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So, there exists :Ro C N! ,. :Ro. ,. CM, J = contradicting the finiteness of M. Both assertions follow from the non-existence of a T as above for any pair of admissible decompositions. 3. 6. If 11, N TI M are both non-zero and finite, let [M/ N ] denote the uniquely determined integer card I, as in Prop. 5. 0 Note that in the example M = l(Jf), [M/ N] is the greatest integer which does not exceed dim M/dim N, so the similarity with the notation for the greatest integer function ({t] = n iff n , t < n + 1) is not an accident.

If x e M, then x e M+ if and only If CP(x) "'0 for all cP in M. +, and similarly, the dual statement (with the roles of cP and x interChanged) is also valid. (As above, we shall think of the elements of M. ,+' let x e M and let Mo = (x}", the von Neumann algebra generated by x and 1. (a) If x is normal, the equation vX

Before doing that, however, it will help to examine the quantitative aspects of the Euclidean algorithm established earlier (cf. Prop. 3). 5. Let M, N T) M; suppose N '" (0) and M is finite. ,. N for all i E I and :R ~ N (as in Prop. 3), the index set I is finite and its cardinality is independent of the particular decomposition chosen. Proof. ) $:R' is another such decomposition and suppose, if possible, Jthat J there exists a map T: I .. J which is injective but not surjective. ' . So, there exists :Ro C N!