Timeline for Restriction on the coefficients for an operator in the free group factor $ L(\mathbb{F}_2) $
Current License: CC BY-SA 3.0
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| Jan 3, 2014 at 3:41 | vote | accept | Jiang | ||
| Jun 29, 2011 at 20:06 | comment | added | Matthew Daws | @Yemon: Nice Question! In the general case, I don't know. But if $G$ is amenable, then as the trivial rep is contained in the left regular rep, if $f\in C^*(G)$ (with abuse of notation, $f$ is the "Fourier coefficient") is pointwise positive, then $\|f\|_{C^*} = \|f\|_1$. As $\ell^1(G) \not= C^*(G)$ unless $G$ is finite, it cannot be that $f\in C^*(G)$ implies that the absolute value of $f$ is also in $C^*(G)$. | |
| Jun 29, 2011 at 18:59 | comment | added | Yemon Choi |
Matt, do you know of any example of something in the reduced $C^*$ algebra where replacing each "Fourier coefficient" with its absolute value results in an unbounded operator? (Classical Fourier analysis gives examples of this for the group ${\mathbb Z}$, i.e. you can't always tell if a function on the circle is continuous just by knowing the modulus of each Fourier coefficient.)
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| Jun 29, 2011 at 17:11 | history | edited | Matthew Daws | CC BY-SA 3.0 |
Better reference.
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| Jun 29, 2011 at 16:34 | history | answered | Matthew Daws | CC BY-SA 3.0 |