In general there seems no way to naturally define the tensor product of two $A_\infty$ algebras $A$ and $B$. But, if $A$ is only a DGA(differential graded algebra) and $B$ 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. Thanks!

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!