Let's say that a (countable, discrete) group $G$ has property $\Gamma$ iff $L(G)$ has property $\Gamma$, i.e. if for each ${x_{1},\dots,x_{n}}\subseteq L(G)$ and $\varepsilon>0$ there's a unitary $u\in L(G)$ such that $\tau(u)=0$ and $||ux_{j}-x_{j}u||{2}<\varepsilon$ for all $x{1},\dots,x_{n}$. This is a purely operator algebraic property; is there a group theoretic way to talk about this?
#Group Theory and Property Gamma
storm rampart
manic juncoBOT
Let's say that a (countable, discrete) group $G$ has property $\Gamma$ iff $L(G)...
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undone quailBOT
Pear Category Theorem
storm rampart
My first instinct is perhaps some generalization of amenability, seeing as the hyperfinite II_1 factor has property Gamma, but I can't seem to make much progress here.
tulip arrow
Why are there distance metrics on group objects now ðŸ˜ðŸ˜ðŸ˜
storm rampart
Why are there distance metrics on group objects now ðŸ˜ðŸ˜ðŸ˜
(a) That's not what I'm asking about; this metric is on the von neumann algebra L(G), not the group itself. It's a somewhat involved construction, but suffice to say that it's very natural.
(b) Distance metrics come up in group theory literally all the time. That's what ggt is about.
tulip arrow
(a) That's not what I'm asking about; this metric is on the von neumann algebra ...
Dude what’s a Von Neumann Algebra ðŸ˜ðŸ˜ðŸ˜
And I haven’t done enough math to see distance metrics explicitly used on group objects ðŸ˜ðŸ˜ðŸ˜
storm rampart
Dude what’s a Von Neumann Algebra ðŸ˜ðŸ˜ðŸ˜
A banach algebra that's closed under a funny topology.
tulip arrow
A banach algebra that's closed under a funny topology.
Banach space??
And it’s closed under a weird topology
tulip arrow
And a star.
What
storm rampart
Like a complex conjugate.
tulip arrow
Dude I struggle in Baby Rudin ðŸ˜ðŸ˜ðŸ˜
I would like to understand but the amount of prerequisites itself would take me like at minimum a year
Magma, ZFN or Apex could help @storm rampart
Magma should be able to
gloomy escarp
storm rampart
https://arxiv.org/pdf/1308.0942v4
Maybe?
Okay, this is very nontrivial, and there isn't a straightforward answer. Thanks.
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drifting kettleBOT
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