6. Nonseparable C*algebras

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.
[FarahKatsura] 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.