I'm reading about Cuntz-Pimsner algebras at the moment, and something simple albeit annoying has been bothering me.
Given a $C^{*}$-Correspondence $(\sf{X},\mathfrak{A})$, Pimsner defines an 'augmented Cuntz-Pimsner algebra' as 'the $C^{*}$-algebra generated by $\mathcal{O}_{\sf{X}}$ and $\mathfrak{A}$'.
Preseumably he means the augmented algebra is $C^{*}(\mathcal{O}_{\sf{X}},\mathfrak{A})$, but where would this be taking place? You need some ambient $C^{*}$-algebra $\mathfrak{B}$ where $\mathcal{O}_{\sf{X}},\mathfrak{A} \subset \mathfrak{B}$ so that $C^{*}(\mathcal{O}_{\sf{X}},\mathfrak{A})$ makes any sense, right?
This seems to be a type of language researchers use alot, but because of my above quarrel of needing some ambient $C^{*}$-algebra floating around to make sense of $C^{*}(\text{stuff})$, it makes reading papers confusing at times about what they really mean.
Pimsner also mentions that if the action $\phi: \mathfrak{A} \longrightarrow \mathcal{L}(\sf{X})$ is injective, then $\overset{\sim}{\mathcal{O}_{\sf{X}}} \cong \mathcal{O}_{\sf{X}}$. Could someone also explain why this is true?
If you're following Pimsner's paper then he defines $\mathcal{O}_E$ as a subalgebra of $M(\mathcal{E}_+)/J(\mathcal{E}_+)$. Since $A \subseteq \mathcal{E}_+ \subseteq M(\mathcal{E}_+)$ it makes sense to talk about $\widetilde{\mathcal{O}}_E$ as the algebra generated by $\mathcal{O}_E$ and $A$ in $M(\mathcal{E}_+)/J(\mathcal{E}_+)$. These days it's much more common to build $\mathcal{O}_E$ as a quotient like in Theorem 3.13 of Pimsner's paper, and then show it satisfies the appropriate universal property (aka Theorem 3.12 of Pimsner).
As for your second question, if $a \in \ker(\phi)$ then $\phi(a) = 0$ is compact. So condition (4) of Theorem 3.12 becomes $\sigma(a) = \sigma^{(1)} \circ \phi(a) = 0$. In other words, $\sigma \colon A \to \mathcal{O}_E$ will not be injective. If we manually insert $A$ as in $\widetilde{\mathcal{O}}_E$ then there's space to represent $A$ in the algebra. So whether you use the augmented algebra or not really depends on whether you think a representation of a correspondence in a $C^*$-algebra should include the coefficient algebra $A$ or not.
If you want to deal with general $\phi \colon A \to \mathcal{L}(E)$ then I'd suggest looking at the approach found in "On $C^*$-algebras associated with $C^*$-correspondences" by Katsura. This generalises Pimsner's approach and these days is usually deemed the "correct" way of doing things since $A$ always includes in $\mathcal{O}_E$.