Ternary rings of operators arising from inverse semigroups

TL;DR

This paper investigates the properties of ternary rings of operators generated by inverse semigroup subsets, identifying semiheaps in the extended bicyclic semigroup and establishing their generated W*-TROs as injective operator spaces.

Contribution

It characterizes semiheaps within the extended bicyclic semigroup and proves that their generated W*-TROs are injective, advancing understanding of operator space structures from inverse semigroups.

Findings

Identified all semiheaps in the extended bicyclic semigroup.

Proved that W*-TROs generated by semiheaps are injective.

Established properties of ternary rings of operators from inverse semigroups.

Abstract

We are interested in properties, especially injectivity (in the sense of category theory), of the ternary rings of operators generated by certain subsets of an inverse semigroup via the regular representation. We determine all subsets of the extended bicyclic semigroup which are closed under the triple product $xy^*z$ (called semiheaps) and show that the weakly closed ternary rings of operators generated by them (W*-TROs) are injective operator spaces.