On $C^*$-algebras with Local Lifting Property and Weak Expectation Property

TL;DR

The paper addresses the existence of a non-nuclear $C^*$-algebra with both LLP and WEP. It develops a simplified constructive framework based on an isometric embedding $F: C \to \mathcal{L}(C)$ and a completely contractive, self-adjoint lift $f$ with a finitary asymptotic local isometry condition, producing $Z$ and $A=Q(Z)$ whose LLP follows from the LP on $C$; if $F$ is $D$-nuclear for some separable $D$, then $A$ inherits $D$-nuclearity. Cone algebras are employed to generate explicit $F$ that are $\mathscr{C}$-nuclear (with $\mathscr{C}=C^*(\mathbb{F}_\infty)$), ensuring that $A$ has both WEP and LLP while remaining non-nuclear, and the local-embedding perspective clarifies how LLP interacts with mb-/max-norm control. An appendix links LLP to $j$-nuclearity for embeddings into a WEP algebra, strengthening the theoretical bridge between local lifting properties and global nuclearity notions. Overall, the work provides a transparent route to non-nuclear LLP+WEP C*-algebras and highlights the role of cone algebras and local embeddings in this landscape.

Abstract

We give a new, somewhat simpler, presentation of the author's recent construction of a non-nuclear $C^*$-algebra which has both the local lifting property (LLP) and the weak expectation property (WEP).

Theorems & Definitions (53)

• Theorem 1.1
• Proposition 1.2
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• Definition 2.4
• Proposition 2.5
• Remark 2.6
• proof
• Definition 2.7