  Let $F$ be a field and let $F \langle X \rangle$ be the free unital
associative $F$-algebra on the free generating set $X = \{ x_1, x_2, \dots \}$.
A subalgebra (a vector subspace) $V$ in $F \langle X \rangle$ is called a
$T$-subalgebra (a $T$-subspace) if $\phi (V) \subseteq V$ for all endomorphisms
$\phi$ of $F \langle X \rangle$. For an algebra $G$, its central polynomials
form a $T$-subalgebra $C(G)$ in $F \langle X \rangle$. Over a field of
characteristic $p > 2$ there are algebras $G$ whose algebras of all central
polynomials $C (G)$ are not finitely generated as $T$-subspaces in $F \langle X
\rangle$. However, no example of an algebra $G$ such that $C(G)$ is not
finitely generated as a $T$-subalgebra is known yet.
  In the present paper we construct the first example of a $2$-graded unital
associative algebra $B$ over a field of characteristic $p>2$ whose algebra $C_2
(B)$ of all $2$-graded central polynomials is not finitely generated as a
$T_2$-subalgebra in the free $2$-graded unital associative $F$-algebra $F
\langle Y,Z \rangle$. Here $Y = \{ y_1, y_2, \dots \}$ and $Z = \{ z_1, z_2,
\dots \}$ are sets of even and odd free generators of $F \langle Y,Z \rangle$,
respectively. We hope that our example will help to construct an algebra $G$
whose algebra $C(G)$ of (ordinary) central polynomials is not finitely
generated as a $T$-subalgebra in $F \langle X \rangle$.
