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The pure infiniteness transfer problem

  • Arindam Sutradhar
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Abstract

Problem 8.4 of Pino, Goodearl, Perera and Molina in [arxiv.org/abs/0806.4156] asks whether a dense subalgebra A0 of a C*-algebra A that is purely infinite as a ring forces A to be purely infinite as a C*-algebra. We call this the pure infiniteness transfer problem, the transfer being from the ring to its C*-completion. The problem is open, even when A0 is unital and simple. We settle it under two hypotheses: A has real rank zero, and A0 is closed under holomorphic functional calculus. Under these hypotheses A0 is purely infinite simple as a ring if and only if A is purely infinite simple as a C*-algebra. We also prove an obstruction: a unital C*-algebra with a nonzero finite projection has no dense hfc-closed purely infinite simple unital subring. Finally, the hypotheses hold for proper subalgebras, for the gauge action of 𝕋 on a Cuntz algebra π’ͺn , the smooth subalgebra π’ͺn∞ is a proper dense hfc-closed subalgebra that is purely infinite simple as a ring.

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Cite

BibTeXRIS
BibTeX
@misc{sutradhar2026pure,
  title = {{The pure infiniteness transfer problem}},
  author = {Arindam Sutradhar},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.30906},
  eprint = {2608.30906},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30906},
}
RIS
TY  - UNPB
TI  - The pure infiniteness transfer problem
AU  - Arindam Sutradhar
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2608.30906
UR  - https://arxiv.org/abs/2608.30906
AB  - Problem 8.4 of Pino, Goodearl, Perera and Molina in [arxiv.org/abs/0806.4156] asks whether a dense subalgebra A_0 of a C^*-algebra A that is purely infinite as a ring forces A to be purely infinite as a C^*-algebra. We call this the pure infiniteness transfer problem, the transfer being from the ring to its C^*-completion. The problem is open, even when A_0 is unital and simple. We settle it under two hypotheses: A has real rank zero, and A_0 is closed under holomorphic functional calculus. Under these hypotheses A_0 is purely infinite simple as a ring if and only if A is purely infinite simple as a C^*-algebra. We also prove an obstruction: a unital C^*-algebra with a nonzero finite projection has no dense hfc-closed purely infinite simple unital subring. Finally, the hypotheses hold for proper subalgebras, for the gauge action of 𝕋 on a Cuntz algebra π’ͺ_n , the smooth subalgebra π’ͺ_n^∞ is a proper dense hfc-closed subalgebra that is purely infinite simple as a ring.
ER  -