The inverse problem for primitive ideal spaces
Hergen Harnisch, Eberhard Kirchberg
TL;DR
The paper provides a pure topological characterization of primitive ideal spaces of separable nuclear C*-algebras via point-complete second countable T0 spaces $X$ and locally compact Polish surjections $\pi: P\to X$ that are pseudo-open and pseudo-epimorphic. It develops a framework linking lattice morphisms of ideals to Hilbert bimodules and Cuntz–Pimsner algebras, realizing $X$ as the primitive ideal space of a CP-algebra $\mathcal{O}(\mathcal{H})$ that is KK-equivalent to $C_0(P)$. Central to the approach is constructing $A\cong C_0(P)\otimes\mathbb{K}$ and a nondegenerate $h:A\to\mathcal{M}(A)$ from a lattice map $\Psi$, then analyzing the associated crossed product $[D]_{\sigma}\rtimes_{\sigma}\mathbb{Z}$ and the CP-algebra $\mathcal{O}(\mathcal{H})$ as a full hereditary subalgebra. The results yield a KK-theoretic bridge between topological data and nuclear C*-algebras, with a detailed description of ideal lattices via semi-invariant/Cancellation conditions and regular abelian subalgebras, and establish equivalent descriptions of primitive ideal spaces among Dini spaces and pseudo-open images of Polish spaces.
Abstract
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 $KK(X;.,.)$-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 $\mathcal{O}_2$.