2. Calkin algebra

Problem 2.1.
Is there a model of set theory in which the Calkin algebra has an automorphism sending the unilateral shift $S$ to $S^{\ast }$? Is the analogous fact for $\ell ^{\infty }\left/ c_{0}\right. $ true? 
Problem 2.2.
Is there a model of set theory in which the Calkin algebra has a not approximately inner automorphism preserving Ktheory? 
Problem 2.3.
Is it consistent that, for a nonseparable Hilbert space $\mathcal{H}$, $\mathcal{B}\left( \mathcal{H}\right) \left/ \mathcal{K}\left( \mathcal{H}% \right) \right. $ has an outer automorphism? 
Problem 2.4.
[Anderson] Does every masa in the Calkin algebra that is generated by its projections lift to a masa in $\mathcal{B}\left( H\right) $?
Cite this as: AimPL: Set theory and C* algebras, available at http://aimpl.org/settheorycstar.