A non-nuclear C*-algebra with the weak expectation property and the local lifting property

• 成果类型: Article
• 署名作者: Pisier, Gilles
• 署名单位: Texas A&M University System; Texas A&M University College Station
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00977-4
• 发表日期: 2020
• 页码: 513-544
• 关键词: extensions; conjecture

摘要

We construct the first example of a C*-algebra A with the properties in the title. This gives a new example of non-nuclear A for which there is a unique C*-norm on A circle times A(op). This example is of particular interest in connection with the Connes-Kirchberg problem, which is equivalent to the question whether C* (F-2), which is known to have the LLP, also has the WEP. Our C*-algebra A has the same collection of finite dimensional operator subspaces as C* (F-2) or C* (F-infinity). In addition our example can be made to be quasidiagonal and of similarity degree (or length) 3. In the second part of the paper we reformulate our construction in the more general framework of a C*-algebra that can be described as the limit both inductive and projective for a sequence of C*-algebras (C-n) when each Cn is a subquotient of Cn+1. We use this to show that for certain local properties of injective (non-surjective) *-homomorphisms, there are C*-algebras for which the identity map has the same properties as the *-homomorphisms.