Representing conditional expectations as elementary operators

Let $\mathcal{A}$ be a $C^{*}$-algebra and let $\mathcal{B}$ be a $C^{*}$-subalgebra of $\mathcal{A}$. We call a linear operator from $\mathcal{A}$ to $\mathcal{B}$ an elementary conditional expectation if it is simultaneously an elementary operator and a conditional expectation of $\mathcal{A}$ onto $\mathcal{B}$. We give necessary and sufficient conditions for the existence of a faithful elementary conditional expectation of a prime unital $C^{*}$-algebra onto a subalgebra containing the identity element. We give a description of all faithful elementary conditional expectations. We then use these results to give necessary and sufficient conditions for certain conditional expectations to be index-finite (in the sense of Watatani) and we derive an inequality for the index.