By Kenneth I. Joy

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**Example text**

1 to conclude that each Gn is nonvoid; also the elements of {Gn : n = 1 , . . } are ordered by reverse inclusion. Thus there is a projection p in the intersection of all the G n 's. (p - a) n—*oo in the weak* topology. Since ||* n (p — c)lloo < ^°° for each n, and since the norm is weak* lower semicontinuous, we get \\^(p — c)||oo < ^°°, as desired. Next observe that the sets of symmetries and unitaries in M are also weak* closed. In addition, E((pMq)i) = pqE({M)\) and so it too is weak* closed.

1] the least cardinal whose power set admits a nontrivial, countably addititive two-valued measure is measurable. It is known that if a measurable cardinal exists, then it must be quite large. )) 1 In practice, most operator algebraists (willing to assume the continuum hypothesis) will never encounter a von Neumann algebra that fails to satisfy the condition. In particular, if M. is countably decomposable, then, assuming the continuum hypothesis, M** is essentially countably decomposible. 52 C.

Let J\f denote the direct sum of k copies of Af and write a — a\ 0 • • • 0a*;. Let Q be as defined as above. Let $ stand for the direct sum of k copies of ^ and observe that $ has range in Ckn and is weak* continuous. 6 to get an extreme point p of Q such that $(p) = $(a). C. A. 1 p has the form p = p\ 0 • • • 0 p&, where each pj is a projection. Moreover p\ + \-pk = \ and ^(pj) = ^((tj) for each index j . • We now turn our attention to the finite atomic case. Recall that we define the constant 6 as follows: 8 = 6(ty) = 5wp{||^(^)||i : q is a minimal projection in Af}.