Let $X$ be a linear space and let $\mathscr{L}(X)$ be the set of all linear operators on $X$, then take $\mathfrak{A}(\mathscr{L}(X))$ as the algebra generated by the set of all linear operators on $X$. A representation of an algebra $\mathfrak{A}$ on the linear space $X$ is any homomorphism of $\mathfrak{A}$ into $\mathfrak{A}(\mathscr{L}(X))$. Let $\mathfrak{L}$ be an ideal of $\mathfrak{A}$ and consider the representation $a{\rightarrow}A_a^{\mathfrak{A}-\mathfrak{L}}$. Let $\mathfrak{L}_1$ be any ideal in $\mathfrak{A}$ which contains the ideal $\mathfrak{L}$. Take $\mathfrak{L}_1^{\prime}$ as the image of $\mathfrak{L}_1$ in $\mathfrak{A}-\mathfrak{L}$.
My question is, what does the previous sentence mean? What image, under what mapping? Isn't the image of $\mathfrak{L}_1$, that is $\mathfrak{L}_1^{\prime}$, a set of vectors in the linear space $X$? Aren't we mixing apples and oranges when we say that $\mathfrak{L}_1^{\prime}$ is the image of $\mathfrak{L}_1$ in $\mathfrak{A}-\mathfrak{L}$ in the sense that $\mathfrak{L}_1^{\prime}$ is a set of vectors in $X$ and $\mathfrak{A}-\mathfrak{L}$ is an operator space?
The above is loosely based on Theorem 2.2.1 page 50 of Rickart "$\textit{General Theory of Banach Algebras,}$" 1960.