Two types of ∗-anti-automorphisms
of a von Neumann algebra A acting on a Hilbert space ℋ leaving the center of A
elementwise fixed are discussed, those of order two and those of the form
A → V−1A∗V,V being a conjugate linear isometry of ℋ onto itself such
that V2∈ 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
ℋ 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 → JA∗J. 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 JA∗J = A for all A in the center of
A.
