Let $A = \bigoplus_{k \in \mathbb{Z}} A_k$ be a not necessarily commutative $\mathbb{Z}$-graded unital algebra over a field $\mathbb{K}$, and assume that it is strongly graded:
$$
A_kA_l = A_{k+l}.
$$
In general can it happen that the multiplication does **not** give an isomorphism
$$
A_k \otimes_{A_0} A_l \simeq A_{k+l}?
$$
The map will be surjective since we are assuming the gradind to be strong, but will it be injective?