# Do extensions of pure states separate points?

Let $$B$$ be a unital C*-algebra and let $$A⊆B$$ be a closed *-subalgebra containing the unit of $$B$$. I am mostly interested in the case that $$A$$ is abelian but, for the strict purpose of stating my question, this does not seem to matter much.

Let us say that the inclusion "$$A⊆B$$" has property (EPS) (for Extended Pure states Separate) provided the set of all state extensions of pure states on $$A$$ separates points of $$B$$ in the sense that, if $$b∈ B$$, and $$\psi(b^*b)=0$$, for all such states $$\psi$$, then $$b=0$$.

An example failing this property is $$A=C([0,1])$$ (represented on $$L^2[0, 1]$$ as multiplication operators), and $$B=A+K$$ (compact operators), but I suspect the failure is due to the fact that $$A$$ is not a regular subalgebra of $$B$$ (normalizers, in the sense of Kumjian, do not span $$B$$). Nevertheless there are regular counter-examples as well.

For obvious reasons any inclusion of the form "$$ℂ⊆B$$" satisfies (EPS) and so does "$$ℂ^n⊆B$$".

Other situations in which (EPS) holds are:

• $$B$$ is abelian,
• $$A$$ is abelian, $$B=A\rtimes G$$, and $$G$$ is amenable.

My questions are:

Questions. Has property (EPS) been studied before? Can it be characterized in some sensible way?

I suspect that this might have something to do with nuclearity, so here is another:

Question. If $$A$$ is abelian and regular and $$B$$ is nuclear, can one show that (EPS) holds?

• Is the paper " Extensions of Pure states and projections of norm one" by R. Archbold relevant? Oct 12, 2021 at 21:47
• Archbold's paper is definitely in the same ball park but I can't find any direct application of the results in that paper to my question, mainly due to their strong hypotheses. Nevertheless there are many ideas stemming from there which I'm going to try to pursue. Thanks very much for suggesting it!
– Ruy
Oct 13, 2021 at 17:27

Thanks to user @DarthVader I was led to consider a special case of my question, namely when every pure state of $$A$$ admits a unique extension to a state on $$B$$, and under this hypothesis I was able to answer my last question affirmatively: when $$B$$ is nuclear then the property (EPS) does hold. See https://arxiv.org/abs/2110.09445