Derivations on ternary rings of operators

TL;DR

This paper explores derivations on ternary rings of operators, establishing their spatial nature and examining conditions under which they are inner, thus advancing understanding of their algebraic structure.

Contribution

It introduces p-derivations associated with projections and relates them to triple derivations, providing new insights into derivations on ternary rings of operators.

Findings

Every derivation on a ternary ring of operators is spatial.

The paper investigates conditions for derivations to be inner.

Relations between p-derivations and triple derivations are established.

Abstract

To each projection $p$ in a $C^*$-algebra $A$ we associate a family of derivations on $A$, called $p$-derivations, and relate them to the space of triple derivations on $p A (1-p)$. We then show that every derivation on a ternary ring of operators is spatial and we investigate whether every such derivation on a weakly closed ternary ring of operators is inner.