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2 results

  • The generator rank of subhomogeneous $C^*\!$-algebras

  • Part of
  • Hannes Thiel
  • 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
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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.

  • Idempotent-generated semigroups and pseudovarieties

  • Part of
  • J. Almeida, A. Moura
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 54 / Issue 3 / October 2011
    Published online by Cambridge University Press:
    20 June 2011, pp. 545-568
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  • The operator that constructs the pseudovariety generated by the idempotent-generated semi-groups of a given pseudovariety is investigated. Several relevant examples of pseudovarieties generated by their idempotent-generated elements are given, as well as some properties of this operator. Particular attention is paid to the pseudovarieties in {J, R, L, DA} concerning this operator and their generator ranks and idempotent-generator ranks.