A $*$-homomorphism from the CAR algebra to $\mathfrak B(\mathcal H)$

Question

Could a $*$-homomorphism $\pi:\text{CAR}\to\mathfrak B(\mathcal H)$ exist (with $\mathcal H$ separable) such that there is a compact and positive element $h\in\mathcal K$ commuting with the image of $\pi$? Some naive considerations have led me to conclude that this shouldn't be the case, but I hope I'm wrong.

Answer 1

No. The CAR algebra $A$ is simple, so any representation $\pi$ is necessarily injective. Assume $\pi$ is unital. If there is a compact $h$ in the commutant $\pi(A)'$, then the commutant contains a finite dimensional projection $p$ and $p \pi(\cdot) p$ would be a representation of $A$ on a finite dimensional Hilbert space, which is impossible.

Hope this helps.