be an ideal in $\phi_X^{-1}({{\mathcal{K}}}(X))$. A representation $(\pi,t)$ is called *$J$-covariant* if $$\psi_t(\phi_X(a)) = \pi(a) {\text{ for all }}a\in J.$$ The representations $(\pi,t)$ that are $J_{X}$-covariant for *Katsura’s ideal* $$J_X := \ker\phi_X^\bot \cap \phi_X^{-1}({{\mathcal{K}}}(X)),$$ are called *covariant representations*. The ideal $J_X$ is the largest ideal on which the restriction of $\phi_X$ is injective. An example of a covariant representation is given by taking the quotient map with respect to ${{\mathcal{K}}}({{\mathcal{F}}}(X) J_X)$ on the Fock representation [@Kat04].

The *Toeplitz-Pimsner algebra* ${{\mathcal{T}}}_X$ is the universal C\*-algebra with respect to the Toeplitz representations of ${}_A X_A$. The *$J$-relative Cuntz-Pimsner algebra* ${{\mathcal{O}}}(J,X)$ is the universal C\*-algebra with respect to the $J$-covariant representations of ${}_A X_A$. The *Cuntz-Pimsner algebra* ${{\mathcal{O}}}_X$ is the universal C\*-algebra with respect to the covariant representations of ${}_A X_A$. The *tensor algebra* ${{\mathcal{T}}}_X^+$ in the sense of Muhly-Solel [@MuhSol98] is the algebra generated by the copies of $A$ and $X$ inside ${{\mathcal{T}}}_X$.

Due to the Fock representation, the copies of $A$ and $X$ inside ${{\mathcal{T}}}_X$ are isometric. In addition ${{\mathcal{T}}}_X^+$ is embedded completely isometrically in ${{\mathcal{O}}}_X$, and ${\mathrm{C}^*_{\textup{env}}}({{\mathcal{T}}}_X^+) \simeq {{\mathcal{O}}}_X$. This was accomplished under certain conditions by Fowler-Muhly-Raeburn [@FMR03]; all assumptions were finally removed by Katsoulis-Kribs [@KatKri06]. Furthermore Kakariadis-Peters [@KakPet14] have 