the adelic group \(G({\mathbb {A}}_F)\) and the Plancherel measure \(\nu _{G({\mathbb {A}}_F)}\) on it can be determined by the Plancherel measure of its local groups \(G(F_v).\) Given a subset \(X\subset \widehat{G({\mathbb {A}}_F)}\) of finite Plancherel measure, let \(H_X\) be the direct integral of the irreducible representations in X. Besides a \(G({\mathbb {A}}_F)\) -module and a G(F)-module, \(H_X\) is also a module over the group von Neumann algebra \({\mathcal {L}}(G(F)),\) hence there is a canonical dimension \(\dim _{{\mathcal {L}}(G(F))}H_X\in [0,\infty ).\) It is proved that the Plancherel measure of \(G({\mathbb {A}}_F)\) coincides with the dimension over the algebra \({\mathcal {L}}(G(F))\) : \(\begin{aligned} \dim _{{\mathcal {L}}(G(F))}H_X=\nu _{G({\mathbb {A}}_F)}(X), \end{aligned}\) if G is semisimple, simply connected and \(G({\mathbb {A}}_F)\) is equipped with the Tamagawa measure.
