Glimm criteria for separable C star algebras and type I groups

Put $A=C^{\ast }(G)$. By [F1] it is separable, and its nondegenerate representation classes, kernels and generated algebras agree with those of $G$. By [F4] this correspondence identifies the actual Mackey Borel structures, not just the underlying class sets.

We prove the carrier reduction needed for the cited implication. Let $\rho$ be a nonzero factor representation of $A$ on arbitrary $H$, let $M=\rho (A{)}^{\mathrm{\prime }\mathrm{\prime }}$, choose $\xi \mathrm{\ne }0$, and let $K=\overline{\rho (A)\xi }$. Nondegeneracy makes $K\mathrm{\ne }0$, and separability of $A$ makes $K$ separable. It reduces $\rho (A)$, so its projection lies in $M^{\mathrm{\prime }}$. Restriction $\mathrm{\Phi }:M\to B(K)$ is therefore a unital star-homomorphism. Its kernel is a weakly closed ideal $J_M$ of $M$. A positive approximate unit of $J_M$ converges strongly to its support $z$: convergence holds on $J_{M}H$ by norm approximation and on its orthogonal complement by annihilation. That support reduces $M$ and $M^{\mathrm{\prime }}$, hence $z\in Z(M)$; weak closedness puts $z\in J_M$, and $J_M=Mz$. Since restriction is nonzero and $M$ is a factor, $z=0$, so $\mathrm{\Phi }$ is injective and isometric.

Conversely, if $A$ is GCR, [F2] makes every factor generated algebra type I. For separable-carrier group representations [F1] and [F3] identify this with the multiple-of-an-irreducible condition, so (i) follows. Also [F2,F4] give standardness and countable separation of the group Mackey dual. For its topology, let $S\subseteq \widehat{G}$. By [F5], $\pi \in \bar{S}$ exactly when ${\bigcap }_{\sigma \in S}\ker \sigma \subseteq \ker \pi$; the intersection is the kernel of the class direct sum. This is exactly the primitive hull-kernel closure rule. GCR makes the kernel map bijective by [F2], so that rule proves it is a Fell-to-Jacobson homeomorphism. Hence the topology Borel structure equals the standard Mackey Borel structure, proving (ii), (iii) and (iv).

We also justify its von Neumann image. The unit ball of $B(H)$ is compact in WOT: encode bounded sesquilinear forms by their values on all vector pairs in the corresponding compact scalar discs, impose the closed linearity and norm bounds, and use product compactness and Riesz from [F6]. The product compactness here is exactly the earlier Tychonoff theorem of [F6], with our stated AC hypothesis; no Boolean prime ideal/product equivalence is needed. The unit ball of $M$ is a closed subset and is compact. Restriction is WOT-continuous, so its image unit ball is compact and WOT-closed in $B(K)$. It is the unit ball of $\mathrm{\Phi }(M)$ by isometry. Bounded density [F2] applied to the concrete unital C*-algebra $\mathrm{\Phi }(M)$ now makes its generated von Neumann unit ball strongly approximable by that same closed ball; hence $\mathrm{\Phi }(M)$ is von Neumann. Finally $\rho (A)$ is boundedly strongly dense in $M$, so restrictions show $\mathrm{\Phi }(M)=(\rho \mathrm{\mid }K)(A{)}^{\mathrm{\prime }\mathrm{\prime }}$. It is a factor isomorphic to $M$.

Suppose (i), the stated separable-carrier group type-I convention. By [F1], $\rho \mathrm{\mid }K$ corresponds to a strongly continuous factor representation of $G$ on separable $K$. Its generated algebra $\mathrm{\Phi }(M)$ is type I by (i) and [F3]. An inverse image under the isomorphism of a minimal projection is minimal in $M$. Thus every arbitrary-carrier factor representation of $A$ is type I. The one cited fact [F7] therefore gives that $A$ is GCR. This is the only original-source cited implication used.

If (iii) holds, [F4] transports its countable separation to the C*-Mackey dual, so [F2] gives GCR. If (iv) holds, kernel injectivity and [F1,F2] give GCR. If (ii) holds, its standard Mackey structure is countably separated and the same argument applies. Combined with steps 4.1 and 2.2, these implications prove the full four-clause equivalence. The factor/multiplicity, arbitrary-carrier reduction, Borel, topology and all assembling steps are local; only the explicitly identified implication [F7] is cited.