One-parameter groups of orthogonality preservers on JB$^*$-algebras

TL;DR

This paper characterizes one-parameter groups of orthogonality-preserving operators on JB$^*$-algebras, enhancing understanding of their structure and properties in the context of JB$^*$-algebras and triples.

Contribution

It provides a complete characterization of all one-parameter groups of orthogonality-preserving operators on JB$^*$-algebras, and clarifies conditions for bijective orthogonality-preserving operators.

Findings

Orthogonality-preserving bijections are equivalent to biorthogonality-preserving and zero-triple-product preserving.

Complete characterization of one-parameter groups of such operators.

Enhanced understanding of operator structure on JB$^*$-algebras.

Abstract

In a first objective we improve our understanding about surjective and bijective bounded linear operators preserving orthogonality from a JB$^*$-algebra $\mathcal{A}$ into a JB$^*$-triple $E$. Among many other conclusions, it is shown that a bounded linear bijection $T: \mathcal{A}\to E$ is orthogonality preserving if, and only if, it is biorthogonality preserving if, and only if, it preserves zero-triple-products in both directions (i.e., $\{a,b,c\}=0 \Leftrightarrow \{T(a),T(b),T(c)\}=0$). In the second main result we establish a complete characterization of all one-parameter groups of orthogonality preserving operators on a JB$^*$-algebra.