▁ 1 . ▁The ▁M - dim ension ▁is ▁also ▁called ▁the ▁coupling ▁constant ▁of ▁the ▁module ▁H . ▁Type ▁II ∞ : ▁The ▁M - dim ension ▁can ▁be ▁anything ▁in ▁[ 0 , ▁ ∞ ]. ▁There ▁is ▁in ▁general ▁no ▁canonical ▁way ▁to ▁normal ize ▁it ; ▁the ▁factor ▁may ▁have ▁outer ▁autom orph isms ▁multip lying ▁the ▁M - dim ension ▁by ▁constants . ▁The ▁standard ▁representation ▁is ▁the ▁one ▁with ▁M - dim ension ▁ ∞ . ▁Type ▁III : ▁The ▁M - dim ension ▁can ▁be ▁ 0 ▁or ▁ ∞ . ▁Any ▁two ▁non - zero ▁modules ▁are ▁is omorphic , ▁and ▁all ▁non - zero ▁modules ▁are ▁standard . ▁ ▁A men able ▁von ▁Ne umann ▁al gebras ▁ ▁and ▁others ▁proved ▁that ▁the ▁following ▁conditions ▁on ▁a ▁von ▁Ne umann ▁algebra ▁M ▁on ▁a ▁separ able ▁Hil bert ▁space ▁H ▁are ▁all ▁equivalent : ▁▁ ▁M ▁is ▁hyper finite ▁or ▁AF D ▁or ▁approximately ▁finite ▁dimensional ▁or ▁approximately ▁finite : ▁this ▁means ▁the ▁algebra ▁contains ▁an ▁asc ending ▁sequence ▁of ▁finite ▁dimensional ▁sub al gebras ▁with ▁dense ▁union . ▁( Warning : ▁some ▁authors ▁use ▁" hyper finite " ▁to ▁mean ▁" AF D ▁and ▁finite ". ) ▁M ▁is ▁am enable : ▁this ▁means ▁that ▁the ▁deriv ations ▁of ▁M ▁with ▁values ▁in ▁a ▁normal ▁dual ▁Ban ach ▁b im od ule ▁are ▁all ▁inner . ▁M ▁has ▁Schw art z ' s ▁property ▁P : ▁for ▁any ▁bounded ▁operator ▁T ▁on ▁H