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I have always been happy with the proof of the functoriality of the Cuntz semigroup $\mathsf{Cu}$ given in arXiv:0902.3381, where the isomorphism $$\mathsf{Cu}(A)\cong W(A\otimes K)$$ is used, $A$ being any C*-algebra, and the functor can be though of as going from the category of C*-algebras to that of partially ordered monoids. I went back to the paper arXiv:0705.0341, where the stabilised functor $\mathsf{Cu}$ was introduced because I wanted to see how the functoriality of this map was proven within their approach. I might be overlooking something, for the argument that is given there does not seem complete to me.

To abstract a bit, consider a set $S_A$, depending on a C*-algebra $A$, where a relation $\ll$ and an equivalence $\cong$ are defined. We say that $s,t\in S$ are such that $s\precsim t$ if

$$\forall s'\ll s\qquad\exists t'\ll t\qquad |\qquad s'\cong t'.$$

Denote by $\sim$ the antisymmetrisation of $\precsim$, that is say that $s\sim t$ if $s\precsim t$ and $t\precsim s$.

The functor $\mathsf{Cu}$ sends a $*$-homomorphism $f:A\to B$ between C*-algebras to a map $\mathsf{Cu} : \mathsf{Cu}(A) \to \mathsf{Cu}(B)$, where $\mathsf{Cu}(A)$ and $\mathsf{Cu}(B)$ arise as sets of equivalence classes of objects from a set of the kind above w.r.t. the relation $\sim$ (countably generated Hilbert $A$- and $B$-modules in this case). Between $f$ and $\mathsf{Cu}(f)$ there is an intermediate map $f_*:S_A\to S_B$ that is then pushed towards classes to yield $\mathsf{Cu}(f)$.

Now it seems to me that the argument in the second cited paper goes like this:

  • prove that $f_*(a)\in S_B$ for any $a\in S_A$;
  • prove that $f_*$ preserves the relation $\ll$;
  • prove that $f_*$ preserves the relation $\cong$;
  • conclude that $a\precsim b\ \Rightarrow\ f_*(a)\precsim f_*(b)$.

But in order to reach the conclusion of the last point using the results obtained at the previous ones, I believe that one has to make sure that

$$f_*(a)^\ll \subset f_*(a^\ll)$$

(and that, in fact, they are the same set), where

$$a^\ll = \{b \in S_A\ |\ b\ll a\}.$$

However, as far as I can tell, this is not stated anywhere in the second cited paper, nor it is clear to me that this is the case in general. Rather than modules though, I was considering the parallel situation one has in the open projections picture of the Cuntz semigroup described in arXiv:1008.3497, the results of which allow to recycle all the symbolism above with obvious meaning. From this approach, it seems to me that this can be related to whether the image under a $*$-homomorphism of an hereditary C*-subalgebra is still hereditary.

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  • $\begingroup$ You need to check that $f_*$ preserves some sort of suprema. There is some flexibility in how to do this. Say you already know that for each $s\in S_A$ there exists an increasing sequence $s_1\ll s_2\ll s_3\cdots$ with supremum $s$. (In the picture with countably generated Hilbert modules it is clear how to do this.) Check now that the $f_*(s_i)$s have supremum $f_*(s)$. (In the modules picture, that the union of the modules $f_*(s_i)$ is dense in $f_*(s)$). Now you get preservation of $\precsim$. $\endgroup$
    – Leonel Robert
    Commented Dec 10, 2015 at 14:48
  • $\begingroup$ @LeonelRobert Many thanks for your comment. The method of proof you suggest seems to involve the knowledge of extra structure of the objects of the category $\mathrm{Cu}$ (e.g. every element is the sup of a rapidly increasing sequence). However here I am interested in showing that $\mathrm{Cu}$ is a functor from the category of C*-algebras to that of positively ordered monoids (perhaps I should have specified this in the question), without any further axioms on the target category. This should be possible, since it is so for $W$. $\endgroup$
    – Phoenix87
    Commented Dec 10, 2015 at 22:29

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