Uniqueness of $A_{\mathrm{\infty }}$-structures and Hochschild cohomology

This paper investigates if a differential graded algebra can have more than one $A_{\mathrm{\infty }}$-structure extending the given differential graded algebra structure. We give a sufficient condition for uniqueness of such an $A_{\mathrm{\infty }}$-structure up to quasi-isomorphism using Hochschild cohomology. We then extend this condition to Sagave’s notion of derived $A_{\mathrm{\infty }}$-algebras after introducing a notion of Hochschild cohomology that applies to this.