Small masas of the Calkin algebra in the Cohen model

Abstract

We show that maximal abelian C*-subalgebras (masas) of the Calkin algebra (the algebra of all bounded operators on the separable Hilbert space modulo compact operators) may consistently have their densities strictly less than continuum and we describe many isomorphism types of such masas. Specifically, we prove that after adding any number of Cohen reals to a model of CH the algebra C(K_𝒜) of all complex-valued continuous functions on the Stone space K_𝒜 of a Boolean algebra 𝒜 of cardinality ω₁ is *-isomorphic to a masa of the Calkin algebra if and only if 𝒜 does not admit a countably generated ultrafilter. Moreover, for every such Boolean algebra we obtain ω₂ pairwise unitarily non-equivalent such masas, none of which has a commutative lift. We also show in ZFC that if a C*-algebra of the form C(K) for any compact Hausdorff K is *-isomorphic to a masa of the Calkin algebra, then no point of K may have character smaller than p. Therefore, consistently, there may not be any masa of the Calkin algebra of density less than continuum.

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BibTeX RIS

BibTeX

@misc{koszmider2026small,
  title = {{Small masas of the Calkin algebra in the Cohen model}},
  author = {Piotr Koszmider},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12916},
  eprint = {2609.12916},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12916},
}

RIS

TY  - UNPB
TI  - Small masas of the Calkin algebra in the Cohen model
AU  - Piotr Koszmider
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12916
UR  - https://arxiv.org/abs/2609.12916
AB  - We show that maximal abelian C*-subalgebras (masas) of the Calkin algebra (the algebra of all bounded operators on the separable Hilbert space modulo compact operators) may consistently have their densities strictly less than continuum and we describe many isomorphism types of such masas. Specifically, we prove that after adding any number of Cohen reals to a model of CH the algebra C(K_𝒜) of all complex-valued continuous functions on the Stone space K_𝒜 of a Boolean algebra 𝒜 of cardinality ω_1 is *-isomorphic to a masa of the Calkin algebra if and only if 𝒜 does not admit a countably generated ultrafilter. Moreover, for every such Boolean algebra we obtain ω_2 pairwise unitarily non-equivalent such masas, none of which has a commutative lift. We also show in ZFC that if a C*-algebra of the form C(K) for any compact Hausdorff K is *-isomorphic to a masa of the Calkin algebra, then no point of K may have character smaller than p. Therefore, consistently, there may not be any masa of the Calkin algebra of density less than continuum.
ER  -

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