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| Հիմնական հեղինակներ: | , |
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| Ձևաչափ: | Preprint |
| Հրապարակվել է: |
2024
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| Խորագրեր: | |
| Առցանց հասանելիություն: | https://arxiv.org/abs/2401.05917 |
| Ցուցիչներ: |
Ավելացրեք ցուցիչ
Չկան պիտակներ, Եղեք առաջինը, ով նշում է այս գրառումը!
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| _version_ | 1866929207125213184 |
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| author | Harnisch, Hergen Kirchberg, Eberhard |
| author_facet | Harnisch, Hergen Kirchberg, Eberhard |
| contents | 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 $\mathcal{O}_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05917 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The inverse problem for primitive ideal spaces Harnisch, Hergen Kirchberg, Eberhard Operator Algebras 46L35, 46L80, 06D99, 54D10 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 $\mathcal{O}_2$. |
| title | The inverse problem for primitive ideal spaces |
| topic | Operator Algebras 46L35, 46L80, 06D99, 54D10 |
| url | https://arxiv.org/abs/2401.05917 |