The strong Morita equivalence for inclusions of $C^{\ast }$-algebras and conditional expectations for equivalence bimodules

We shall introduce the notions of the strong Morita equivalence for unital inclusions of unital $C^{\ast }$-algebras and conditional expectations from an equivalence bimodule onto its closed subspace with respect to conditional expectations from unital $C^{\ast }$-algebras onto their unital $C^{\ast }$-subalgebras. Also, we shall study their basic properties.