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Let $G$ be a real Lie group and $A(G)$ be its Fourier algebra. Let us call a linear continuous functional $f:A(G)\to{\mathbb C}$ a tangent vector of $A(G)$ in the point $a\in G$, if it satisfies the Leibniz identity $$ f(u\cdot v)=u(a)\cdot f(v)+f(u)\cdot v(a),\qquad u,v\in A(G) $$ and preserves involution $$ f(\overline{u})=\overline{f(u)},\qquad u\in A(G). $$

Is anything known about the tangent space $T_a(A(G))$ (i.e. the set of tangent vectors) of $A(G)$?

Is it possible, that $T_a(A(G))$ coincides with the usual tangent space $T_a(G)$ of the Lie group $G$?

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Without the condition on involutions, this is the space of continuous point derivations on A(G) and it always vanishes. This seems to have first been observed by Brian Forrest although the necessary ideas were probably known earlier. See Proposition 1 here.

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  • $\begingroup$ Yemon, how can this be? There are non-smooth functions in $A(G)$? $\endgroup$ Sep 27, 2014 at 12:30
  • $\begingroup$ I thought, when $G$ is not discrete there must be at least usual derivatives $\frac{\partial}{\partial x_i}$... They are not continuous here? $\endgroup$ Sep 27, 2014 at 12:40
  • $\begingroup$ Many of them, Sergei. When G=T or G=R one can write down explicit examples (I will try to provide these later) and then for any G which contains a closed copy of T or R, one can extend the previous examples to elements of A(G) by Herz's restriction theorem $\endgroup$
    – Yemon Choi
    Sep 27, 2014 at 12:44
  • $\begingroup$ @SergeiAkbarov Some of the background can be found in fields.utoronto.ca/programs/scientific/13-14/harmonicanalysis/… CAVEAT: the theorem on page 2 is not quite stated correctly, but will work provided A is unital. The theorem of Brian Forrest is true for all groups, not just the compact ones. $\endgroup$
    – Yemon Choi
    Sep 27, 2014 at 12:47
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    $\begingroup$ Further to my previous comment: for $G={\bf R}$ one way to get functions that aren't everywhere differentiable is to consider $u_\delta$, which takes the value $\delta^{-1/2}$ on the interval $[-\delta/2, \delta/2]$ and is $0$ everywhere else, and then consider $f=u_\delta*u_\delta \in A({\bf R})$ $\endgroup$
    – Yemon Choi
    Sep 28, 2014 at 1:27

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