▁C * - al gebras ▁and ▁locally ▁compact ▁Haus dor ff ▁spaces . ▁ ▁Every ▁commut ative ▁von ▁Ne umann ▁algebra ▁is ▁is omorphic ▁to ▁L ∞ ( X ) ▁for ▁some ▁measure ▁space ▁( X , ▁ μ ) ▁and ▁convers ely , ▁for ▁every ▁ σ - finite ▁measure ▁space ▁X , ▁the ▁* - algebra ▁L ∞ ( X ) ▁is ▁a ▁von ▁Ne umann ▁algebra . ▁ ▁Due ▁to ▁this ▁anal ogy , ▁the ▁theory ▁of ▁von ▁Ne umann ▁al gebras ▁has ▁been ▁called ▁non comm ut ative ▁measure ▁theory , ▁while ▁the ▁theory ▁of ▁C * - al gebras ▁is ▁sometimes ▁called ▁non comm ut ative ▁topology ▁. ▁ ▁Pro jections ▁ ▁Oper ators ▁E ▁in ▁a ▁von ▁Ne umann ▁algebra ▁for ▁which ▁E ▁= ▁E E ▁= ▁E * ▁are ▁called ▁pro jections ; ▁they ▁are ▁exactly ▁the ▁operators ▁which ▁give ▁an ▁orthogonal ▁projection ▁of ▁H ▁onto ▁some ▁closed ▁sub space . ▁A ▁sub space ▁of ▁the ▁Hil bert ▁space ▁H ▁is ▁said ▁to ▁belong ▁to ▁the ▁von ▁Ne umann ▁algebra ▁M ▁if ▁it ▁is ▁the ▁image ▁of ▁some ▁projection ▁in ▁M . ▁This ▁establish es ▁a ▁ 1 : 1 ▁correspond ence ▁between ▁pro jections ▁of ▁M ▁and ▁sub spaces ▁that ▁belong ▁to ▁M . ▁Inform ally ▁these ▁are ▁the ▁closed ▁sub spaces ▁that ▁can ▁be ▁described ▁using ▁elements ▁of ▁M , ▁or ▁that ▁M ▁" kn ows " ▁about . ▁ ▁It ▁can ▁be ▁shown ▁that ▁the ▁closure ▁of ▁the ▁image ▁of ▁any ▁operator ▁in