ELEMENTARY PROOF OF SCHWEITZER'S THEOREM ON HILBERT C~*-MODULES IN WHICH ALL CLOSED SUBMODULES ARE ORTHOGONALLY CLOSED

摘要

Let A and B be C~*-algebras and let X be an A-B-imprimitivity bimodule. Schweitzer showed the theorem that if every closed right B-submodule of X is orthogonally closed, then there are families {H_i}_(i ∈ I), {K_i}_(i ∈ I) of Hilbert spaces such that A (resp. B) is isomorphic to the c_0-direct sum ∑_(i ∈ I)~⊕ C(H_i) of all compact operators C(H_i) on H_i (resp. ∑_(i ∈ I)~⊕ C(K_i) of all compact operators C(K_i) on K_i) as a C~*-algebra, and X is isomorphic to the C_0-direct sum ∑_(i ∈ I)~⊕ C(K_i, H_i) as a Hilbert C~*-module, where C(K_i, H_i) denotes the Hilbert C~*-module consisting of all compact operators from K_i into H_i. In this paper, we give an alternative proof, of this theorem, which is shorter and more elementary than the original one.