Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by finite-finite group.
Denote the (Banach) space (in fact an algebra under convolution), $l^1(G)=\{\sum_{g\in G}\lambda_g g| \sum_{g\in G}|\lambda_g|<\infty, \lambda_g\in \mathbb{C}\}$.$$\ell^1(G)=\left\{\sum_{g\in G}\lambda_g g\;\Big|\; \sum_{g\in G}|\lambda_g|<\infty, \,\lambda_g\in \mathbb{C}\right\}.$$
Note that $\mathbb{Z}G\subset l^1(G)$$\mathbb{Z}G\subset \ell^1(G)$ and we can consider $l^1(G)$$\ell^1(G)$ as a right $\mathbb{Z}G$-module, my. My question is:
$\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?
$\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $\ell^1(G)$ is a flat $\mathbb{Z}G$-(right) module?
If it is true, any reference (maybe for related topics)? Otherwise, any counterexample? Especially, is it true for the Heisenberg group?
RK: I have asked a general one in MSE but no answer appeared, so I asked here a more specific one.
