Cup product on $A_{\mathrm{\infty }}$-cohomology and deformations

We propose a simple method for constructing formal deformations of differential graded algebras in the category of minimal $A_{\mathrm{\infty }}$-algebras. The basis for our approach is provided by the Gerstenhaber algebra structure on the $A_{\mathrm{\infty }}$-cohomology, which we define in terms of the brace operations. As an example, we construct a minimal $A_{\mathrm{\infty }}$-algebra from the Weyl-Moyal $\ast$-product algebra of polynomial functions.