Is the Hochschild chain complex $C_*(A, A)$ a $B_\infty$-module over the Hochschild cochain complex $C^*(A, A)$?

Question

Let $A$ be a dg-algebra, or more generally an $A_\infty$-algebra. Then it is well known that the Hochschild cochain complex $C^*(A, A)$ computing Hochschild cohomology is a $B_\infty$-algebra, see for example, the paper of Bernhard Keller "Derived invariance of higher structures on the Hochschild complex" available on his pageweb.

I would like to know whether the Hochschild chain complex $C_*(A, A)$ (which computes Hochschild homology) is a $B_\infty$-module over the Hochschild cochain complex $C^*(A, A)$?

Can someone give me the precise defintion, or a precise reference, of the action of $C^*(A,A)$

on $C_*(A, A)$ if the answer is Yes?

Answer 1

This is the subject of Section 2 in that paper (sorry for self-promotion). Chains actually have two $B_\infty$-module structures (over cochains). Those two module structures are moreover compatible (see Theorem 2.4 of the above paper for a precise statement).