Let $A = \bigoplus\_{i \in \mathbb{Z}} A\_i$ be a strongly graded unital algebra over $\mathbb{C}$, with no zero divisors. Is it always true that 
$$
m: A\_i \otimes\_{A\_0} A\_j \to A\_{i+j}
$$
is an isomorphism?



Strongly graded algebras with no zero divisors
Yes this is always an isomorphism of $A\_0$-bimodules. 


It is a general result for strongly graded rings. It holds for an arbitrary grading group $G$ (not necessarily $\mathbb{Z}$) and does not depend on the presence of zero divisors. For a proof, see Corollary 3.1.2, p.82, from [Methods of Graded Rings](https://www.springer.com/gp/book/9783540207467). 


