Let $ A$ and $ B$ be $ A_\infty$ -algebras. It’s true, but it’s a quite nontrivial fact, that the tensor product $ A \otimes B$ can be given the structure of $ A_\infty$ -algebra, too. What is much easier to prove is that the tensor product of an $ A_\infty$ -algebra with a dg-algebra is again an $ A_\infty$ -algebra.
One can ask similarly whether the tensor product of a $ C_\infty$ -algebra and an $ L_\infty$ -algebra is an $ L_\infty$ -algebra, generalizing the simple fact that the tensor product of a commutative algebra and a Lie algebra is again a Lie algebra. But in fact I can’t even prove what should be a much simpler statement: that the tensor product of a $ C_\infty$ -algebra with a Lie algebra is an $ L_\infty$ -algebra. Is this true?
Remark. What is quite clear is that the tensor product of a commutative algebra and an $ L_\infty$ -algebra is an $ L_\infty$ -algebra.
