Let $G$ be a Lie group and $C_r^*(G)$ and $C^*(G)$ be its reduced and maximal group $C^*$-algebras respectively. The left-regular representation of a group $G$ induces a surjective map

$$\lambda_G:C^*(G)\rightarrow C^*_r(G),$$

so $C_r^*(G)$ is a quotient of $C^*(G)$. I am wondering whether $C_r^*(G)$ is in fact a direct summand of $C^*(G)$?

Thanks for your help.

**Edit**: As YCor commented, more precisely what I am looking for is a $*$-homomorphism $f:C_r^*(G)\rightarrow C^*(G)$ such that $\lambda_G\circ f=\text{id}_{C_r^*(G)}$ and $f(x)y=0$ for all $x\in C_r^*(G)$ and $y\in\ker\lambda_G$. That is, a direct summand in the sense of $C^*$-algebra direct sums.

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