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.

中文翻译：

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具有弱期望性质和局部提升性质的非核 $$C^*$$-代数

我们用标题中的属性构造 $C^*$-algebra $A$ 的第一个例子。这给出了非核 $A$ 的新示例，其中在 $A \otimes A^{op}$ 上存在唯一的 $C^*$-范数。这个例子与 Connes-Kirchberg 问题有关，它等价于 $C^*({\bb F}_2)$（已知有 LLP）是否也有 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，我们的例子可以是拟对角的，相似度（或长度）为 3。在论文的第二部分，我们在更一般的 $C^*$-代数框架中重新表述了我们的构造，该代数可以描述为 $C^*$- 序列的 \emph {limit both inductive and projective}代数 $(C_n)$ 当每个 $C_n$ 是 $C_{n+1}$ 的 \emph{subquotient} 时。我们用它来证明对于单射（非满射）$*$-同态的某些局部性质，存在 $C^*$-代数，其身份映射与 $*$-同态具有相同的性质。