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Let $\mathbb{G}$ be a discrete quantum group, and consider the non-commutative Furstenberg boundary $\partial_F\mathbb{G}$ with function algebra $C(\partial_F \mathbb{G}) = I_{\mathbb{G}}(\mathbb{C})$. Here, $I_\mathbb{G}(\mathbb{C})$ is injective so has a $C^*$-algebra structure coming from the Choi-Effros product.

Is something known about nuclearity of this $C^*$-algebra? Of course, if $\mathbb{G}$ is a classical discrete group, then $C(\partial_F\mathbb{G})$ is commutative, so nuclear. Are there non-trivial examples known of genuine discrete quantum groups where this $C^*$-algebra is nuclear or not?

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If there is a bound on the size of matrix blocks in $\ell_\infty(G)$, then this algebra is nuclear by a result of S. Wassermann [1], which would imply the nuclearity of $I_G(\mathbb{C})$. Without such a bound, his result says that $\ell_\infty(G)$ is not nuclear, and I don't see any reason why $I_G(\mathbb{C})$ should be nuclear then (unless $G$ is amenable).

  1. Wassermann, Simon, On tensor products of certain group (C^*)-algebras, J. Funct. Anal. 23, 239-254 (1976). ZBL0358.46040.
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  • $\begingroup$ This is nice, thanks! It might be interesting to note that if there is a bound on the size of the matrix blocks, we are in the Kac-case, see arxiv.org/pdf/1710.04863.pdf $\endgroup$
    – J. De Ro
    Commented Dec 7, 2023 at 15:02
  • $\begingroup$ It might be worth noting that if $\ell_\infty(G)$ is subhomogeneous then it is not just nuclear, it is Type I, in which case every Cstar-subalgebra is itself Type I and hence nuclear. $\endgroup$
    – Yemon Choi
    Commented Dec 7, 2023 at 18:34
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    $\begingroup$ @YemonChoi I don't think $C(\partial_F G)$ is a subalgebra of $\ell_\infty(G)$ though. The $C^*$-algebra structure comes from the Choi-Effros product. $\endgroup$
    – J. De Ro
    Commented Dec 7, 2023 at 18:41
  • $\begingroup$ @J.DeRo Oops, you are quite right. I suspect that the range of a conditional expectation on a Type I Cstar algebra, equipped with the Choi-Effros product, might still be Type I, but I admit that I do not have a proof or reference for such a claim. $\endgroup$
    – Yemon Choi
    Commented Dec 7, 2023 at 19:22
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    $\begingroup$ @YemonChoi I'm not sure about Type I but the range equipped with the CE product is at least nuclear since it satisfies the CPAP $\endgroup$ Commented Dec 8, 2023 at 14:45

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