Timeline for Relation between tracial norm and operator norm on a von Neumann algebra
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Feb 7, 2018 at 8:49 | comment | added | H1ghfiv3 | This comment answers basically all of my questions (apart from whether $3 \implies 2$). I think you should post this as an answer. | |
| Feb 6, 2018 at 21:51 | comment | added | Jesse Peterson | The question of whether every II$_1$ factor has an orthonormal basis consisting of unitaries is a well known open problem which was first asked by Kadison. See: On “Problems on von Neumann Algebras by R. Kadison, 1967”, by Ge. | |
| Jan 27, 2018 at 12:48 | history | edited | H1ghfiv3 | CC BY-SA 3.0 |
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| Jan 27, 2018 at 12:47 | comment | added | H1ghfiv3 | If I understand you correctly, your consider 2 x 2 matrices with complex entries. Perhaps I should have made this clearer, but in requiring $l^2(\mathcal A)$ to be infinite-dimensional (as a $\mathbb C$-vector space), I am also avoiding these "silly" examples. | |
| Jan 26, 2018 at 22:43 | comment | added | Caleb Eckhardt | $2\times 2$ matrices are an example of a finite von Neumann algebra that satisfies (1) but is not a group vNa. If you restrict to $II_1$-factors you probably eliminate these silly examples (but I'm no expert) | |
| Jan 26, 2018 at 16:48 | answer | added | Adrián González Pérez | timeline score: 1 | |
| Jan 26, 2018 at 9:15 | comment | added | Mateusz Wasilewski | Well, that's one of the reasons that I put this remark in a comment rather than in an answer; it's not that easy to prove that something is not a group von Neumann algebra. One property of group von Neumann algebras is that they are isomorphic to their opposite algebras (with reversed multiplication), which one can achieve using the inverse map on the group. But examples that do not have this property are, as far as I can tell, hard to obtain. I am sure that experts in the field know way more about von Neumann algebras that are not group algebras. | |
| Jan 26, 2018 at 0:15 | history | edited | Martin Sleziak | CC BY-SA 3.0 |
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| Jan 25, 2018 at 22:13 | comment | added | H1ghfiv3 | Thanks for your detailed answer and various hints. As a beginner in this field, I wonder how to show that a given finite von Neumann algebra is not (isomorphic to) a group von Neumann algebra ... | |
| Jan 25, 2018 at 21:53 | comment | added | Mateusz Wasilewski | Actually, any non-atomic measure space $(X,\mu)$ is isomorphic to the unit circle endowed with the Lebesgue measure, so we can transfer the trigonometric basis in this case. It means that any measure-preserving action on a non-atomic measure space gives rise to a von Neumann algebra that satisfies (1); many of these are not group von Neumann algebras. | |
| Jan 25, 2018 at 18:42 | comment | added | Mateusz Wasilewski | An idea for possible examples of von Neumann algebras that satisfy (1): consider measure preserving actions on measure spaces and form the crossed product algebra. This is a finite von Neumann algebra and the associated Hilbert space is of the form $L^{2}(X,\mu)\otimes \ell^2(G)$, so if $L^{2}(X,\mu)$ admits an ONB formed by function with absolute value $1$, then this von Neumann algebra satisfies (1). There should be examples, when these algebras are not group von Neumann algebras. | |
| Jan 25, 2018 at 15:06 | answer | added | Mateusz Wasilewski | timeline score: 4 | |
| Jan 25, 2018 at 14:16 | history | asked | H1ghfiv3 | CC BY-SA 3.0 |