▁the ▁von ▁Ne umann ▁algebra ▁of ▁all ▁bounded ▁operators ▁on ▁it , ▁since ▁it ▁is ▁is omet r ically ▁is omorphic ▁to ▁a ▁proper ▁subset ▁of ▁itself . ▁However ▁it ▁is ▁possible ▁for ▁infinite ▁dimensional ▁sub spaces ▁to ▁be ▁finite . ▁ ▁Orth ogonal ▁pro jections ▁are ▁non comm ut ative ▁analog ues ▁of ▁indicator ▁functions ▁in ▁L ∞ ( R ). ▁L ∞ ( R ) ▁is ▁the ▁|| · || ∞ - closure ▁of ▁the ▁sub space ▁generated ▁by ▁the ▁indicator ▁functions . ▁Similarly , ▁a ▁von ▁Ne umann ▁algebra ▁is ▁generated ▁by ▁its ▁pro jections ; ▁this ▁is ▁a ▁consequence ▁of ▁the ▁spectral ▁theorem ▁for ▁self - ad joint ▁operators . ▁ ▁The ▁pro jections ▁of ▁a ▁finite ▁factor ▁form ▁a ▁continuous ▁geometry . ▁ ▁Fact ors ▁ ▁A ▁von ▁Ne umann ▁algebra ▁N ▁whose ▁center ▁consists ▁only ▁of ▁multi ples ▁of ▁the ▁identity ▁operator ▁is ▁called ▁a ▁factor . ▁▁ ▁showed ▁that ▁every ▁von ▁Ne umann ▁algebra ▁on ▁a ▁separ able ▁Hil bert ▁space ▁is ▁is omorphic ▁to ▁a ▁direct ▁integral ▁of ▁factors . ▁ ▁This ▁decomposition ▁is ▁essentially ▁unique . ▁ ▁Thus , ▁the ▁problem ▁of ▁class ifying ▁isomorphism ▁classes ▁of ▁von ▁Ne umann ▁al gebras ▁on ▁separ able ▁Hil bert ▁spaces ▁can ▁be ▁reduced ▁to ▁that ▁of ▁class ifying ▁isomorphism ▁classes ▁of ▁factors . ▁▁ ▁showed ▁that ▁every ▁factor ▁has ▁one ▁of ▁ 3 ▁types ▁as ▁described ▁below . ▁ ▁The ▁type ▁classification ▁can ▁be ▁extended ▁to ▁von ▁Ne umann ▁al gebras ▁that ▁are ▁not ▁factors ,