On Kirchberg's Embedding Problem

By Isaac Goldbring, Thomas Sinclair. 2014

48 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c024a3c57dd358dba57b24f7e7197712cfd2db62

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In Algebra and Number Theory

1. Selccuk Barlak
2. G'abor Szab'o

In Mathematical Physics

1. Selccuk Barlak
2. G'abor Szab'o
3. David Kerr

Citations in the corpus

• Sequentially split ∗-homomorphisms between C*-algebras (2015, background): “Existential embeddings Let us briefly recall the notion of an existential embedding, which originally stems from model theory of metric structures and was introduced for C∗-algebras in [22].”