Relative Weak Injectivity for Operator Systems

TL;DR

This paper explores the concept of relative weak injectivity in operator systems, providing new characterizations of nuclearity and establishing equivalences between different nuclearity notions, advancing understanding in operator algebra theory.

Contribution

It introduces new characterizations of nuclearity and weak injectivity, and demonstrates equivalences between various nuclearity concepts in operator systems.

Findings

Characterization of the weak expectation property.

(c,max)-nuclearity characterizes Kirchberg and Wasserman's C*-systems.

Quasi-nuclearity and nuclearity are shown to be equivalent.

Abstract

We investigate the notion of relative weak injectivity and its nuclearity related properties in the category of operator systems. We obtain several characterizations of the weak expectation property. We show that (c,max)-nuclearity characterizes Kirchberg and Wasserman's C*-systems. Namioka and Phelps' test systems, which detects nuclear C*-algebras, is shown to characterize nuclear C*-systems. We study quasi-nuclearity in the operator system setting and prove that quasi-nuclearity and nuclearity are equivalent, in other words, (er,max)-nuclearity and (min,max)-nuclearity are equivalent.