Split property for type III algebras entails practical separability

Question

I am reading Halvorson's thesis (http://philsci-archive.pitt.edu/346/1/main-new.pdf), however I don't understand a proof at p.50 where he tries to explain why the split property allows a local agent to disentangle his system from its environment for all practical purposes. He proves that $P_iXP_i=c_iP_i$ on $P_i\mathcal{H}$ hence $T(X)=\rho(X)1$ and then says that by summing over $i$ one gets the result. However, the $1$ in the previous equality is the identity on $P_i\mathcal{H}$, so I don't understand how he can perform the sum. Any insights?

This post imported from StackExchange Physics at 2014-10-11 09:56 (UTC), posted by SE-user Issam Ibnouhsein