Jordan Isomorphisms on Nest Subalgebras

This paper is devoted to the study of Jordan isomorphisms on nest subalgebras of factor von Neumann algebras. It is shown that every Jordan isomorphismϕbetween the two nest subalgebrasalgMβandalgMγis either an isomorphism or an anti-isomorphism.

Invertibility-preserving maps ofC∗-algebras with real rank zero

In 1996, Harris and Kadison posed the following problem: show that a linear bijection betweenC∗-algebras that preserves the identity and the set of invertible elements is a Jordan isomorphism. In this paper, we show that ifAandBare semisimple Banach algebras andΦ:A→Bis a linear map ontoBthat preserves the spectrum of elements, thenΦis a Jordan isomorphism if eitherAorBis aC∗-algebra of real rank zero. We also generalize a theorem of Russo.

Commutativity Preserving Mappings of Von Neumann Algebras

A map θ: M —> N where M and N are rings is said to preserve commutativity in both directions if the elements a,b ∊ M commute if and only if θ(a) and θ(b) commute. In this paper we show that if M and N are von Neumann algebras with no central summands of type I1 or I2 and θ is a bijective additive map which preserves commutativity in both directions then θ(x) = cφ(x) +f(x) where c is an invertible element in ZN, the center of N, φ M —> N is a Jordan isomorphism of M onto N, and f is an additive map of M into ZN.

Hilbert space methods in the theory of Jordan algebras. II

In this paper we consider the classification problem for separable special simple J*-algebras (cf. (8)). We show, using a result of Ancochea, that if is the (finite-dimensional) Jordan algebra of all complex n × n matrices and ø a Jordan isomorphism of onto a special J*-algebra J then An can be given the structure of an H*-algebra such that ø is a *-preserving isomorphism of the J*-algebra onto J. This result enables us to construct explicitly a canonical basis for a finite-dimensional simple special J*-algebra isomorphic to a Jordan algebra of type I from which we also obtain canonical bases for special simple finite-dimensional J*-algebras isomorphic to Jordan algebras of type II and III.