We consider two inclusions of $C^{*}$-algebras whose small $C^{*}$-algebras have approximate units of the large $C^{*}$-algebras and their two spaces of all bounded bimodule linear maps. We suppose that the two inclusions of $C^{*}$-algebras are strongly Morita equivalent. In this paper, we shall show that there exists an isometric isomorphism from one of the spaces of all bounded bimodule linear maps to the other space and we shall study the basic properties about the isometric isomorphism. And, using this isometric isomorphism, we define the Picard group for a bimodule linear map and discuss the Picard group for a bimodule linear map.

We shall introduce the notions of 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.

Let A be a unital C*-algebra. Let (B,E) be a pair consisting of a unital C*-algebra B containing A as a C*-subalgebra with a unit that is also the unit of B, and a conditional expectation E from B onto A that is of index-finite type and of depth 2. Let B1 be the C*-basic construction induced by (B,E). In this paper, we shall show that any such pair (B,E) satisfying the conditions that A′∩B=ℂ1 and that A′∩B1 is commutative is constructed by a saturated C*-algebraic bundle over a finite group. Furthermore, we shall give a necessary and sufficient condition for B to be described as a twisted crossed product of A by its twisted action of a finite group under the condition that A′∩B1 is commutative.

We develop some approaches to the characterization of distributions of real-valued random variables, useful in practical applications, in terms of conditional expectations and hazard measures. We prove several representation theorems generalizing earlier results, and establish stability theorems for two general characteristics introduced in this paper.