Relative Weak Injectivity for C*-algebras

TL;DR

This paper explores the order theoretic aspects of relative weak injectivity in C*-algebras, connecting it with extension theorems, interpolation properties, and Connes' embedding problem, with examples showing limitations in operator systems.

Contribution

It establishes new characterizations of relative weak injectivity in C*-algebras via order properties and links these to fundamental problems like Connes' embedding problem.

Findings

Arveson's extension theorem relates to w.r.i. under order assumptions.

(2,3)-Riesz-Arveson property is equivalent to w.r.i.

Order properties fail in general operator systems.

Abstract

We study the order theoretic properties of relative weak injectivity, w.r.i., in short, in the category of C*-algebras. We prove that Arveson's extension theorem, with additional order assumption on the morphisms, is tightly connected with relative weak injectivity. We prove that (2,3)-Riesz-Arveson property, defined below, is equivalent to w.r.i. Likewise tight Riesz interpolation property yields another characterization of w.r.i. We exhibit, with examples, that these C*-algebraic properties fail in general operator systems. Several order theoretic characterization of Connes' embedding problem is given.