In general there seems no way to naturally define the tensor product of two $A\_\infty$ algebras $A$ and $B$. But, if $(A, m^A\_1,m^A\_2)$ is only a DGA(differential graded algebra) and $(B, m^B\_k, k\ge 1) $ is an $A\_\infty$ algebra, then is there a natural way to get an $A\_\infty$ algebra structure on the tensor product $A\otimes B$?


I guess this should be correct. But for the safety, I was wondering if there is a standard reference for this fact.


Moreover, if this is right, what I really want is an explicit formula of the $A\_\infty$ algebra structure on $A\times B$ in terms of $m^A$ and $m^B$. Thank you!



Tensor product of a DGA and an $A_\infty$ algebra
In fact the tensor product of two $A\_\infty$ algebras can be made into an $A\_\infty$ algebra in an explicit way: there are two constructions, one by Saneblidze-Umble and one by Loday. See the paper <https://arxiv.org/abs/0710.0572>


(For cofibrancy reasons one also knows abstractly that there is such a tensor product, but this of course doesn't give a formula.)


