Exactness and LLP Results via Operator System Methods

TL;DR

This paper uses operator system techniques to provide direct proofs of exactness and LLP properties of group C*-algebras, offering a new approach that avoids traditional heavy machinery.

Contribution

It introduces operator system methods to analyze C*-algebra properties, simplifying proofs of exactness and LLP without relying on classical C*-algebra theory.

Findings

C*-algebras with n unitaries and joint numerical radius less than n lack certain properties.

Operator system techniques can replace traditional heavy machinery in proofs.

Results apply to group C*-algebras and their structural properties.

Abstract

In this paper, we employ operator system techniques to investigate structural properties of C*-algebras. In particular, we provide more direct proofs of results concerning exactness and the local lifting property (LLP) of group C$^*$-algebras that avoid relying on the traditional heavy machinery of C$^*$-algebra theory. Briefly, these methods allow us to deduce that any C*-algebra containing $n$ unitaries whose $\textit{ joint numerical radius}$, in the sense defined by \cite{FKP}, is strictly less than $n$, must fail certain of these properties.