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)?
Best Answer
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.