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

7. Borel complexity


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      Problem 7.05.

      Is conjugacy by automorphism of masas of the hyperfinite II_{1} factor classifiable by countable structures?


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          Problem 7.1.

          What is the right set-theoretic framework to deal with functorial classification?


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              Problem 7.15.

              Is there a constructive Borel proof of the \mathcal{O}_{2} embedding theorem?


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                  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?


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                      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?


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                          Problem 7.3.

                          Is there a Borel inverse of the classification functor?


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                              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?


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                                  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.


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                                          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.

                                              Add remark

                                              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.