Some results on natural Hilbert modules

In this paper, ideal subbimodule and bi-unitary operator of Hilbert bimodules are cosidered and the canonical Hilbert bimodule structure on the quotient of a Hilbert bimodule over an ideal subbimodule is described. Let φ : A → C and ψ : B → D be injective morphisms of C∗-algebras and U : M → N be a surjective (φ, ψ)-morphism. Applying U, we demonstrate an isomorphism between the corresponding C-algebras B (M) and B (N) of adjointable operators. Given ideals I of A and J of B, and ideal submodules IMJ and φ(I)Nψ(J), we associate with U a natural bi-unitary operator Ub:MIMJ → Nφ(I)Nψ(J) of the quotient of Hilbert bimodules over their associated ideal subbimodule to prove that BMIMJ and BN φ(I)Nψ(J) are isomorphic C∗-algebras.