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

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:

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.

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.

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:

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 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 fact strict? If so, what are the corresponding models of $\text{ZFC}$ proving these facts?

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.

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

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:

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.

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:

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 $\text{BDF}$?

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.

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:

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 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 fact strict? If so, what are the corresponding models of $\text{ZFC}$ proving these facts?

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.