7. Borel complexity

Problem 7.05.
Is conjugacy by automorphism of masas of the hyperfinite II$_{1}$ factor classifiable by countable structures? 
Problem 7.1.
What is the right settheoretic framework to deal with functorial classification? 
Problem 7.15.
Is there a constructive Borel proof of the $\mathcal{O}_{2}$ embedding theorem? 
Problem 7.2.
Is the isomorphism relation in the following classes of separable C*algebras: all
 nuclear
 exact
 simple
 simple exact
 nuclear $\mathcal{Z}$stable
 simple nuclear
complete analytic? below a group actions? above all group actions?  all

Problem 7.25.
What is the complexity of isometric isomorphism of direct limits of not necessarily selfadjoint subalgebras of finite dimensional C*algebras? How does it compare to the complexity of isomorphism of AF algebras? 
Problem 7.35.
For a separable C*algebra $A$, what is the complexity of orbit equivalence relations associated to the action $\mathrm{Aut}\left( A\right) $ on itself by conjugacy and to the action of $\mathrm{Inn}\left( A\right) $ on $\mathrm{\mathrm{Au}t}\left( A\right) $ by left translation? Are these action turbulent? How are they related to structural properties of $A$? 
Problem 7.4.
[Diximier] Is the Mackey Borel structure on the spectrum of a simple separable C*algebra always the same when it is not standard?
Remark. [Elliott] All nonstandard spectra of AF algebras are isomorphic.


Problem 7.45.
Does the complexity of the Mackey Borel structure of a simple separable C*algebra increase as one goes from nuclear C*algebras to exact ones to ones that are not even exact? 
If A is a C*algebra, denote by $E_A$ the relation of unitary equivalence of pure states of A.
Problem 7.5.
Assume $A$ and $B$ are C*algebras and $E_A$ is Borelreducible to $E_B$. What does this fact imply about the relation between $A$ and $B$?
Cite this as: AimPL: Set theory and C* algebras, available at http://aimpl.org/settheorycstar.