On Kirchberg's Embedding Problem

Isaac Goldbring; Thomas Sinclair

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arXiv · 1404.1861

On Kirchberg's Embedding Problem

Isaac Goldbring·Thomas Sinclair

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Abstract

Kirchberg's Embedding Problem (KEP) asks whether every separable C$^*$ algebra embeds into an ultrapower of the Cuntz algebra $\mathcal{O}_2$. In this paper, we use model theory to show that this conjecture is equivalent to a local approximate nuclearity condition that we call the existence of good nuclear witnesses. In order to prove this result, we study general properties of existentially closed C$^*$ algebras. Along the way, we establish a connection between existentially closed C$^*$ algebras, the weak expectation property of Lance, and the local lifting property of Kirchberg. The paper concludes with a discussion of the model theory of $\mathcal{O}_2$. Several results in this last section are proven using some technical results concerning tubular embeddings, a notion first introduced by Jung for studying embeddings of tracial von Neumann algebras into the ultrapower of the hyperfinite II$_1$ factor.

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Isaac Goldbring, Thomas Sinclair. 2015-02-27. On Kirchberg's Embedding Problem. https://arxiv.org/abs/1404.1861

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