Conjugacy of involutive antiautomorphisms of von Neumann algebras (Q1820397)

A central involution of a von Neumann algebra is a *-antiautomorphism of order 2 leaving the center elementwise fixed. It is proved that two central involutions of a von Neumann algebra are conjugate via a *- automorphism leaving the center elementwise fixed if and only if the corresponding Jordan algebras of selfadjoint fixed points are isomorphic via an isomorphism leaving the center elementwise fixed. Since there are von Neumann algebras with many conjugacy classes of central involutions and the Jordan algebra of selfadjoint fixed points usually generates the von Neumann algebra it is thus shown that there may be even an uncountable number of non-isomorphic JW-algebras generating the same von Neumann algebra. The paper also discusses the problem if two close central involutions are conjugate.