▁ ▁is ▁a ▁von ▁Ne umann ▁algebra , ▁non - comm ut ative ▁if ▁the ▁Hil bert ▁space ▁has ▁dimension ▁at ▁least ▁. ▁ ▁Von ▁Ne umann ▁al gebras ▁were ▁first ▁studied ▁by ▁ ▁in ▁ 1 9 2 9 ; ▁he ▁and ▁Francis ▁Murray ▁developed ▁the ▁basic ▁theory , ▁under ▁the ▁original ▁name ▁of ▁rings ▁of ▁operators , ▁in ▁a ▁series ▁of ▁papers ▁written ▁in ▁the ▁ 1 9 3 0 s ▁and ▁ 1 9 4 0 s ▁( ; ▁), ▁repr int ed ▁in ▁the ▁collected ▁works ▁of ▁. ▁ ▁Int ro duct ory ▁accounts ▁of ▁von ▁Ne umann ▁al gebras ▁are ▁given ▁in ▁the ▁online ▁notes ▁of ▁ ▁and ▁ ▁and ▁the ▁books ▁by ▁, ▁, ▁ ▁and ▁. ▁The ▁three ▁volume ▁work ▁by ▁ ▁gives ▁an ▁en cyc lop edic ▁account ▁of ▁the ▁theory . ▁The ▁book ▁by ▁ ▁discuss es ▁more ▁advanced ▁topics . ▁ ▁De fin itions ▁There ▁are ▁three ▁common ▁ways ▁to ▁define ▁von ▁Ne umann ▁al gebras . ▁ ▁The ▁first ▁and ▁most ▁common ▁way ▁is ▁to ▁define ▁them ▁as ▁weak ly ▁closed ▁* - al gebras ▁of ▁bounded ▁operators ▁( on ▁a ▁Hil bert ▁space ) ▁containing ▁the ▁identity . ▁In ▁this ▁definition ▁the ▁weak ▁( operator ) ▁topology ▁can ▁be ▁replaced ▁by ▁ ▁many ▁other ▁common ▁top ologies ▁including ▁the ▁strong , ▁ul tr astr ong ▁or ▁ult ra weak ▁operator ▁top ologies . ▁The ▁* - al gebras ▁of ▁bounded ▁operators ▁that ▁are ▁closed ▁in ▁the ▁norm ▁topology ▁are ▁C *