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

6. Nonseparable C*-algebras


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      Problem 6.1.

      Is there a nonseparable AF C*-algebra or W*-algebra not isomorphic to its opposite algebra?


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          Problem 6.2.

          [Naimark] Is the following relatively consistent with ZFC?

          Every C*-algebra $A$ that has a unique irreducible representation up to the unitary equivalence is isomorphic to the algebra of compact operators on some Hilbert space.
              The statement is true for separable C*-algebras. The negation of the statement is relatively consistent with ZFC, since it follows from Jensen’s diamond principle.


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              Problem 6.3.

              [Farah-Katsura] Assume $A$ is a tensor product of algebras of the form $\mathbb{M}_n(\mathbb C)$, for $n\in \mathbb N$, $\kappa<\kappa'$ are cardinals and $\bigotimes_{\kappa'}\mathbb{M}_2(\mathbb C)$ unitally embeds into $A\otimes \bigotimes_{\kappa}\mathbb{M}_2(\mathbb C)$. Can we conclude that there is a unital embedding of $\bigotimes_\kappa \mathbb{M}_2(\mathbb C)$ into $A$?
                  Such $A$ would have to be nonseparable, and the simplest open case is whether $\bigotimes_{\aleph_1} \mathbb{M}_2(\mathbb C)$ unitally embeds into $\bigotimes_{\aleph_0}\mathbb{M}_2(\mathbb C)\otimes \bigotimes_{\aleph_1} \mathbb{M}_3(\mathbb C)$.


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                  Problem 6.4.

                  Is every exact C*-algebra a subalgebra of a nuclear C*-algebra?
                      The answer is positive for separable algebras since every separable exact C*-algebra is a subalgebra of $\mathcal O_2$.


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                      Problem 6.5.

                      Is there a universal nuclear C*-algebra of character density $\aleph_1$? More generally, for which cardinals $\kappa$ is there a universal nuclear C*-algebra of character density $\kappa$? Similar question can be asked for exact algebras.


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                          Problem 6.6.

                          [Phillips] Suppose a C*-algebra has an approximate identity consisting of projections. Does it have an increasing approximate identity (on some index set, possibly different) consisting of projections?
                              The answer is positive in the separable case.

                              Cite this as: AimPL: Set theory and C* algebras, available at http://aimpl.org/settheorycstar.