post-liminal/GCR algebras

Question

from Theorem 5.6.2 [Murphy-C* algebras and operator theory]:

let (H , φ) be an non zero irreducible representation of C* algebra A, and let I be a closed ideal for A. Denote (H',φ') the restriction of (H , φ) to I. Given A/I and I are postliminal algebras, and I is not contained in Ker(φ):

there are two things i find confusing:

1) Why (H',φ') is a non zero representation?

2) When b∊ I, why is φ(b) compact iff φ'(b) is compact?

Thank you for any help!

Answer 1

As far as I can tell,

1. If $(H,\varphi')$ is zero, then $\varphi'(b)=0$ for all $b\in I$, so $I\subset\ker\varphi$.

2. the compactness of an operator does not depend on its "complement". And you have $\varphi(b)=\varphi'(b)\oplus 0$ (from $H=H'\oplus(H')^\perp$).