▁and ▁a ▁von ▁Ne umann ▁algebra ▁is ▁of ▁type ▁X ▁if ▁it ▁can ▁be ▁decom posed ▁as ▁a ▁direct ▁integral ▁of ▁type ▁X ▁factors ; ▁ ▁for ▁example , ▁every ▁commut ative ▁von ▁Ne umann ▁algebra ▁has ▁type ▁I 1 . ▁Every ▁von ▁Ne umann ▁algebra ▁can ▁be ▁written ▁uniqu ely ▁as ▁a ▁sum ▁of ▁von ▁Ne umann ▁al gebras ▁of ▁types ▁I , ▁II , ▁and ▁III . ▁ ▁There ▁are ▁several ▁other ▁ways ▁to ▁divide ▁factors ▁into ▁classes ▁that ▁are ▁sometimes ▁used : ▁▁ ▁A ▁factor ▁is ▁called ▁discrete ▁( or ▁occasionally ▁t ame ) ▁ ▁if ▁it ▁has ▁type ▁I , ▁and ▁continuous ▁( or ▁occasionally ▁wild ) ▁if ▁it ▁has ▁type ▁II ▁or ▁III . ▁ ▁A ▁factor ▁is ▁called ▁semif in ite ▁if ▁it ▁has ▁type ▁I ▁or ▁II , ▁and ▁purely ▁infinite ▁if ▁it ▁has ▁type ▁ ▁III . ▁ ▁A ▁factor ▁is ▁called ▁finite ▁if ▁the ▁projection ▁ 1 ▁is ▁finite ▁and ▁properly ▁infinite ▁otherwise . ▁Fact ors ▁of ▁types ▁I ▁and ▁II ▁may ▁be ▁either ▁finite ▁or ▁properly ▁infinite , ▁but ▁factors ▁of ▁type ▁III ▁are ▁always ▁properly ▁infinite . ▁ ▁Type ▁I ▁factors ▁A ▁factor ▁is ▁said ▁to ▁be ▁of ▁type ▁I ▁if ▁there ▁is ▁a ▁minimal ▁projection ▁E ▁ ≠ ▁ 0 , ▁i . e . ▁a ▁projection ▁E ▁such ▁that ▁there ▁is ▁no ▁other ▁projection ▁F ▁with ▁ 0 ▁< ▁F ▁< ▁E . ▁ ▁Any ▁factor ▁of ▁type ▁I ▁is ▁is omorphic ▁to ▁the ▁von ▁Ne umann ▁algebra ▁of ▁all