The generator problem for 𝒵-stable 𝒞*-algebras

Thiel, Hannes; Winter, Wilhelm

doi:10.1090/S0002-9947-2014-06013-3

The generator problem for $\mathcal {Z}$-stable $C^*$-algebras
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by Hannes Thiel and Wilhelm Winter PDF

Trans. Amer. Math. Soc. 366 (2014), 2327-2343 Request permission

Abstract:

The generator problem was posed by Kadison in 1967, and it remains open today. We provide a solution for the class of $C^*$-algebras absorbing the Jiang-Su algebra $\mathcal {Z}$ tensorially. More precisely, we show that every unital, separable, $\mathcal {Z}$-stable $C^*$-algebra $A$ is singly generated, which means that there exists an element $x\in A$ that is not contained in any proper sub-$C^*$-algebra of $A$.

To give applications of our result, we observe that $\mathcal {Z}$ can be embedded into the reduced group $C^*$-algebra of a discrete group that contains a non-cyclic, free subgroup. It follows that certain tensor products with reduced group $C^*$-algebras are singly generated. In particular, $C^*_r(F_\infty )\otimes C^*_r(F_\infty )$ is singly generated.

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