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\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

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.