Timeline for Projections in the tensor product of von Neumann algebras
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Nov 19, 2023 at 17:30 | comment | added | Masayoshi Kaneda | @Adrián González Pérez: It was about your old question, but I answered in my comments above. Or, maybe you have already figured out. Thank you for asking, though. | |
| Nov 19, 2023 at 3:32 | comment | added | Masayoshi Kaneda | After all, any projection in $M\bar{\otimes}N\setminus M\otimes N$ cannot be the supremum of projections of the form $p\otimes q$ with $p\in M$ and $q\in N$. This gives an alternate and more complete answer to my original question. | |
| Nov 19, 2023 at 3:27 | comment | added | Masayoshi Kaneda | Indeed, if $r$ is a projection in $M\bar{\otimes}N$ such that $p_\alpha\otimes q_\alpha\le r$ for all $\alpha$, then $p_\alpha\mathcal{H}\odot q_\alpha\mathcal{K}\subseteq(p_\alpha\otimes q_\alpha)(\mathcal{H}\otimes\mathcal{K})\subseteq r(\mathcal{H}\otimes\mathcal{K})$ for all $\alpha$. Thus $p\mathcal{H}\odot q\mathcal{K}\subseteq(p\otimes q)(\mathcal{H}\otimes\mathcal{K})\subseteq r(\mathcal{H}\otimes\mathcal{K})$, and hence $p\otimes q\le r$. | |
| Nov 19, 2023 at 3:26 | comment | added | Masayoshi Kaneda | @user92646: Thank you for your interest and your question. I think the answer is no, regardless of diffuseness. Suppose that $\{p_\alpha\otimes q_\alpha\}$ is a set of projections in $M\otimes N$ such that $p_\alpha\in M$ and $q_\alpha\in N$. Let $p$ (resp., $q$) be the supremum of $\{p_\alpha\}$ (resp., $\{q_\alpha\}$) in $M$ (resp., in $N$). Then it is easy to see that $p\otimes q$, which is in $M\otimes N$, serves as the supremum of $\{p_\alpha\otimes q_\alpha\}$ in $M\bar{\otimes}N$. | |
| Nov 9, 2023 at 0:20 | comment | added | user92646 | @MasayoshiKaneda Hi Masayoshi, do you have an answer for diffuse algebras? I am also interested in this question | |
| Jan 25, 2019 at 13:32 | vote | accept | Masayoshi Kaneda | ||
| Jan 23, 2019 at 19:02 | comment | added | Masayoshi Kaneda | @Adrián González-Pérez: Thank you for a good question. I will think about it. | |
| Jan 23, 2019 at 13:00 | comment | added | Adrián González Pérez | Nice. It would be interesting to see if the statement is true in the case of diffuse algebras. | |
| Jan 23, 2019 at 10:04 | history | answered | Masayoshi Kaneda | CC BY-SA 4.0 |