Equivalent characterizations of second-countable type I groups

By [F1], clause (i) is the definition of clause (ii). The parenthetical equivalence in (ii) is precisely the separable factor-to-multiple equivalence discharged in [F1], so it retains every stated multiplicity, including countably infinite multiplicity. These are nonzero factor representations; the zero carrier introduces no additional obligation.

By [F2], the condition in clause (ii) is equivalent to standardness of the Mackey dual together with equality of Mackey and Fell-topology Borel sets, which is clause (iii) under [F3]; it is also equivalent to countable separation in clause (iv) and to the homeomorphism condition in clause (v). In particular, a homeomorphism onto its image is injective and gives the kernel criterion of [F2]; conversely that criterion supplies the asserted homeomorphism. The primitive-kernel map has image all primitive ideals because a primitive ideal is the kernel of an irreducible nondegenerate representation, but the weaker literal “onto its image” formulation is already enough. Thus the implications are in both directions, with the Borel equality and topological assertion included.

Combining step 1.1 and step 2.1 proves the exact five clauses in the Statement. The canonical dual-indexed irreducible-multiplicity decomposition is a subsequent theorem using the now-proved standard dual; it is not a premise of these equivalences. AC is inherited from [F1]–[F3], and this assembly makes no additional field selections.