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 $G$?