Let A be a type $I$ factor on a Hilbert space H. Let $\varphi$ be a semi-finite normal weight on $A^{+}$ is it possible to say that then there exist Hilbert spaces $H_2 \subset H_1$ and an isomorphism of $H$ onto $H_2\otimes H_1$ such that A is isomorphic to $ \mathbb{C}_{H_2}\otimes B(H_1)$ ($\mathbb{C}_{H_2}$ denotes the scalars in $H_2$) and $$\varphi(T)=c\text{Tr}(P_{H_2}T) $$ for some scalar $c$.

  • $\begingroup$ It's a bit unclear to me how you are making sense of the formula. Does $P_{H_2}$ denote the orthogonal projection onto $H_2$? If so, what is the domain, and where does $T$ live? $\endgroup$ – Jesse Peterson Apr 25 '13 at 0:25
  • $\begingroup$ I am sorry for not being clear.Yes, $P_{H_2}$ is the projection onto $H_2$ and $T$ lives in $A^{+}$. Since $A$ should be isomorphic to $B(H_1)$ we identify $T$ with an element in $B(H_1)$ and also call it $T$ by abuse of notation. I hope now is clear. $\endgroup$ – Carlos De la Mora Apr 25 '13 at 2:30
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    $\begingroup$ In that case a simple counter example is to take $A = B(H)$. If the dimension of $H$ is at least 2 then the right hand side of the formula cannot be faithful. Just consider $T = 1 - P_{H_2}$. $\endgroup$ – Jesse Peterson Apr 25 '13 at 3:35
  • $\begingroup$ In order to be a counter example we will need to say why is $1-P_{H_2}\neq 0$. So in other words I to be a counter example you need to show that there are some semi-finite faithful normal weights that are not constant multiples of a trace. However your comment does make me realize that I probably can drop the assumption of $\varphi$ being faithful. $\endgroup$ – Carlos De la Mora Apr 27 '13 at 18:03
  • $\begingroup$ Perhaps I am still confused about your question. When you ask for $H$ to be isomorphic to $H_2 \otimes H_2$ such that $A$ is isomorphic to $\mathbb C_{H_2} \otimes B(H_1)$, am I correct that you mean for the second isomorphism to be implemented by the first? $\endgroup$ – Jesse Peterson Apr 28 '13 at 3:52

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