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Take an unital differential graded (dg) $k$-algebra $A$, we can regard it as $A_\infty$-algebra with $m_1$ as differential and $m_2$ as algebra multiplication, and $m_n=0$ or all $n\geq 0$. Take a dg $A$-module $M$, then we can form the $A_\infty$-algebra $B=End_A(M)$.

We say $B$ is formal if the homology $H^*(B)$ is quasi-isomorphic to $B$. Apparently, formality of $B$ implies derived (dg) equivalence, i.e. equivalence between the derived dg categories $D(A)$ and $D(B)$, and hence with $D(H^*(B))$. Is this true? Where can I find an exact reference which states anything like this? I have looked through Professor Keller's note "Introduction to $A_\infty$-algebra and modules", but doesn't seems to see anything like this.

Moreover, is the converse statement true? i.e. if $B$ is not formal, then there is no derived equivalence between $D(A)$ and $D(H^*(B))$.?

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    Interesting question, Aaron! Without having checked it out myself, I think that you may be able to find such a statement (if it's true) in Loday and Vallete's "Algebraic operads". An online copy is here: http://math.unice.fr/~brunov/Operads.pdf2012-05-15
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    Dag Madsen's PhD thesis "Homological aspects in representation theory" seems related.2012-11-09

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