Does tensor product of C* and Banach *-algebras preserve injectivity of $*$-homomorphism? Let $A$ and $B$ be $C^*$-algebras. Let $A_0\subseteq A$ be a $C^*$-subalgebra and $i\colon A_0\to A$ the inclusion.
Then we have an injective $*$-homomorphism of $*$-algebras,
$$i\odot\text{id}\colon A_0\odot B\to A\odot B,$$
where $\odot$ means the algebraic tensor product.
So one can conclude, by the definition of the maximal tensor product $\otimes_{\text{max}}$, that $i\odot\text{id}$ extends to a $*$-homomorphism $i\otimes_{\text{max}}\text{id}\colon A_0\otimes_{\text{max}}B\to A\otimes_{\text{max}}B$.
Question 1: Is $i\otimes_{\text{max}}\text{id}$ injective?
I would also like to know whether this is still true if $\otimes_{\text{max}}$ is replaced by any other $C^*$-tensor product. That is:
Question 2: Does $i\odot\text{id}$ induce an injective $*$-homomorphism
$$i\otimes\text{id}\colon A_0\otimes B\to A\otimes B,$$
where $\otimes$ is any $C^*$-tensor product?
Question 3: Suppose now that $A_0$, $A$, and $B$ are Banach $*$-algebras. Then to what extent does the conclusion to Q2 hold true (e.g. is it true for the projective tensor product)?
 A: Unfortunately, the answer to question (1) is false.
Consider a discrete group $\Gamma$ and consider the reduced group $C^*$-algebra $C_r^*(\Gamma) \subseteq B(\ell^2(\Gamma)).$ If $\iota: C_r^*(\Gamma) \hookrightarrow B(\ell^2(\Gamma))$ is the inclusion map, then the map
$$\iota \otimes_{\max} \operatorname{id}_{C_r^*(\Gamma)}: C_r^*(\Gamma) \otimes_{\max}C_r^*(\Gamma) \to B(\ell^2(\Gamma)) \otimes_{\max} C_r^*(\Gamma) $$
is injective if and only if $\Gamma$ is amenable. A proof of this fact can be found in the book "C*-algebras and finite-dimensional approximations" by Brown and Ozawa (see proposition 3.6.9, p89). Hence, to get a concrete counterexample to your question (1), simply consider a non-amenable group $\Gamma$, such as $\Gamma = \mathbb{F}_2$ (the free group on two generators).
About question (2): By question (1) this is false, but it is true for the minimal $C^*$-norm.

It is worth pointing out that in some cases, given an inclusion $A \subseteq B$ of $C^*$-algebras, that the natural map $A \otimes_\max C \to B \otimes_\max C$ is injective. For example, when $A$ is a closed ideal in $B$ (or more generally, when $A$ a hereditary $C^*$-subalgebra of $B$) or when $A$ is a nuclear $C^*$-algebra. See corollary 3.6.3 and 3.6.4 in the aforementioned book. A characterisation of when this inclusion is injective is given in proposition 3.6.6.
