The strong Morita equivalence for inclusions of $C^*$-algebras and   conditional expectations for equivalence bimodules
  We shall introduce the notions of the strong Morita equivalence for unital inclusions of unital $C^*$-algebras and conditional expectations from an equivalence bimodule onto its closed subspace with respect to conditional expectations from unital $C^*$-algebras onto their unital $C^*$-subalgebras. Also, we shall study their basic properties. 
Kazunori Kodaka, Tamotsu Teruya
math.OA