The [Calkin algebra](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Open_coloring_axiom) ($\text{OCA}$) (which itself is a combinatorial consequence of [Proper Forcing Axiom](https://en.wikipedia.org/wiki/Proper_forcing_axiom) ($\text{PFA}$)) implies $\text{BDF}$.


As a summary of what is already known:



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



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> **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 $\text{BDF}$? 
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> **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?
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**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](https://zbmath.org/?q=an:0348.46050).
2. *Phillips, N. Christopher; Weaver, Nik*, [**The Calkin algebra has outer automorphisms**](http://dx.doi.org/10.1215/S0012-7094-07-13915-2), Duke Math. J. 139, No. 1, 185-202 (2007). [ZBL1220.46040](https://zbmath.org/?q=an:1220.46040).
3. *Farah, Ilijas*, [**All automorphisms of the Calkin algebra are inner**](http://dx.doi.org/10.4007/annals.2011.173.2.1), Ann. Math. (2) 173, No. 2, 619-661 (2011). [ZBL1250.03094](https://zbmath.org/?q=an:1250.03094).



What is the consistency strength of non-existence of outer automorphisms of Calkin algebra?
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](https://books.google.com/books?id=mPLpBwAAQBAJ&pg=PA142&lpg=PA142&dq=consistency+strength+of+OCA&source=bl&ots=13EWpwogET&sig=rQbd_HV3rd8HFx6Gb4B3EVk-0fI&hl=en&sa=X&ved=2ahUKEwjTktDV0cLcAhVp04MKHRv7CBAQ6AEwBnoECAYQAQ#v=onepage&q=consistency%20strength%20of%20OCA&f=false). 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).


