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The following picture is lemma 4.23 in Lectures on Coarse Geometry by John Roe:

enter image description here

I guess the $E_i$ in the centered formula is $X_i$. Does Roe mean that $X_j\cap \mathrm{Supp}(u)=\emptyset $ implies $\lambda(E_j)u=0$? But I can't work out. Can someone help me? (If more details are needed, let me know.)

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Indeed, I think the $E_i$'s should be $X_i$'s - I can't see what else they would be.

For the last step, the implication needed is that if $x \in X_k$ and $\lambda(X_k)Tu = 0$ then $x \notin Supp(Tu)$.

But this is just a restatement of the definition of $Supp(Tu)$. The actual definition says that $Supp(Tu)$ is the set of all $x$ such that $\lambda(X_j)Tu \neq 0$ whenever $x \in X_j$. It follows that $x \notin Supp(Tu)$ if and only if there is $X_j$ containing $x$ such that $\lambda(X_j)Tu = 0$.

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  • $\begingroup$ The support of $ v\in H $ is the set of points $x\in X$ such that for every $B\in \mathfrak C$ that contains $x$, $\lambda(B)v\neq 0$. I really can't see why $x\notin \operatorname{Supp}(v)$ implies there is a $X_j$ containing $x$ such that $\lambda(X_j)v=0$. This is in fact what I want to ask. $\endgroup$
    – C. Ding
    Commented Apr 26, 2018 at 6:05
  • $\begingroup$ @C.Ding Without having my copy of the book in front of me, I assume that the $\mathfrak{C}$ you're referring to is the family of localizing sets in the definition of "geometric hilbert space over $X$". I assume that "We choose a decomposition $X = \bigcup X_j$ as in Definition 4.20" also refers to the localizing sets, so the $X_j$'s are interchangeable with the $B$'s. $\endgroup$
    – Paul Siegel
    Commented Apr 26, 2018 at 12:35
  • $\begingroup$ If so then you just have to take the definition, which gives necessary and sufficient conditions for $x$ to be in $Supp(v)$, and rewrite it so that it gives necessary and sufficient conditions for $x$ to not be in $Supp(v)$. That's what I wrote in my answer. $\endgroup$
    – Paul Siegel
    Commented Apr 26, 2018 at 12:36

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