Structure of associative subalgebras of Jordan operator algebras

TL;DR

This paper demonstrates that the structure of associative subalgebras in Jordan operator algebras uniquely determines the algebra's Jordan structure, using order isomorphisms and orthogonality relations.

Contribution

It establishes that JBW algebras can be reconstructed from their associative unital JB subalgebras and extends this to abelian subalgebras of von Neumann algebras.

Findings

Order isomorphisms correspond to Jordan isomorphisms.

JBW algebras are characterized by their associative subalgebra structure.

Reconstruction of Jordan structure from orthogonality relations.

Abstract

We show that any order isomorphism between ordered structures of associative unital JB-subalgebras of JBW algebras is implemented naturally by a Jordan isomorphism. Consequently, JBW algebras are determined by the structure of their associative unital JB subalgebras. Further we show that in a similar way it is possible to reconstruct Jordan structure from the order structure of associative subalgebras endowed with an orthogonality relation. In case of abelian subalgebras of von Neuman algebra it is we shown that order isomorphisms of the structure of abelian C*-subalgebras that are well behaved with respect to the structure of two by two matrices over original algebra are implemented by *-isomorphisms.