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Let $F$ be a field and let $F \langle X \rangle$ be the free unitalassociative $F$-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 polynomialsform a $T$-subalgebra $C(G)$ in $F \langle X \rangle$. Over a field ofcharacteristic $p > 2$ there are algebras $G$ whose algebras of all centralpolynomials $C (G)$ are not finitely generated as $T$-subspaces in $F \langle X\rangle$. However, no example of an algebra $G$ such that $C(G)$ is notfinitely generated as a $T$-subalgebra is known yet.  In the present paper we construct the first example of a $2$-graded unitalassociative 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 finitelygenerated as a $T$-subalgebra in $F \langle X \rangle$.

The subalgebra of graded central polynomials of an associative algebra