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Let $\{P_i\}$ be a net of projections on a Hilbert space. Can we show that the WOT limit of this net is a projection too?

I saw below example of a sequence of projections which its wot limit is not a projection.

Example: Let $P_n$ be the infinite matrix whose entries at the positions $(1,1),(1,n),(n,1), (n,n)$ are $\frac{1}{2}$ and whose remaining entries are all $0$ ($n=2,3,...$). Each $P_n$ is a projection and $P_n\to A$ (weakly) when $A$ is a matrix whose only nonzero entry is $\frac{1}{2}$. Clearly $A$ is not a projection.

I'm getting confused about it. How is it possible? Please help me. Thanks.

Tomasz Kania
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niki
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1 Answers1

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SOT-limit of a net of projections is a projection. This is not true for WOT-limit (as your example shows).

Yurii Savchuk
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