Timeline for Weights on Von Neuman factors
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Apr 29, 2013 at 17:19 | comment | added | Carlos De la Mora | Yes, you are correct. Thank you for your help. I need to think more on what is exactly what I want to ask. | |
| Apr 28, 2013 at 4:00 | comment | added | Jesse Peterson | In either case, if $H = \mathbb C^2$, and $A = \mathbb M_2(\mathbb C)$. Then requiring $A$ to be isomorphic to $\mathbb C_{H_2} \otimes B(H_1)$ means that $H_1$ is two dimensional. Since $H$ is isomorphic to $H_2 \otimes H_1$ you then know that $H_2$ is one dimensional. Thus $P_{H_2} \not= 1$. | |
| Apr 28, 2013 at 3:52 | comment | added | Jesse Peterson | 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? | |
| Apr 27, 2013 at 18:04 | history | edited | Carlos De la Mora | CC BY-SA 3.0 |
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| Apr 27, 2013 at 18:03 | comment | added | Carlos De la Mora | 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. | |
| Apr 25, 2013 at 11:52 | history | edited | Chris Heunen |
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| Apr 25, 2013 at 11:52 | history | edited | Chris Heunen |
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| Apr 25, 2013 at 3:35 | comment | added | Jesse Peterson | 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 $T = 1 - P_{H_2}$. | |
| Apr 25, 2013 at 2:30 | comment | added | Carlos De la Mora | 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. | |
| Apr 25, 2013 at 0:25 | comment | added | Jesse Peterson | 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? | |
| Apr 24, 2013 at 23:42 | history | asked | Carlos De la Mora | CC BY-SA 3.0 |