Let $G$ be a real Lie group and $A(G)$ be its [Fourier algebra](http://www.encyclopediaofmath.org/index.php?title=Fourier-algebra(2)). 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 $G$?



Tangent space of the Fourier algebra $A(G)$
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](http://www.ams.org/journals/proc/1988-104-02/S0002-9939-1988-0931730-5/home.html).


