Representing conditional expectations as elementary operators

Let A \mathcal {A} be a C ∗ C^{*} -algebra and let B \mathcal {B} be a C ∗ C^{*} -subalgebra of A \mathcal {A} . We call a linear operator from A \mathcal {A} to B \mathcal {B} an elementary conditional expectation if it is simultaneously an elementary operator and a conditional expectation of A \mathcal {A} onto B \mathcal {B} . We give necessary and sufficient conditions for the existence of a faithful elementary conditional expectation of a prime unital C ∗ 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.