GCR kernel and Mackey Borel characterizations

Let $D\subseteq \mathcal{B}(E)$ be a nonzero irreducible image of a separable C*-algebra. For every nonzero $\xi$, the closure of $D\xi$ is a nonzero reducing subspace, hence all of $E$; applying a countable dense algebra family to $\xi$ shows that $E$ is separable. Its commutant is scalar: a nonscalar self-adjoint $S\in D^{\mathrm{\prime }}$ would, by [F5], have two disjoint nonzero continuous spectral cutoffs; their operators commute with $D$ and have orthogonal nonzero ranges, so the closure of either range is a proper nonzero invariant subspace. Real and imaginary parts then give $D^{\mathrm{\prime }}=\mathbb{C}I$ and $D^{\mathrm{\prime }\mathrm{\prime }}=\mathcal{B}(E)$. If $D$ contains a nonzero compact $x$, then $t=x^{\ast }x\in D$ is positive, compact and nonzero. By [F5,F6], an isolated nonzero spectral value of $t$ yields a nonzero finite-rank projection $p=f(t)\in D$, with the cutoff chosen to vanish at zero. The corner $pDp$ is norm closed: inside the closed algebra $D$ it is defined by the closed equation $d=pdp$. Bounded density [F4] approximates every operator on $pE$ by this corner in norm, since convergence on a finite orthonormal basis controls the operator norm. Thus $pDp=\mathcal{B}(pE)$ and contains a rank-one projection $e$ onto a unit vector $\xi$. For $a,b\in D$, $ae{b}^{\ast }$ is the rank-one map $v\mapsto ⟨v,b\xi ⟩a\xi$. The density of $D\xi$ gives all rank-one maps by norm limits; finite-rank density [F5] gives $\mathcal{K}(E)\subseteq D$. Compacts form a closed two-sided ideal here: compositions preserve compactness by [F6], and closure follows from its norm-limit assertion. In a faithful irreducible representation of $B$, their preimage is therefore a closed ideal $I\cong \mathcal{K}(E)$.

We prove the required amplification for every nonzero nondegenerate representation $R:\mathcal{K}(E)\to \mathcal{B}(K)$, allowing arbitrary $K$. Choose an orthonormal basis $(v_i{)}_{i\in J}$ of the nonzero separable $E$, indexed from zero, and put $e_{ij}v=⟨v,v_j⟩v_i$. The finite initial sums $p_n={\sum }_{i\in J,i\le n}{e}_{ii}$ form a positive contractive two-sided approximate unit: $p_{n}v\to v$ because $p_n$ fixes the increasing finite basis spans, their union is dense, and $\mathrm{\parallel }{p}_n\mathrm{\parallel }\le 1$, the two norm limits follow first for rank-one maps and then for all compacts by finite-rank density. Contractivity and nondegeneracy imply $R(p_n)\to I_K$ strongly, first on $R(\mathcal{K}(E))K$ and then on its dense span. Put $L=R(e_{00})K$. The maps $R(e_{i0})$ are isometries from $L$ onto the mutually orthogonal ranges of $R(e_{ii})$, since $e_{0i}{e}_{i0}=e_{00}$ and $e_{i0}{e}_{0i}=e_{ii}$. Their sum defines an onto unitary from ${\widehat{⨁}}_{i\in J}L$ to $K$, and $L\mathrm{\ne }0$ because these ranges exhaust $K$. Denote this sum model by $E\otimes L$. The matrix-unit relations give $R(e_{ij})=e_{ij}\otimes I_L$. For any $T\in \mathcal{B}(E)$, its scalar matrix acts boundedly on this model: on a finite-coordinate vector, expand its finitely many $L$-components in a finite orthonormal basis of their span; the norm estimate on each scalar column gives $\mathrm{\parallel }T\otimes I_L\mathrm{\parallel }\le \mathrm{\parallel }T\mathrm{\parallel }$, and testing $({\alpha }_i\mathrm{\ell }{)}_i$ for a fixed unit $\mathrm{\ell }\in L$ gives equality. Norm approximation by finite matrix compressions extends the formula $R(a)=a\otimes I_L$ to every compact $a$. An operator commuting with all $e_{ii}\otimes I_L$ is block diagonal, and commuting with the $e_{ij}\otimes I_L$ forces all its diagonal blocks to be one $Q\in \mathcal{B}(L)$; thus $R(\mathcal{K}(E){)}^{\mathrm{\prime }}=I_E\otimes \mathcal{B}(L)$. Conversely, the blocks of any operator commuting with this last algebra commute with every operator on $L$, hence are scalars: commuting with each rank-one projection makes each line an eigenspace, and sums of two independent vectors make the scalar constant. Testing on $({\alpha }_i\mathrm{\ell }{)}_i$ makes this scalar matrix a bounded $T\in \mathcal{B}(E)$. Therefore $R(\mathcal{K}(E){)}^{\mathrm{\prime }\mathrm{\prime }}=\mathcal{B}(E)\otimes I_L$. In particular $R$ is irreducible exactly when $\dim L=1$, so the irreducible representation of $\mathcal{K}(E)$ is unique up to unitary equivalence.

