Exactness and LLP Results via Operator System Methods

TL;DR

This work develops an operator-system–theoretic approach to fundamental properties of C*-algebras, notably exactness and the local lifting property (LLP), with a focus on reduced and full group C*-algebras. By introducing and exploiting the joint cb-numerical radius $w_{cb}$ and related parameters $r_\infty(\mathcal S)$ and $d_\infty(\mathcal S^d)$, the authors derive direct proofs and new obstructions to exactness and quasidiagonality, including a streamlined route to Wassermann’s non-exactness results for free groups and connections to hyperlinearity. The paper also develops matrix-range techniques, defines $k$-max and $k$-min operator-system constructions, and provides quantitative bounds on distances between matrix ranges, yielding new estimates such as $d_\infty(\mathcal S_n) \ge 1/2$ and lower bounds for $d_\infty(\mathcal U_n)$ based on $w=\sqrt{2n-1}$. Together, these results illustrate how operator-system methods can yield concrete, computable criteria for exactness, LLP, and quasidiagonality, offering a more direct and tensor-theoretic perspective on longstanding C*-algebra problems and their group-theoretic manifestations.

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.

Theorems & Definitions (47)

• Proposition 1.1
• Theorem 1.2
• proof
• Theorem 1.3
• proof
• Definition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof