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I was reading the section on Schur Multipliers in Ozawa's book "C*-algebras and Finite-Dimensional Approximations" and I am having troubles understanding the proof of Proposition D.6, namely the part about ultraweak continuity.

I think I can skip most of the context and I am stuck on a particular part, so I'll write it differently.

Let $H$ be a Hilbert (in the problem we have $H=\ell^2(\Gamma)$) and let $T: B(H) \rightarrow B(H)$ be an ultraweak continuous linear operator on $H$, moreover, we know that the restriction $T|_{K(H)}:K(H) \rightarrow K(H) $ is well defined and not only ultraweak continuous, but also norm continuous. To my understanding of the proof in the book, this implies that $T$ is also norm continuous on the whole $B(H)$, however I am not being able to prove this.

I tried using the fact that trace class operators (predual of $B(H)$) are compact, but did not reach any relevant conclusion, moreover I am almost sure the Uniform Boundness Theorem is required.

If this affirmation is false and more context is needed, it can be found in the mentioned book, Proposition D.6.

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    $\begingroup$ This is valid in a more general context. Let $X,Y$ be Banach spaces, $S:Y^*\to X^*$ be linear. $S$ is weak$^*$-to-weak$^*$ continuous iff $\exists$ bounded $T:X\to Y$ such that $S=T^*$. Apply this to $X=Y=$trace class operators, and keep in mind that $X^*=B(H)$ and ultraweek topology is the weak$^*$ topology on $B(H)$. $\endgroup$
    – Onur Oktay
    Mar 3 at 23:37
  • $\begingroup$ @OnurOktay Yes this is very helpful and it is similar to what I was trying to do, It will be very good to keep this mind going foward thank you. $\endgroup$ Mar 3 at 23:44
  • $\begingroup$ @OnurOktay However, I think this is not applicable in this specific case because we only know that $T|_{K(H)}:K(H) \rightarrow K(H)$ and we do not necessarily have $T|_{L^1(B(H))}:L^1(B(H)) \rightarrow L^1(B(H))$, only $T|_{L^1(B(H))}:L^1(B(H)) \rightarrow K(H)$ . Is there a way around this? $\endgroup$ Mar 3 at 23:46
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    $\begingroup$ @TomásPacheco That’s not what Onur meant. What Onur’s comment suggested is that, more generally, if a linear map $T: Y^\ast \to X^\ast$ is weak$^\ast$-to-weak$^\ast$ continuous, without any other assumptions whatsoever, then it is automatically bounded, which can be proved via uniform boundedness principle. $\endgroup$
    – David Gao
    Mar 4 at 0:01
  • $\begingroup$ @DavidGao Ah! and thats because if $S: X \rightarrow Y$ is bounded and such that $S^* = T$ then $T$ is bounded right? $\endgroup$ Mar 4 at 0:04

2 Answers 2

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By Kaplansky density theorem, for any $x \in B(H)$, there exists a net $x_\lambda \in K(H)$ with $x_\lambda \to x$ ultraweakly and $\|x_\lambda\| \leq \|x\|$ for all $\lambda$. Then $\|Tx_\lambda\| \leq \|T|_{K(H)}\|\|x_\lambda\| \leq \|T|_{K(H)}\|\|x\|$. By ultraweak continuity of $T$, we have $Tx_\lambda \to Tx$ ultraweakly, whence $\|Tx\| \leq \|T|_{K(H)}\|\|x\|$. Since $x \in B(H)$ is arbitrary, this means $T$ is bounded and in fact $\|T\| \leq \|T|_{K(H)}\|$.

