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The following statement is being used a lot in the literature, and I wonder how to prove it.

Let $M$ be an infinite-dimensional von Neumann algebra (with unit element), show that there is an increasing series of projections {$p_i$}$_{i=1}^\infty$ such that $\forall n : p_n \neq 1$, and $p_n\stackrel{n\to\infty}\longrightarrow 1$.

My thoughts so far:

I know that in an infinite-dimensional von Neumann algebra there is an infinite set of pairwise orthogonal projections. so taking their partial sums would be an increasing series of projections, however I'm not sure if they converge to 1.

Thanks in advance!

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  • $\begingroup$ Take a unitary element $u$ so that $\sigma(u)=\mathbb T$, the unit circle. Then $\{1,u\}^{\prime\prime}\subset M$ is isomorphic to $L_\infty(\mathbb T)$. Can you do it in $L_\infty(\mathbb T)$? (Simply take an essentially increasing family of sets of positive measure that cover $\mathbb T$ and their indicators are the sought projections.) $\endgroup$
    – Tomasz Kania
    Commented May 27, 2020 at 15:01
  • $\begingroup$ @TomaszKania - Thanks for your comment, could you explain why does {$1,u$}$''$ isomorphic to $L_\infty(\mathbb{T})$? $\endgroup$
    – dreamwave
    Commented May 27, 2020 at 16:09
  • $\begingroup$ That's the spectral theorem for normal elements; see Borel functional calculus in any operator algebra textbook. $\endgroup$
    – Tomasz Kania
    Commented May 27, 2020 at 16:34

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