Let $G$ be a LCH group and $\mu$ be its left Haar measure. Call $\lambda_G : G \to U(L_2(G,\mu))$ the left regular representation. We can define the reduced $C^{\ast}$ algebra and reduced Von Neumann algebra, $C_{\lambda}^{\ast} G $ and $W_{\lambda}^{\ast} G $ respectively, as the smallest $C^{\ast}$, resp. Von Neumann, subalgebras of $B(L_2(G,\mu))$ containing $\lambda_G[C_c(G)]$. It is easy to see that

$$ C_{\lambda}^{\ast} G \subset M(C_{\lambda}^{\ast} G) \subset W_{\lambda}^{\ast} G $$

and that for every $\mu$ in the space of finite Borel measures over $G$, $\lambda(\mu) \in M(C_{\lambda}^{\ast}G )$.

**Question 1** Let $j : H \to G$ be a proper continuous injective homomorphism whose image $j H$ we will denote also by $H$. Let $\nu_{H} \in M_{\text{loc}}(G)$ be the locally finite measure described by its action on the compactly supported continuous functions $C_c(G)$ as:

$$ \nu_H (f) = \int_{H} f |_H (j(h)) d \mu_H (h), $$

where $\mu_H (h)$ is the Haar measure on $H$. Or equivalently $j^{\ast} \mu_H = \nu_H$. For every $\varphi \in C_c(H)$ we can define $\varphi \nu_H$ as $j^{\ast} (\varphi \mu_H)$. Since $\varphi \nu_H \in M(G)$ we have that $\lambda_G (\varphi \nu_H) \in M(C_\lambda^{\ast} G)$. Do the map ${J}$ given by:

$$ J : \lambda_H(\varphi) \mapsto \lambda_G(\varphi \nu_H) $$

extends to a normal $\ast$-homomorphism $J: W_{\lambda}^{\ast} H \to W_{\lambda}^{\ast} G$ ?. Does $J$ extends to a non degenerate $\ast$ homomorphism $J: C_{\lambda}^{\ast} H \to M( C_{\lambda}^{\ast} G )$?

**Question 2** If $\alpha : G \to G$ is a continuous automorphism then, if $\varphi \in C_c(G)$, does the map $\Phi$ defined by:

$$ \Phi : \lambda_G (\varphi) \mapsto \lambda_G (\alpha_{\ast} \varphi), $$

where $\alpha_{\ast} \varphi(t) = \varphi(\alpha(t))$, extends to a normal $\ast$-homomorphism $\Phi: W_{\lambda}^{\ast} G \to W_{\lambda}^{\ast} G$?. Does $\Phi$ extends to a non degenerate $\ast$ homomorphism $\Phi: C_{\lambda}^{\ast} G \to M( C_{\lambda}^{\ast} G )$?

**Question 3** If $q : G \to K$ is a continuous and open surjective group homomorphism and $q^{-1}[\{e\}] = N$ be its kernel. Then given $\phi \in C_c(G)$ we can define $P_N \phi \in C_c(K)$ as:

$$ (P_N f)(\kappa N) = \int_{N} f(\kappa \eta) d \mu_N (\eta) $$

And it is clear (as long as $\Delta_G |_N = \Delta_N$) that $P_N$ is contractive in the $L_1$ norm. Does the map $Q$ given by extension of:

$$ Q : \lambda_G(f) \mapsto \lambda_K(P_N f) $$

extends to a normal $\ast$-homomorphism $Q: W_{\lambda}^{\ast} G \to W_{\lambda}^{\ast} K$, or to a non degenerate $\ast$-homomorphism $Q: C_{\lambda}^{\ast} G \to M( C_{\lambda}^{\ast} K )$ ?.

The first two questions seem to be easy when $\Delta_G |_{j H} = \Delta_H$ and when $\alpha$ is measure preserving, respectively. While the third seems much difficult to me. Indeed, assuming $\Delta_G |_{j H} = \Delta_H$, the $G$-space $X = G/H$ has a $G$ invariant measure $\rho$ such that $(G,\mu_G) = (H \times X,\mu_H \otimes \rho)$, where the equivalence is understood as measurable spaces. That equivalence induces a unitary isometry $u: L_2(G) \to L_2(H) \otimes_2 L_2(X,\rho)$. For every $j(h) \in jH$:

$$ u \lambda_G(j(h)) = \lambda_H (h) \otimes \text{Id}_{L_2(X,\rho)} u. $$

Since $\lambda_G(j(h)) = J(\lambda_H(h))$, we have that $J$ is unitary equivalent to the tensor amplification. A very similar argument works for the second question. Just by constructing a unitary $u: L_2(G) \to L_2(G)$ given by $u(f)(t) = f(\phi(t))$ and see that $u$ intertwines $\Phi$ and the identity. Can this type of arguments be extended to the general case? Or are $\Delta_G |_{j H} = \Delta_H$ and $\alpha_{\ast}\mu_G = \mu_G$ necessary conditions for question 1 and question 2 respectively?.

I am aware that similar questions have been posted on this site, see [1] and [2] but in the context of discrete groups. In that setting the first and second questions have always a positive answer while the third has a positive answer for $C^{\ast}$ algebras if and only if $N$ is amenable and for Von Neumann algebras if and only if $N$ is finite. The converse in the $C^{\ast}$ algebra case seems to be related to the problem of characterizing amenable groups as those groups for which the co-unit $\mathcal{E} : C_\lambda^{\ast} G \to \mathbb{C}$ given by

$$ \mathcal{E}(\lambda_G(f)) = \int_G f d \mu_G $$

is bounded. Are there references for this characterization in the case of non discrete groups?

[1] The functoriality of group C* algebra structure

[2] Is the group von Neumann algebra construction functorial?