Is the C*-algebra of compact operators nuclear?

Question

Is the algebra of compact operators a nuclear C*-algebra in the sense that there exists a unique C*-norm on the algebraic tensor product $A\odot\mathcal K$ for all C*-algebras $A$? If so, where can I find a reference?

Answer 1

Yes, the $C^*$-algebra of compact operators is nuclear. There is a proof in chapter 6 of Murphy's $C^*$-algebras and Operator Theory. The idea of the proof is that $M_n(\mathbb C)$ is nuclear for all $n$, $\mathcal K$ is the direct limit of the $M_n(\mathbb C)$ under upper left corner inclusions, and nuclearity is preserved under direct limits.