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The Calkin algebra $C(H)$ is the quotient of $B(H)$, the ring of bounded linear operators on a separable infinite-dimensional Hilbert space $H$, by the ideal $K(H)$ of compact operators.

In 1977, Brown, Douglas, and Fillmore [1] asked whether it is possible for a Calkin algebra to have no outer automorphism. The statement that we denote as $\text{BDF}$.

In 2007, Phillips and Weaver [2] proved that assuming the Continuum Hypothesis, $\text{BDF}$ fails badly and in fact, there are $2^{\aleph_{1}}$ outer automorphisms of $C(H)$.

In 2011, Farah [3] completed this independence result by focusing on the other direction. He proved that Todorčević's Open Coloring Axiom ($\text{OCA}$) (which itself is a combinatorial consequence of Proper Forcing Axiom ($\text{PFA}$)) implies $\text{BDF}$.

As a summary of what is already known:

Theorem. $\text{PFA}\Rightarrow \text{OCA}\Rightarrow \text{BDF}\Rightarrow \neg \text{CH}$

So we currently know that $\text{BDF}$ is independent of $\text{ZFC}$ and follows from a forcing axiom of high large cardinal strength such as $\text{PFA}$.

My questions are about the possible consistency strength of $\text{BDF}$ as well as the strictness of the arrows in the presented theorem:

Question 1. What is the exact consistency strength of $\text{BDF}$, the assertion that "all automorphisms of Calkin algebra are inner"? Is there any known lower bound for the consistency strength of $\text{BDF}$?

Question 2. Are all the arrows in the above theorem strict? If so, what are the corresponding models of $\text {ZFC}$ proving that $\neg\text {CH}+\neg\text{BDF}$, $\text {BDF}+\neg\text{OCA}$, and $\text {OCA}+\neg\text{PFA}$ are consistent?


References.

  1. Douglas, R. G., Extensions of $C^*$-algebras and K-homology, $K$-Theory Oper. Algebr., Proc. Conf. Athens/Georgia 1975, Lect. Notes Math. 575, 44-52 (1977). ZBL0348.46050.

  2. Phillips, N. Christopher; Weaver, Nik, The Calkin algebra has outer automorphisms, Duke Math. J. 139, No. 1, 185-202 (2007). ZBL1220.46040.

  3. Farah, Ilijas, All automorphisms of the Calkin algebra are inner, Ann. Math. (2) 173, No. 2, 619-661 (2011). ZBL1250.03094.

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    $\begingroup$ "But this chain of theorems says nothing about the possible consistency strength of BDF". Not at all, the chain actually answers your question, we know exactly what its consistency strength is. $\endgroup$ – Andrés E. Caicedo Jul 28 '18 at 20:15
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Perhaps surprisingly, the Open Coloring Axiom (even OCA + MA$_{\aleph_1}$) has no additional consistency strength. The situation is described in Velickovic's paper "Applications of the open coloring axiom," and you can see the relevant bit here. So sine OCA implies BDF, the answer to your question is "none."

I believe, meanwhile, that most of the "natural" principles which outright imply OCA do have nontrivial consistency strength. This situation happens occasionally; e.g. "Every projective set of countable ordinals contains or is disjoint from a club" isn't known to me to be implied by any mild large cardinals, but its consistency strength is at worst an ineffable (this was proved by Harrington).

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  • $\begingroup$ (+1) Interesting, Noah! Thanks for pointing out to this fact! Indeed, it is surprising that OCA has no consistency strength! By the way, due to its Ramsey-like formulation my initial guess was that it might be as strong as some Ramsey type large cardinals in the hierarchy at least! (e.g. Ramsey cardinals or weakly compacts) $\endgroup$ – Morteza Azad Jul 29 '18 at 3:21

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