Assume $A$, $B$ to be $\mathbb C$-Algebras.
Is there ( under certain conditions) a Künneth theorem or a Künneth spectral sequence relating the Hochschild homology $HH(A\otimes_\mathbb C B) $ with $HH(A)$ and $HH(B)$.
Since JHF kindly answered my question I'd like to ask the follow up one:
If $B$ is a bialgebra and $A$ a $B$-module algebra via $\triangleright:B\otimes A \rightarrow A $ we can equip $B\otimes A$ with the smash product algebra structure given by
$(a\otimes b)(c\otimes d)=a(b_{(1)}\triangleright c)\otimes b_{(2)}d$
This algebra will be denoted $A\#B$. It would be rather interesting to know the relation between $HH (A\#B)$ and $HH(A)$ as well as $HH(B)$.
Since you're working over the field $\mathbb{C}$, the shuffle map induces an isomorphism $$HH_*(A) \otimes HH_*(B) \xrightarrow{\cong} HH_*(A \otimes B).$$ This is theorem 4.2.5 in Loday's Cyclic homology.