1 ▁then ▁the ▁factor ▁is ▁of ▁type ▁III 0 , ▁if ▁the ▁Con nes ▁spectrum ▁is ▁all ▁integral ▁powers ▁of ▁ λ ▁for ▁ 0   <   λ   <   1 , ▁then ▁the ▁type ▁is ▁III λ , ▁and ▁if ▁the ▁Con nes ▁spectrum ▁is ▁all ▁positive ▁re als ▁then ▁the ▁type ▁is ▁III 1 . ▁( The ▁Con nes ▁spectrum ▁is ▁a ▁closed ▁subgroup ▁of ▁the ▁positive ▁re als , ▁so ▁these ▁are ▁the ▁only ▁possibilities .) ▁The ▁only ▁trace ▁on ▁type ▁III ▁factors ▁takes ▁value ▁ ∞ ▁on ▁all ▁non - zero ▁positive ▁elements , ▁and ▁any ▁two ▁non - zero ▁pro jections ▁are ▁equivalent . ▁At ▁one ▁time ▁type ▁III ▁factors ▁were ▁considered ▁to ▁be ▁int ract able ▁objects , ▁but ▁Tom ita – T akes aki ▁theory ▁has ▁led ▁to ▁a ▁good ▁structure ▁theory . ▁In ▁particular , ▁any ▁type ▁III ▁factor ▁can ▁be ▁written ▁in ▁a ▁canonical ▁way ▁as ▁the ▁crossed ▁product ▁of ▁a ▁type ▁II ∞ ▁factor ▁and ▁the ▁real ▁numbers . ▁ ▁The ▁pre du al ▁Any ▁von ▁Ne umann ▁algebra ▁M ▁has ▁a ▁pre du al ▁M ∗ , ▁which ▁is ▁the ▁Ban ach ▁space ▁of ▁all ▁ult ra weak ly ▁continuous ▁linear ▁function als ▁on ▁M . ▁As ▁the ▁name ▁suggests , ▁M ▁is ▁( as ▁a ▁Ban ach ▁space ) ▁the ▁dual ▁of ▁its ▁pre du al . ▁The ▁pre du al ▁is ▁unique ▁in ▁the ▁sense ▁that ▁any ▁other ▁Ban ach ▁space ▁whose ▁dual ▁is ▁M ▁is ▁canon ically