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Let $\omega$ be a positive linear functional on $M$ which is a Von Neumann Algebra. Suppose $\omega$ is completely additive (i.e. $\omega$ applied to a strongly convergent sum of mutually orthogonal projections is the same as the convergent complex valued sum of $\omega$ applied to the projections). Then supposedly, $\omega$ is the pointwise convergent sum:

$\sum\limits_{i=0}^\infty \omega_{\xi_n} \text{ where } \omega_{\xi_n}(x)=\langle\xi_n, x(\xi_n)\rangle \qquad\forall x \in M$

Please prove this for me, but beware to not use the Cauchy Schwarz inequality in an incorrect way. The reason I am without a proof is that the proof in a document I am reading is actually incorrect, and uses Cauchy Schwarz incorrectly.

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    Okay I will see what I can find in the "math library" at my school. I don't know any of those things you mentioned, but I am aware they are true. I am unwilling to use them because in my resource, the equivalence of 3 of the 4 statements you mentioned, plus the one I mentioned in this post, is proven. The one implication I asked about in this post is the last one to establish this equivalence, but as far as I'm concerned right now none of these things are equivalent. Thanks.2012-06-04

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A concise proof of this fact is given as Theorem 46.4 in Conway's book "A course in operator theory" (2000).

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    I actually e$n$ded up chasi$n$g through a $n$ot so concise proof that was obtained through a composition of Dixmier's book with other resources.2012-08-10