▁the ▁weak ▁operator ▁closed ▁convex ▁h ull ▁of ▁the ▁elements ▁u Tu * ▁contains ▁an ▁element ▁comm uting ▁with ▁M . ▁M ▁is ▁sem id is crete : ▁this ▁means ▁the ▁identity ▁map ▁from ▁M ▁to ▁M ▁is ▁a ▁weak ▁point wise ▁limit ▁of ▁completely ▁positive ▁maps ▁of ▁finite ▁rank . ▁M ▁has ▁property ▁E ▁or ▁the ▁H ak eda – Tom iy ama ▁extension ▁property : ▁this ▁means ▁that ▁there ▁is ▁a ▁projection ▁of ▁norm ▁ 1 ▁from ▁bounded ▁operators ▁on ▁H ▁to ▁M ▁'. ▁M ▁is ▁inject ive : ▁any ▁completely ▁positive ▁linear ▁map ▁from ▁any ▁self ▁ad joint ▁closed ▁sub space ▁containing ▁ 1 ▁of ▁any ▁un ital ▁C * - algebra ▁A ▁to ▁M ▁can ▁be ▁extended ▁to ▁a ▁completely ▁positive ▁map ▁from ▁A ▁to ▁M . ▁ ▁There ▁is ▁no ▁generally ▁accepted ▁term ▁for ▁the ▁class ▁of ▁al gebras ▁above ; ▁Con nes ▁has ▁suggested ▁that ▁am enable ▁should ▁be ▁the ▁standard ▁term . ▁ ▁The ▁am enable ▁factors ▁have ▁been ▁class ified : ▁there ▁is ▁a ▁unique ▁one ▁of ▁each ▁of ▁the ▁types ▁In , ▁I ∞ , ▁II 1 , ▁II ∞ , ▁III λ , ▁for ▁ 0 ▁< ▁ λ ▁ ≤ ▁ 1 , ▁and ▁the ▁ones ▁of ▁type ▁III 0 ▁correspond ▁to ▁certain ▁er god ic ▁flows . ▁( For ▁type ▁III 0 ▁calling ▁this ▁a ▁classification ▁is ▁a ▁little ▁mis leading , ▁as ▁it ▁is ▁known ▁that ▁there ▁is ▁no ▁easy ▁way ▁to ▁class ify ▁the ▁corresponding ▁er god ic ▁flows