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!