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Let $k$ be a field of characteristic $0$ and $(A,d_A)$, $(B,d_B)$ be two differential graded (dg) algebras over $k$. Let $f: A\to B$ be a closed degree $0$ map of dg-algebras and $g: B\to A$ be a map of dg-vectors spaces such that $gf-id_A\sim 0$ and $fg-id_B\sim 0$.

My question is: could we extend $g$ to an $A_{\infty}$-map $B\to A$? Is it a consequence of the Homotopy Transfer Theorem?

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Yes, see this paper of Martin Markl. Since $B$ is homotopy equivalent to an $A_\infty$-algebra, you can endow it with an $A_\infty$-structure and extend your maps to $A_\infty$-morphisms that yield a homotopy equivalence of $A_\infty$-algebras. The fact that $f$ is already a map of algebras and $A$ and $B$ are dg-algebras will simplify the formulas of the paper significantly.

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