一个具有弱期望性质和局部提升性质的非核 $C^*$-代数
A non-nuclear $C^*$-algebra with the Weak Expectation Property and the Local Lifting Property
AI总结：
本文构造了首个同时具有弱期望性质和局部提升性质的非核 $C^*$-代数，证明了其上存在唯一的 $C^*$-范数，并探讨了其与 Connes-Kirchberg 问题的联系。
摘要：
我们构造了首个具有标题所述性质的 $C^*$-代数 $A$ 的例子。这给出了一个新的非核 $A$ 的例子，使得在 $A \otimes A^{op}$ 上存在唯一的 $C^*$-范数。这个例子在与 Connes-Kirchberg 问题的联系中尤为引人关注，该问题等价于已知具有 LLP 的 $C^*({\bb F}_2)$ 是否也具有 WEP 的问题。我们的 $C^*$-代数 $A$ 具有与 $C^*({\bb F}_2)$ 或 $C^*({\bb F}_\infty)$ 相同的有限维算子子空间集合。此外，我们的例子可以被构造为拟对角的，并且具有相似度（或长度）3。在论文的第二部分，我们在更一般的框架下重新表述了我们的构造，即一个 $C^*$-代数可以被描述为 $C^*$-代数序列 $(C_n)$ 的归纳与射影极限，其中每个 $C_n$ 是 $C_{n+1}$ 的子商。我们利用这一点表明，对于单射（非满射）$*$-同态的某些局部性质，存在一些 $C^*$-代数，其恒等映射具有与这些 $*$-同态相同的性质。
英文摘要：
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 $A \otimes A^{op}$. This example is of particular interest in connection with the Connes-Kirchberg problem, which is equivalent to the question whether $C^*({\bb F}_2)$, 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 $C^*({\bb F}_2)$ or $C^*({\bb F}_\infty)$. 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 \emph{limit both inductive and projective} for a sequence of $C^*$-algebras $(C_n)$ when each $C_n$ is a \emph{subquotient} of $C_{n+1}$. 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.
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