The [Calkin algebra][4] $C(H)$ 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][5] ($\text{OCA}$) (which itself is a combinatorial consequence of [Proper Forcing Axiom][6] ($\text{PFA}$)) implies $\text{BDF}$.

As a summary of what is already known:

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

But this chain of theorems says nothing about the possible *consistency strength* of $\text{BDF}$. All what we currently know is that $\text{BDF}$ is independent of $\text{ZFC}$ and follows from a forcing axiom of high large cardinal strength such as $\text{PFA}$. So here is the question:

> **Question.** 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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**References.**

1. <cite authors="Douglas, R. G.">_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).</cite>

2. <cite authors="Phillips, N. Christopher; Weaver, Nik">_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).</cite>

3. <cite authors="Farah, Ilijas">_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).</cite>

  [4]: https://en.wikipedia.org/wiki/Calkin_algebra
  [5]: https://en.wikipedia.org/wiki/Open_coloring_axiom
  [6]: https://en.wikipedia.org/wiki/Proper_forcing_axiom