Tensor product of filtered $A_{\mathrm{\infty }}$-algebras

We define the tensor product of filtered $A_{\mathrm{\infty }}$-algebras. establish some of its properties and give a partial description of the space of bounding cochains in the tensor product. Furthermore we show that in the case of classical $A_{\mathrm{\infty }}$-algebras our definition recovers the one given by Markl and Shnider. We also give a criterion that implies that a given $A_{\mathrm{\infty }}$-algebra is quasi-isomorphic to the tensor product of two subalgebras. This will be used in a sequel to prove a K"unneth Theorem for the Fukaya algebra of a product of Lagrangian submanifolds.