▁has ▁a ▁standard ▁representation , ▁which ▁is ▁unique ▁up ▁to ▁isomorphism . ▁The ▁standard ▁representation ▁has ▁an ▁an til ine ar ▁in volution ▁J ▁such ▁that ▁J M J ▁= ▁M ′ . ▁For ▁finite ▁factors ▁the ▁standard ▁module ▁is ▁given ▁by ▁the ▁G NS ▁construction ▁applied ▁to ▁the ▁unique ▁normal ▁tra cial ▁state ▁and ▁the ▁M - dim ension ▁is ▁normal ized ▁so ▁that ▁the ▁standard ▁module ▁has ▁M - dim ension ▁ 1 , ▁while ▁ ▁for ▁infinite ▁factors ▁the ▁standard ▁module ▁is ▁the ▁module ▁with ▁M - dim ension ▁equal ▁to ▁ ∞ . ▁ ▁The ▁possible ▁M - dim ensions ▁of ▁modules ▁are ▁given ▁as ▁follows : ▁Type ▁In ▁( n ▁finite ): ▁The ▁M - dim ension ▁can ▁be ▁any ▁of ▁ 0 / n , ▁ 1 / n , ▁ 2 / n , ▁ 3 / n , ▁... , ▁ ∞ . ▁The ▁standard ▁module ▁has ▁M - dim ension ▁ 1 ▁( and ▁complex ▁dimension ▁n 2 .) ▁Type ▁I ∞ ▁The ▁M - dim ension ▁ ▁can ▁be ▁any ▁of ▁ 0 , ▁ 1 , ▁ 2 , ▁ 3 , ▁... , ▁ ∞ . ▁The ▁standard ▁representation ▁of ▁B ( H ) ▁is ▁H ⊗ H ; ▁its ▁M - dim ension ▁is ▁ ∞ . ▁Type ▁II 1 : ▁The ▁M - dim ension ▁can ▁be ▁anything ▁in ▁[ 0 , ▁ ∞ ]. ▁It ▁is ▁normal ized ▁so ▁that ▁the ▁standard ▁module ▁has ▁M - dim ension