The CAR algebra is simple and nuclear

Question

It is a fact that $CAR$ algebra is simple and nuclear.How to show this conclusion?

Answer 1

Let $I$ be an ideal in the CAR-algebra. Then you should convince yourself that $$ I = \overline{\bigcup_{i=1}^\infty I \cap M_{2^n}}. $$ Now $I \cap M_{2^n}$ is an ideal in $M_{2^n}$ and you use that $M_{2^n}$ is simple.

For nuclearity, note that there are conditional expectations $M_{2^\infty} \to M_{2^n}$.