On anti-automorphisms of von Neumann algebras

Two types of * -anti-automorphisms of a von Neumann algebra % acting on a Hubert space 3f leaving the center of % elementwise fixed are discussed, those of order two and those of the form A->V~A*V, V being a conjugate linear isometry of <%^ onto itself such that V 2 e 1. The latter antiautomorphisms are called inner, and are the composition of inner ^-automorphisms and * -anti-automorphisms of the form A -> JA*J, where J is a conjugation, i.e. a conjugate linear isometry of f onto itself such that J 2 = I. The former anti-automorphisms are also closely related to conjugations; they are almost, and in many cases exactly of the form A --> JA*J. Moreover, the existence of * -anti-automorphisms of order two leaving the center fixed implies the existence of a conjugation J such that J%J -, and such that JA*J = A for all A in the center of 21.

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