How to find extreme point in Banach Algebra

Question

Let $X$ is Banach Algebra, $Y$ is subspace of $X$ and $A = \big \{ f \in X^* , \Vert f\Vert\leqslant1,f(xy)=f(yx) , x \in X, y \in Y \big\}$ and $A$ separates the points in $X$. How we can show there is no a point $x_0 \in X$ that isn't zero, with this property: for every $f\in A$, if $f(x_0)$ doesn't equal zero, then there are $g,h$ in $A$ such that $f =(g+h)/2$ .