\colon A \to {{\mathcal{L}}}(X)$. It is called *injective* (resp. *non-degenerate*) if $\phi_X$ is injective (resp. non-degenerate). It is called *full* if ${\left\langleX,X\right\rangle} := {\overline}{{\operatorname{span}}}\{{\left\langlex,y\right\rangle} \mid x,y \in X\} = B$. It is called an *imprimitivity bimodule* (or *equivalence bimodule*) if it is full, injective and $\phi_X(A) = {{\mathcal{K}}}(X)$.

A *(Toeplitz) representation* $(\pi,t)$ of ${}_A X_A$ on a Hilbert space $H$, is a pair of a $*$-homomorphism $\pi\colon A \rightarrow {{\mathcal{B}}}(H)$ and a linear map $t\colon X \rightarrow {{\mathcal{B}}}(H)$, such that $$\pi(a)t(x)=t(\phi_X(a)(x)) {\quad\text{and}\quad}t(x)^*t(y)=\pi({\left\langlex,y\right\rangle}_X)$$ for all $a\in A$ and $x,y\in X$. A representation $(\pi,t)$ automatically satisfies $$t(x) \pi(a) = t(x a) {\text{ for all }}x \in X, a \in A.$$ Moreover there exists a $*$-homomorphism $\psi_t \colon {{\mathcal{K}}}(X) \to {{\mathcal{B}}}(H)$ such that $\psi_t(\Theta_{x, y}) = t(x) t(y)^*$ [@KPW98]. A representation $(\pi , t)$ is said to be *injective* if $\pi$ is injective. If $(\pi,t)$ is injective then $t$ is an isometry and $\psi_t$ is injective.

A crucial remark made by Katsura [@Kat04] is that $a \in \ker\phi_X^\perp \cap \phi_X^{-1}({{\mathcal{K}}}(X))$ whenever $\pi(a) \in \psi_t({{\mathcal{K}}}(X))$ for an injective representation $(\pi,t)$ of ${}_A X_A$. This completes the analysis of Muhly-Solel [@MuhSol98] on covariant representations, by covering also the non-injective cases. In more details let $J$ 