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  • $\begingroup$ Thank you so much! I have a question, is it the case that $\|Tx_\lambda\| \rightarrow \|Tx\|$ ? If yes why does ultraweak convergence imply it? I am quite new into the ultraweak topology, do you recommend a reference where I can learn it more? Thank you so much for the reply $\endgroup$ Mar 3 at 23:38
  • $\begingroup$ @TomásPacheco No, it is not true in general that $\|Tx_\lambda\| \to \|Tx\|$, but it is true that $\|Tx\| \leq \lim \sup \|Tx_\lambda\|$, which is enough to prove the claim. $\endgroup$
    – David Gao
    Mar 3 at 23:51
  • $\begingroup$ @TomásPacheco I’m not sure if there’s any reference specifically on ultraweak topology, but most introductory textbooks on operator algebras cover this. See Chapter 4 of Murphy’s $C^\ast$-algebras and operator theory, and Chapters 2 and 3 of Conway’s A course in operator theory, for example. $\endgroup$
    – David Gao
    Mar 3 at 23:55
  • $\begingroup$ Thank you so much for the book recomendations, it was what I was looking for. Regarding the inequality, I am having troubles seeing where it comes from, do you have a hint? Thank you again $\endgroup$ Mar 4 at 0:01
  • $\begingroup$ @TomásPacheco Use the definition of ultraweak convergence to prove the following: if $x_\lambda \to x$ ultraweakly and $x_\lambda$ are all in the unit ball, then $x$ is in the unit ball as well. Hint: $\|x\| = \sup_{\varphi \in (B(H)_\ast)_1} |\varphi(x)| = \sup_{\varphi \in (B(H)_\ast)_1} \lim_\lambda |\varphi(x_\lambda)|$. $\endgroup$
    – David Gao
    Mar 4 at 0:06
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I’m adding another answer, as community wiki, to expand upon @OnurOktay ’s comment:

Claim: If $T: X^\ast \to Y^\ast$ is weak$^\ast$-to-weak$^\ast$ continuous, where $X$ and $Y$ are Banach spaces, then $T$ is bounded. In fact, there exists bounded $S: Y \to X$ s.t. $T = S^\ast$.

Proof: For each $y \in (Y)_1$, we let $T_y: X^\ast \to \mathbb{C}$ be defined by $T_y(\phi) = [T(\phi)](y)$. We observe that $T_y$ is bounded. Indeed, evaluation at $y$ is a weak$^\ast$ continuous linear functional on $Y^\ast$, so by weak$^\ast$-to-weak$^\ast$ continuity of $T$, $T_y$ is a weak$^\ast$ continuous linear functional on $X^\ast$, so it is given by evaluation at some $x \in X$ and thus in particular bounded.

Consider the collection of maps $\{T_y: y \in (Y)_1\}$. For each fixed $\phi \in X^\ast$, $|T_y(\phi)| = |[T(\phi)](y)| \leq \|T(\phi)\|$ since $\|y\| \leq 1$, i.e., $\sup_{y \in (Y)_1} |T_y(\phi)| < \infty$ for any fixed $\phi$. So by uniform boundedness principle, $\sup_{y \in (Y)_1, \phi \in (X^\ast)_1} |T_y(\phi)| < \infty$, but,

$$\sup_{y \in (Y)_1, \phi \in (X^\ast)_1} |T_y(\phi)| = \sup_{\phi \in (X^\ast)_1} \sup_{y \in (Y)_1} |[T(\phi)](y)| = \sup_{\phi \in (X^\ast)_1} \|T(\phi)\| = \|T\|$$

So $\|T\| < \infty$, i.e., $T$ is bounded.

Finally $S$ is simply defined as the adjoint of $T$ restricted to $Y \subset Y^{\ast\ast}$. As we have seen in the first paragraph of the proof, composing $T$ with evaluation at $y \in Y$ gives evaluation at some $S(y) \in X$. Since $S$ is a restriction of $T^\ast$ and $T$ is bounded, $S$ is bounded as well. That $T$ is the adjoint of $S$ follows easily from the definition. $\square$

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  • $\begingroup$ I was quite close to proving this on my own, so $S$ isn't needed to prove that $T$ is bounded. I think I got it except one tiny thing. I am probably wrong but when you say that $T_y$ has to be an evaluation at some $x \in X$, isn't this the case only if $X$ is reflexive? And in case I haven't said it enough, thank you so much. $\endgroup$ Mar 4 at 0:43
  • $\begingroup$ @TomásPacheco $T$, by assumption, is weak$^\ast$-to-weak$^\ast$ continuous. $T_y$ is $T$ composed with evaluation at $y$, which is weak$^\ast$ continuous, so $T_y$ is weak$^\ast$ continuous as well. The only weak$^\ast$ continuous linear functionals on $X^\ast$ are evaluation at some $x \in X$. This does not require any reflexivity assumptions. (It would require reflexivity if $T_y$ is just bounded, but here I’m saying it’s weak$^\ast$ continuous, which is a stronger condition.) $\endgroup$
    – David Gao
    Mar 4 at 0:48
  • $\begingroup$ Ah, this is very subtle but I get it now, and allow me to say one last thank you so much. $\endgroup$ Mar 4 at 0:52

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