The inverse problem for primitive ideal spaces

A pure topological characterization of primitive ideal spaces of separable nuclear C*-algebras is given. We show that a $T_0$-space $X$ is a primitive ideal space of a separable nuclear C*-algebra $A$ if and only if $X$ is point-complete second countable, and there is a continuous pseudo-open and pseudo-epimorphic map from a locally compact Polish space $P$ into $X$. We use this pseudo-open map to construct a Hilbert bi-module $\mathcal{H}$ over $C_0(X)$ such that $X$ is isomorphic to the primitive ideal space of the Cuntz–Pimsner algebra ${\mathcal{O}}_{\mathcal{H}}$ generated by $\mathcal{H}$. Moreover, our ${\mathcal{O}}_{\mathcal{H}}$ is $KK(X;\mathrm{.},\mathrm{.})$-equivalent to $C_0(P)$ (with the action of $X$ on $C_0(P)$ given be the natural map from $\mathbb{O}(X)$ into $\mathbb{O}(P)$, which is isomorphic to the ideal lattice of $C_0(P)$. Our construction becomes almost functorial in $X$ if we tensor ${\mathcal{O}}_{\mathcal{H}}$ with the Cuntz algebra ${\mathcal{O}}_2$.