Representing conditional expectations as elementary operators

Abstract:

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.