Timeline for Monoidal structures on von Neumann algebras
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Aug 17, 2010 at 6:32 | comment | added | André Henriques | Nice! I like your idea to use the Thomson group F. Do you know a coordinate independent way of defining your Thomson-group-direct-sum operation? | |
| Aug 16, 2010 at 5:19 | comment | added | Andreas Thom | What if you consider the group measure space construction of the action (of some element in) Thompsons group $F$ on $[0,1]$? There, you have an explicit element of $F$ which maps $[0,1]$ to itself (piecewise linear) such that $[0,1/2],[1/2,3/4],[3/4,1]$ is mapped to $[0,1/4],[1/4,1/2],[1/2,1]$. I think that this algebra (at least the crossed product by this specific automorphism) is a factor of type $III_{1/2}$. This should satisfy the pentagon axiom. In order to make everything (also the tensor product) work at the same time, more work is necessary. | |
| Aug 15, 2010 at 22:16 | comment | added | André Henriques | Picking an inner automorphism of R that maps the triple p,psi(p),psi(1−p) to phi(p),phi(1−p),1−p is always possible. But picking one that satisfies the pentagon axiom is much more tricky... But maybe it's possible. | |
| Aug 15, 2010 at 10:52 | comment | added | Andreas Thom | I think that there is still some misunderstanding. The $\sum$-operation applied to two automorphisms would not be an automorphisms. (This is analogous to the fact that the sum of two line-bundles is not a line-bundle.) However, as far as I understand, your construction on the level of algebras would always yield that the sum of two automorphisms is an automorphism. | |
| Aug 15, 2010 at 10:33 | history | edited | Andreas Thom | CC BY-SA 2.5 |
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| Aug 15, 2010 at 10:10 | comment | added | Andreas Thom | My reply was to long to for a comment. I edited the original answer instead. | |
| Aug 15, 2010 at 10:08 | history | edited | Andreas Thom | CC BY-SA 2.5 |
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| Aug 15, 2010 at 7:05 | comment | added | André Henriques | Maybe there is a way of picking \iota so that the operation [+] is coherently associative... Is there one? Can one then also arrange it so that [+] distributes over (x) ? It would be great if this worked... | |
| Aug 15, 2010 at 0:15 | comment | added | André Henriques | I'll rephrase your construction in more invariant terms. Given M and N, you construct a new factor M[+]N as follows. First, you pick a Morita equivalence between M and N (given isos M=R and N=R, that's what you get from \iota). That Morita eq. can be interpreted as a choice of 2x2 matrix algebra, with upper-left and lower-right corners given by M and N respectively. Then define M[+]N to be that 2x2 matrix algebra. It comes with two nonunital inclusions M --> M[+]N and N --> M[+]N. I would be surprised if this construction was (coherently) associative (M [+] N) [+] P =?= M [+] (N [+] P). Is it? | |
| Aug 14, 2010 at 16:07 | history | answered | Andreas Thom | CC BY-SA 2.5 |