There is definition of "$A_\infty$ Centre" in article The A_\infty-Centre of the Yoneda Algebra and the Characteristic Action of Hochschild Cohomology on the Derived Category at p.28. It can be defined at two equivalent ways.
$Z_\infty(A) := \operatorname{Im}(\Pi : HH^*(A,A) \to H^*(A))$, where morphism $\Pi$ is just factorprojection $HH^*(A,A) \to H^*(A,A)/F^1 HH^*(A,A)$ where $F^i HH^*(A,A)$ is just standart tensor filtration on Hochschild cohomology.
and second definition
$$a \in Z_\infty(A) \leftrightarrow [a;-]_{1,n} = \sum_{r+s+t = n} m_{r+1+s} (1^{\otimes s} \otimes p_s \otimes 1^{\otimes t}) - (-1)^{|a|} (-1)^{rs+t} p_{r+1+t} (1^{\otimes r} \otimes m_s \otimes 1^{\otimes t})$$ where $p_i : A^i \to A$ some maps of degree $|a|-i$ and $$[a,x_1 \otimes ... \otimes x_n]_{1,n}= m_{n+1}(a \otimes x_1 \otimes ... \otimes x_n + (-1)^{|a| |x_1|} x_1 \otimes a \otimes x_2 \otimes ... \otimes x_n + ... + (-1)^{|a|(|x_1| + ... + |x_n|)} x_1 \otimes ... \otimes x_n \otimes a)$$
the intuition is that $[-,-]_{p,q}$ is analogue of $A_\infty$-commutator and condition $[a,-]_{1,q} = m \circ h + h \circ m$ is "commutator is zero up to homotopy". But what if we require it to be just zero (which is much more easy for computation) and define $$a \in Z_{\infty,strict} (A) \leftrightarrow [a,-]_{1,n} = 0$$ What we can say about such set? Does it invariant under quasiisomorphisms, for example?
