Vol. 21, No. 2, 1967

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On anti-automorphisms of von Neumann algebras

Erling Stormer

Vol. 21 (1967), No. 2, 349–370

Two types of -anti-automorphisms of a von Neumann algebra A acting on a Hilbert space leaving the center of A elementwise fixed are discussed, those of order two and those of the form A V 1AV,V being a conjugate linear isometry of onto itself such that V 2 A. 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 onto itself such that J2 = I. The former anti-automorphisms are also closely related to conjugations; they are almost, and in many cases exactly of the form A JAJ. Moreover, the existence of -anti-automorphisms of order two leavimg the center fixed implies the existence of a conjugation J such that JAJ = A, and such that JAJ = A for all A in the center of A.

Mathematical Subject Classification
Primary: 46.65
Received: 4 March 1966
Published: 1 May 1967
Erling Stormer