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The generator rank of subhomogeneous
$C^*\!$-algebras
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 75 / Issue 4 / August 2023
- Published online by Cambridge University Press:
- 14 June 2022, pp. 1314-1342
- Print publication:
- August 2023
-
- Article
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- You have access
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We compute the generator rank of a subhomogeneous
$C^*\!$-algebra in terms of the covering dimension of the pieces of its primitive ideal space corresponding to irreducible representations of a fixed dimension. We deduce that every 
$\mathcal {Z}$-stable approximately subhomogeneous algebra has generator rank one, which means that a generic element in such an algebra is a generator.This leads to a strong solution of the generator problem for classifiable, simple, nuclear
$C^*\!$-algebras: a generic element in each such algebra is a generator. Examples of Villadsen show that this is not the case for all separable, simple, nuclear 
$C^*\!$-algebras.
