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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Preliminary 251 Construction of crossed products . . . . . . . . . . . . . . . . . . . . . . . 253 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Dual weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Bi-dual weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 Subgroups and subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 284 The structure of a v o n Neumann algebra of type I I I . . . . . . . . . . . . . . 286 Algebraic invariants S ( ~ ) and T ( ~ ) of A. Connes . . . . . . . . . . . . . . . 294 Induced action and crossed products . . . . . . . . . . . . . . . . . . . . . 297 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

Duality for crossed products and the structure of von Neumann algebras of type III