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? -
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. -
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? -
Problem 6.4.
Is every exact C*-algebra a subalgebra of a nuclear C*-algebra? -
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. -
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?
Cite this as: AimPL: Set theory and C* algebras, available at http://aimpl.org/settheorycstar.