# $\mathrm{\u25c7}$

###### Definition 1.

Let $S\mathrm{\subseteq}\kappa $ be a stationary set. Then the combinatorial principle ${\mathrm{\u25c7}}_{S}$ holds if and only if there is a sequence ${\mathrm{\u27e8}{A}_{\alpha}\mathrm{\u27e9}}_{\alpha \mathrm{\in}S}$ such that each ${A}_{\alpha}\mathrm{\subseteq}\alpha $ and for any $A\mathrm{\subseteq}\kappa $, $\mathrm{\{}\alpha \mathrm{\in}S\mathrm{\mid}A\mathrm{\cap}\alpha \mathrm{=}{A}_{\alpha}\mathrm{\}}$ is stationary.

To get some sense of what this means, observe that for any $$, $\{\lambda \}\subseteq \kappa $, so the set of ${A}_{\alpha}=\{\lambda \}$ is stationary (in $\kappa $). More strongly, suppose $\kappa >\lambda $. Then any subset of $T\subset \lambda $ is bounded in $\kappa $ so ${A}_{\alpha}=T$ on a stationary set. Since $|S|=\kappa $, it follows that ${2}^{\lambda}\le \kappa $. Hence ${\mathrm{\u25c7}}_{{\mathrm{\aleph}}_{1}}$, the most common form (often written as just $\mathrm{\u25c7}$), implies CH.

C. Akemann and N. Weaver used $\mathrm{\u25c7}$ to construct a ${C}^{*}$-algebra serving as a counterexample to Naimark’s problem.

## References

- 1 Akemann, C., and N. Weaver, Consistency of a counterexample to Naimark’s problem. Preprint available on the arXiv at http://arxiv.org/abs/math.OA/0312135http://arxiv.org/abs/math.OA/0312135.

Title | $\mathrm{\u25c7}$ |
---|---|

Canonical name | Diamond |

Date of creation | 2013-03-22 12:53:49 |

Last modified on | 2013-03-22 12:53:49 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 8 |

Author | Henry (455) |

Entry type | Definition |

Classification | msc 03E65 |

Synonym | diamond |

Related topic | Clubsuit |

Related topic | DiamondIsEquivalentToClubsuitAndContinuumHypothesis |

Related topic | ProofOfDiamondIsEquivalentToClubsuitAndContinuumHypothesis |

Related topic | CombinatorialPrinciple |

Defines | ${\mathrm{\u25c7}}_{S}$ |