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It is shown that if T is a ternary ring of operators (TRO), X is anondegenerate sub-TRO of T and there exists a contractive idempotent surjectivemap P:T-->X, then P has a unique, explicitly described extension to aconditional expectation between the associated linking algebras. A version ofthe result for W*-TROs is also presented and some applications mentioned.

Inclusions of ternary rings of operators and conditional expectations