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Separable D-stability of the hyperfinite II1 factor

Determine whether the hyperfinite II1 factor R, regarded as a C*-algebra, is separably D-stable with respect to the universal UHF algebra (that is, whether every separable C*-subalgebra of R is contained in a separable C*-subalgebra of R that tensorially absorbs the universal UHF algebra).

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Background

The paper constructs an example showing that separable D-stability can fail to imply D-saturation of the corona in the non–sigma-unital case by considering the tracial ultrapower Rω of the hyperfinite II1 factor R and hereditary subalgebras therein. This motivates a related unresolved question about the D-stability of R itself when viewed as a C*-algebra.

Here D is taken to be the universal UHF algebra, a strongly self-absorbing C*-algebra. Separable D-stability means that every separable C*-subalgebra is contained in a separable D-stable subalgebra. The authors note that their results do not address whether R has this separable D-stability property.

References

In relation to this example, we ought to mention that it is not known whether R itself is (as a C*-algebra) separably D-stable. Our results do not shed light on this problem, however.

— Coronas and strongly self-absorbing C*-algebras (2411.02274 - Farah et al., 4 Nov 2024) in Example 3 (labelled ex:1-dim-corona), Section 3: On D-saturation of coronas