Let $A$ be a unital $C^{\ast}$-algebra and $\{ f\_i: A \rightarrow A\_i \}\_i$ a finite collection of morphisms of unital $C^{\ast}$-algebras, such that the associated map $A \rightarrow \prod\_i A\_i$ is injective. Let $g: A \rightarrow B$ be a morphism of unital $C^{\ast}$-algebras. Then, there is an induced collection of maps $\{ B \rightarrow A\_i \ast\_A B \}\_i$, where $A\_i \ast\_A B$ is the amalgamated free product (pushout). This collection induces a map $f: B \rightarrow \prod\_i A\_i \ast\_A B$. Is $f$ injective?


$\textbf{Motivation:}$ I am trying to use the extensive Grothendieck topology in the category of unital $C^{\ast}$-algebras.



Property of pushouts in the category of unital $C^{\ast}$-algebras
Unfortunately, the assertion is, in general, not true even if we just have a single (injective) unital $\*$-homomorphism $f\_1\colon A\to A\_1$ and another (non-injective) unital $\*$-homomorphism $g\colon A\to B$. The reason is that amalgamated free products with non-injective maps can be trivial: $B\*\_A A\_1=0$ in certain cases.


For an example, take any simple unital $C^\*$-algebra $A\_1$, for example $A\_1=M\_2(\mathbb{C})$ the $2\times 2$-matrices, and $A$ any unital $C^\*$-subalgebra of $B$ with $\dim(A)>1$ and which has a character $\chi\colon A\to \mathbb{C}$, for example the diagonal matrices $D\cong \mathbb{C}^2$ in $M\_2(\mathbb{C})$.


Now take $B=\mathbb{C}$ with the character $g=\chi\colon A\to B$ above, and take $f\_1$ to be the embedding $A\subseteq A\_1$.


In this situation, the amalgamated free product


$$A\_1\*\_A B$$


is canonically isomorphic to the quotient $A\_1/J$, where $J$ is the ideal of $A\_1$ generated by the differences $\chi(a)1\_{A\_1}-a$ with $a\in A$. Since $\dim(A)>1$, this ideal is not zero, so that $J=A\_1$ because $A\_1$ is simple and therefore


$$A\_1\*\_A B\cong A\_1/J =0.$$


In particular, for the specific example mentioned above:


$$M\_2(\mathbb{C})\*\_{\mathbb{C}^2}\mathbb{C}=0.$$


