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I've added some material which partly explains the importance of the bicommutant, although I admit it's not particularly accessible to the layman. Looking...

reverted explicitly said much of the theory is still applicable in general (bicommutant theorem, Kaplansky density theorem etc.) The important fact to note is...

I think the prove gives here can not work when H is inseaperateable, because the definiton of strong topology. The end of the proof that ii)=> iii) doen...