.) ▁The ▁ones ▁of ▁type ▁I ▁and ▁II 1 ▁were ▁class ified ▁by ▁, ▁and ▁the ▁remaining ▁ones ▁were ▁class ified ▁by ▁, ▁except ▁for ▁the ▁type ▁III 1 ▁case ▁which ▁was ▁completed ▁by ▁Ha ager up . ▁ ▁All ▁am enable ▁factors ▁can ▁be ▁constructed ▁using ▁the ▁group - measure ▁space ▁construction ▁of ▁Murray ▁and ▁von ▁Ne umann ▁for ▁a ▁single ▁er god ic ▁transformation . ▁In ▁fact ▁they ▁are ▁precisely ▁the ▁factors ▁ar ising ▁as ▁crossed ▁products ▁by ▁free ▁er god ic ▁actions ▁of ▁Z ▁or ▁Z / n Z ▁on ▁ab elian ▁von ▁Ne umann ▁al gebras ▁L ∞ ( X ). ▁Type ▁I ▁factors ▁occur ▁when ▁the ▁measure ▁space ▁X ▁is ▁atomic ▁and ▁the ▁action ▁trans itive . ▁When ▁X ▁is ▁diff use ▁or ▁non - atomic , ▁it ▁is ▁equivalent ▁to ▁[ 0 , 1 ] ▁as ▁a ▁measure ▁space . ▁Type ▁II ▁factors ▁occur ▁when ▁X ▁adm its ▁an ▁equivalent ▁finite ▁( II 1 ) ▁or ▁infinite ▁( II ∞ ) ▁measure , ▁invariant ▁under ▁an ▁action ▁of ▁Z . ▁Type ▁III ▁factors ▁occur ▁in ▁the ▁remaining ▁cases ▁where ▁there ▁is ▁no ▁invariant ▁measure , ▁but ▁only ▁an ▁invariant ▁measure ▁class : ▁these ▁factors ▁are ▁called ▁Krie ger ▁factors . ▁ ▁T ensor ▁products ▁of ▁von ▁Ne umann ▁al gebras ▁The ▁Hil bert ▁space ▁tensor ▁product ▁of ▁two ▁Hil bert ▁spaces ▁is ▁the ▁completion ▁of ▁their ▁algebraic ▁tensor ▁product . ▁One ▁can ▁define ▁a ▁tensor ▁product ▁of ▁von ▁Ne umann ▁al gebras ▁( a ▁completion ▁of ▁the