Suppose irreducible $\tau (A)$ contains a nonzero compact and put $B=A\mathrm{/}\ker \tau$. Step 1.1 gives its elementary ideal $I\cong \mathcal{K}(E)$. Every other faithful irreducible $\sigma$ of $B$ is nonzero on $I$. The closure of $\sigma (I)H$ is a nonzero reducing subspace for $\sigma (B)$, hence all of $H$. Thus the restriction to $I$ is nondegenerate, and its positive contractive approximate unit satisfies $\sigma (e_t)\to I_H$ strongly by the dense-span argument of step 1.2. For $b\in B$, $b{e}_t\in I$ and $\sigma (b{e}_t)\to \sigma (b)$ strongly. Consequently the restriction and the full representation have the same commutant, so the restriction is irreducible. Step 1.2 makes the restrictions of $\tau$ and $\sigma$ equivalent; their intertwining unitary also intertwines every $b\in B$ by these same strong limits. Hence equal primitive kernels under GCR give equivalent irreducibles. The elementary ideal is taken in $A\mathrm{/}\ker \tau$, which avoids any assumption on arbitrary representations of its preimage in $A$.

Now let $\rho$ be a nonzero nondegenerate factor representation, with $M=\rho (A{)}^{\mathrm{\prime }\mathrm{\prime }}$ and $J=\ker \rho$. The support of a represented ideal lies in $M$ as the strong limit of its approximate unit, and in $M^{\mathrm{\prime }}$ because its range reduces $\rho (A)$. Thus it is a central projection, either0 or1. Two nonzero quotient ideals with zero product would have two nonzero orthogonal such supports, impossible in a factor. Hence $J$ is proper and prime; [F1] makes it primitive. Choose a separate faithful irreducible $\tau$ of $B=A\mathrm{/}J$. GCR passes to this quotient, so step 1.1 gives an elementary ideal $I\cong \mathcal{K}(E)$ in $B$. The original faithful factor representation of $B$ is nonzero on $I$; its support is1, so $\rho \mathrm{\mid }I$ is nondegenerate. This does not turn $\rho$ into an irreducible representation.

Under GCR the kernel map $\kappa :\widehat{A}\to \mathrm{Prim}(A)$ is bijective by step 2.1. The pure-state class map $q:P(A)\to \widehat{A}$ is onto, and its equivalence fibres are internal-unitary orbits by [F4]. Its quotient topology makes it continuous and open, since the saturation of a pure-state open set is the union of its unitary translates. The composite $\kappa q$ is continuous and open by [F1]. Surjectivity and the quotient property make $\kappa$ continuous; if $V$ is open in $\widehat{A}$, $(\kappa q)(q^{-1}V)=\kappa (V)$ is open. Thus $\kappa$ is a homeomorphism, not merely a continuous bijection. By [F3], its class-fibre saturated Borel images identify the Mackey quotient with the standard primitive-code Borel structure, which [F1] identifies with topology Borel sets. Hence the dual is standard Borel and countably separated.

If the kernel map is injective or the Mackey dual is countably separated, then $A$ is GCR: otherwise [F2] gives inequivalent irreducibles with one primitive kernel and also gives a failure of countable separation. A standard Borel space is countably separated, since a countable basis of a Polish presentation separates its points. Combining these implications with step 3.1 proves all four equivalences and the Borel equality. For $A=0$, there are no nonzero irreducible or factor representations, the dual and primitive spaces are empty standard Borel spaces, and all clauses hold.

By the explicit matrix-unit proof of step 1.2, $\rho \mathrm{\mid }I$ is $a\mapsto a\otimes I_L$ on $E\otimes L$ for an arbitrary nonzero Hilbert multiplicity space $L$. Its generated algebra is $B(E)\otimes I_L$. Moreover $\rho (I{)}^{\mathrm{\prime }\mathrm{\prime }}=\rho (B{)}^{\mathrm{\prime }\mathrm{\prime }}$: ideal inclusion gives one direction, and $\rho (b{e}_t)\to \rho (b)$ strongly gives the other. A rank-one projection on $E$ tensored with $I_L$ is therefore a nonzero minimal projection of $M$. Thus every factor representation is type I, with no separability restriction on its multiplicity carrier and no appeal to the cited Glimm converse.