▁can ▁vary ▁dram atically ▁as ▁the ▁states ▁are ▁changed ; ▁for ▁example , ▁the ▁infinite ▁tensor ▁product ▁of ▁an ▁infinite ▁number ▁of ▁type ▁I 2 ▁factors ▁can ▁have ▁any ▁type ▁depending ▁on ▁the ▁choice ▁of ▁states . ▁In ▁particular ▁ ▁found ▁an ▁un count able ▁family ▁of ▁non - is omorphic ▁hyper finite ▁type ▁III λ ▁factors ▁for ▁ 0 ▁< ▁ λ ▁< ▁ 1 , ▁called ▁Pow ers ▁factors , ▁by ▁taking ▁an ▁infinite ▁tensor ▁product ▁of ▁type ▁I 2 ▁factors , ▁each ▁with ▁the ▁state ▁given ▁by : ▁▁▁ ▁All ▁hyper finite ▁von ▁Ne umann ▁al gebras ▁not ▁of ▁type ▁III 0 ▁are ▁is omorphic ▁to ▁Ara ki – W ood s ▁factors , ▁but ▁there ▁are ▁un count ably ▁many ▁of ▁type ▁III 0 ▁that ▁are ▁not . ▁ ▁B im od ules ▁and ▁sub fact ors ▁A ▁b im od ule ▁( or ▁correspond ence ) ▁is ▁a ▁Hil bert ▁space ▁H ▁with ▁module ▁actions ▁of ▁two ▁comm uting ▁von ▁Ne umann ▁al gebras . ▁B im od ules ▁have ▁a ▁much ▁rich er ▁structure ▁than ▁that ▁of ▁modules . ▁Any ▁b im od ule ▁over ▁two ▁factors ▁always ▁gives ▁a ▁sub factor ▁since ▁one ▁of ▁the ▁factors ▁is ▁always ▁contained ▁in ▁the ▁commut ant ▁of ▁the ▁other . ▁There ▁is ▁also ▁a ▁subt le ▁relative ▁tensor ▁product ▁operation ▁due ▁to ▁Con nes ▁on ▁b im od ules . ▁The ▁theory ▁of ▁sub fact ors , ▁initi ated ▁by ▁V aug han ▁Jones , ▁recon c iles ▁these