Local boundary representations of locally C*-algebras

TL;DR

This paper introduces local boundary representations for local operator systems in locally C*-algebras, providing an intrinsic invariant and characterizing these representations using a notion of purity for local completely positive maps.

Contribution

It develops the concept of local boundary representations in the setting of locally C*-algebras and establishes their role as intrinsic invariants for local operator systems.

Findings

Local boundary representations serve as intrinsic invariants.

An analog of purity characterizes local boundary representations.

The framework extends non-commutative Choquet boundary concepts.

Abstract

We initiate a study of non-commutative Choquet boundary for spaces of unbounded operators. We define the notion of local boundary representations for local operator systems in locally C$^*$-algebras and prove that local boundary representations provide an intrinsic invariant for a particular class of local operator systems. An appropriate analog of purity of local completely positive maps on local operator systems is used to characterize local boundary representations for local operator systems in Frechet locally C$^*$-algebras.