▁M ▁and ▁the ▁kernel ▁of ▁any ▁operator ▁in ▁M ▁belongs ▁to ▁M . ▁Also , ▁the ▁closure ▁of ▁the ▁image ▁under ▁an ▁operator ▁of ▁M ▁of ▁any ▁sub space ▁belonging ▁to ▁M ▁also ▁belongs ▁to ▁M . ▁ ▁( Th ese ▁results ▁are ▁a ▁consequence ▁of ▁the ▁polar ▁decomposition ). ▁ ▁Com parison ▁theory ▁of ▁pro jections ▁The ▁basic ▁theory ▁of ▁pro jections ▁was ▁worked ▁out ▁by ▁. ▁Two ▁sub spaces ▁belonging ▁to ▁M ▁are ▁called ▁( M ur ray – von ▁Ne umann ) ▁equivalent ▁if ▁there ▁is ▁a ▁partial ▁is ometry ▁mapping ▁the ▁first ▁is omorph ically ▁onto ▁the ▁other ▁that ▁is ▁an ▁element ▁of ▁the ▁von ▁Ne umann ▁algebra ▁( in form ally , ▁if ▁M ▁" kn ows " ▁that ▁the ▁sub spaces ▁are ▁is omorphic ). ▁This ▁indu ces ▁a ▁natural ▁equivalence ▁relation ▁on ▁pro jections ▁by ▁defining ▁E ▁to ▁be ▁equivalent ▁to ▁F ▁if ▁the ▁corresponding ▁sub spaces ▁are ▁equivalent , ▁or ▁in ▁other ▁words ▁if ▁there ▁is ▁a ▁partial ▁is ometry ▁of ▁H ▁that ▁maps ▁the ▁image ▁of ▁E ▁is omet r ically ▁to ▁the ▁image ▁of ▁F ▁and ▁is ▁an ▁element ▁of ▁the ▁von ▁Ne umann ▁algebra . ▁Another ▁way ▁of ▁stating ▁this ▁is ▁that ▁E ▁is ▁equivalent ▁to ▁F ▁if ▁E = u u * ▁and ▁F = u * u ▁for ▁some ▁partial ▁is ometry ▁u ▁in ▁M . ▁ ▁The ▁equivalence ▁relation ▁~ ▁thus ▁defined ▁is ▁add itive ▁in ▁the ▁following ▁sense : ▁Suppose ▁E 1 ▁~ ▁F 1 ▁and ▁E 2