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Consider the following fragments from Murphy's book "$C^*$-algebras and operator theory":

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To make the post more self-contained, here are the theorems to which the above proof refers.

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Question: Can someone explain more in detail why the marked line is true? I don't see how to connect the situation with strongly continuous functionals on $B(H)$.

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First we check a small detail: The linear map $B(H)\to \Bbb C, v\mapsto \langle v(x), y\rangle$ for some $x,y\in H$ is of the form $v\mapsto \mathrm{Tr}(av)$, where $a= \|x\|\cdot y\otimes x^*$ (this is defined to be the map $H\to H, z\mapsto \|x\|\langle z,x\rangle\cdot y $). This is a rank one map, in particular $a\in L^1(H)$.

Suppose $u(w)=0$ for all $w\in A^\perp$ and $u\notin A$. Note that $A$ is strong closed in $B(H)$ so by A.9 you get a strongly continous functional $\xi: B(H)\to \Bbb C$ with $\xi\lvert_A=0$ and $\xi(u)=1$. By Theorem 4.2.6. you have that $\xi$ is of the form: $$\xi(v)= \sum_{i=1}^n\langle v(x_i), y_i\rangle$$ for all $v\in B(H)$. From what we have checked at the beginning you have that $\xi\in L_1(H)$ follows. Now $\xi$ necessarily vanishes on all of $A$ by construction, hence is an an element of $A^\perp$. But $u(\xi) =1$, contradicting $u(w)=0$ for all $w\in A^\perp$.

What this checks is that a strongly closed sub-space is uniquely determined by its pre-annihilator (via $A= \{ u \mid u(w)=0 \text{ for all $w\in A^\perp$}\}$).

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  • $\begingroup$ I missed the crucial observation you make in the first paragraph. Many thanks! Completely clear! $\endgroup$ – user745578 Sep 9 '20 at 19:19
  • $\begingroup$ Although there are minor mistakes: $\xi$ is not an operator $H \to H$, so it can't be an element of $L_1(H)$ for example, but your answer solved my question. $\endgroup$ – user745578 Sep 9 '20 at 20:03
  • $\begingroup$ You are right, $\xi$ is an element of $B(H)^*$ and the statement $\xi\in L^1(H)$ should be more accurately written as $\xi\in J(L^1(H))$ where $J:L^1(H)\to L^1(H)^{**}=B(H)^*$ is the canonical inclusion. $\endgroup$ – s.harp Sep 9 '20 at 21:03

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