The stable exotic Cuntz algebras are higher-rank graph algebras

Boersema, Jeffrey; Browne, Sarah; Gillaspy, Elizabeth

doi:10.1090/bproc/180

The stable exotic Cuntz algebras are higher-rank graph algebras
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by Jeffrey L. Boersema, Sarah L. Browne and Elizabeth Gillaspy

Proc. Amer. Math. Soc. Ser. B 11 (2024), 47-62

DOI: https://doi.org/10.1090/bproc/180

Published electronically: March 5, 2024

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Abstract:

For each odd integer $n \geq 3$, we construct a rank-3 graph $\Lambda _n$ with involution $\gamma _n$ whose real $C^*$-algebra $C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n)$ is stably isomorphic to the exotic Cuntz algebra $\mathcal E_n$. This construction is optimal, as we prove that a rank-2 graph with involution $(\Lambda ,\gamma )$ can never satisfy $C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )\sim _{ME} \mathcal E_n$, and Boersema reached the same conclusion for rank-1 graphs (directed graphs) in [Münster J. Math. 10 (2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank-1 graph with involution $(\Lambda , \gamma )$ whose real $C^*$-algebra $C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )$ is stably isomorphic to the suspension $S \mathbb {R}$. In the Appendix, we show that the $i$-fold suspension $S^i \mathbb {R}$ is stably isomorphic to a graph algebra iff $-2 \leq i \leq 1$.

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