I have read a theorem which says that if $H$ and $G$ are both discrete groups, $G\leq H$, then $C^*(G)$ is a subalgebra of $C^*(H)$ and $C^*_r(G)$ is a subalgebra of $C^*_r(H)$.
For now, what I can show is that we can get a map from $\mathbb{C}[G]$ to $\mathbb{C}[H]$ easily by the inclusion, which will be norm decreasing if we give $\mathbb{C}[G]$ and $\mathbb{C}[H]$ the norm induced by $C^*(G)$ and $C^*(H)$. Hence we will get a map from $C^*(G)$ to $C^*(H)$ by considering the Cauchy sequences.
My question is,
1: How to prove that the map is injective?
2: How to prove that the map is also norm decreasing if we give $\mathbb{C}[G]$ and $\mathbb{C}[H]$ the norm induced by $C^*_r(G)$ and $C^*_r(H)$ instead of $C^*(G)$ and $C^*(H)$? I think it is not obvioulsly for the reduced group $C^*$- algebra.
Any help will be truly grateful!
These are non-obvious results.
An answer to both questions can be found in the book "C*-algebras and finite-dimensional approximations" by Brown and Ozawa (see p46 and surrounding pages, in particular proposition 2.5.8).
Question 2 is also treated in detail in the (extremely well-written and accessible!) notes "C*-simplicity of discrete groups" by Dan Ursu (see lemma 6.4).
I hope these references can help you out.