# $\Diamond$

###### Definition 1.

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

To get some sense of what this means, observe that for any $\lambda<\kappa$, $\{\lambda\}\subseteq\kappa$, 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 $2^{\lambda}\leq\kappa$. Hence $\Diamond_{\aleph_{1}}$, the most common form (often written as just $\Diamond$), implies CH.

C. Akemann and N. Weaver used $\Diamond$ 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 $\Diamond$ Diamond 2013-03-22 12:53:49 2013-03-22 12:53:49 Henry (455) Henry (455) 8 Henry (455) Definition msc 03E65 diamond Clubsuit DiamondIsEquivalentToClubsuitAndContinuumHypothesis ProofOfDiamondIsEquivalentToClubsuitAndContinuumHypothesis CombinatorialPrinciple $\Diamond_{S}$