On the Shilov boundary ideal for Fr\'{e}chet local operator systems

TL;DR

This paper characterizes the Shilov boundary ideal for separable Fréchet local operator systems as the intersection of kernels of all their boundary representations, advancing understanding in operator system theory.

Contribution

It provides a new description of the Shilov boundary ideal in the context of separable Fréchet local operator systems using boundary representations.

Findings

The Shilov boundary ideal equals the intersection of kernels of boundary representations.

This characterization applies specifically to separable Fréchet local operator systems.

It deepens the theoretical understanding of boundary ideals in operator systems.

Abstract

We show that the Shilov boundary ideal for a separable Fr\'{e}chet local operator system is given by the intersection of the kernels of all its $\Gamma$-boundary representations.