
## 7. Borel complexity

1. #### 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 set-theoretic 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?
• #### Problem 7.25.

What is the complexity of isometric isomorphism of direct limits of not necessarily self-adjoint subalgebras of finite dimensional C*-algebras? How does it compare to the complexity of isomorphism of AF algebras?
• #### Problem 7.3.

Is there a Borel inverse of the classification functor?
• #### 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?
1. 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 Borel-reducible 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.