▁algebraic ▁tensor ▁product ▁of ▁the ▁al gebras ▁considered ▁as ▁rings ), ▁which ▁is ▁again ▁a ▁von ▁Ne umann ▁algebra , ▁and ▁act ▁on ▁the ▁tensor ▁product ▁of ▁the ▁corresponding ▁Hil bert ▁spaces . ▁ ▁The ▁tensor ▁product ▁of ▁two ▁finite ▁al gebras ▁is ▁finite , ▁and ▁the ▁tensor ▁product ▁of ▁an ▁infinite ▁algebra ▁and ▁a ▁non - zero ▁algebra ▁is ▁infinite . ▁The ▁type ▁of ▁the ▁tensor ▁product ▁of ▁two ▁von ▁Ne umann ▁al gebras ▁ ▁( I , ▁II , ▁or ▁III ) ▁is ▁the ▁maximum ▁of ▁their ▁types . ▁The ▁commut ation ▁theorem ▁for ▁tensor ▁products ▁states ▁that ▁ ▁where ▁M ′ ▁denotes ▁the ▁commut ant ▁of ▁M . ▁ ▁The ▁tensor ▁product ▁of ▁an ▁infinite ▁number ▁of ▁von ▁Ne umann ▁al gebras , ▁if ▁done ▁na ively , ▁is ▁usually ▁a ▁rid icul ously ▁large ▁non - separ able ▁algebra . ▁Instead ▁ ▁showed ▁that ▁ ▁one ▁should ▁choose ▁a ▁state ▁on ▁each ▁of ▁the ▁von ▁Ne umann ▁al gebras , ▁use ▁this ▁to ▁define ▁a ▁state ▁on ▁the ▁algebraic ▁tensor ▁product , ▁which ▁can ▁be ▁used ▁to ▁produce ▁a ▁Hil bert ▁space ▁and ▁a ▁( reason ably ▁small ) ▁von ▁Ne umann ▁algebra . ▁ ▁studied ▁the ▁case ▁where ▁all ▁the ▁factors ▁are ▁finite ▁matrix ▁al gebras ; ▁these ▁factors ▁are ▁called ▁Ara ki – W ood s ▁factors ▁or ▁I TP FI ▁factors ▁( IT P FI ▁stands ▁for ▁" in finite ▁tensor ▁product ▁of ▁finite ▁type ▁I ▁factors "). ▁ ▁The ▁type ▁of ▁the ▁infinite ▁tensor ▁product