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 Dec 10 revised Functoriality of $\mathsf{Cu}$ added 113 characters in body Dec 10 comment Functoriality of $\mathsf{Cu}$ @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 $W$. Dec 9 asked Functoriality of $\mathsf{Cu}$ Oct 23 comment Closed containment of open projections in C*-algebras I haven't put too much thought into this at the minute, but consider $p$ a rank one projection in $M_2$, e.g. $p=e_{11}$, and $q$ the unit of $M_2$. Then $p\leq q$ and for $r$ you can take the rank one projection generated by the vector $(\cos\theta,\sin\theta)$. Then $r\wedge p = 0$ unless $\theta = 0,\pi$, and $r\vee q = 1$. However $rp\neq 0$ unless $\theta = \pi/2,\ldots$. Aug 3 awarded Yearling Aug 1 answered examples of completely positive order zero maps to demonstrate a theorem Aug 1 awarded Citizen Patrol Jul 18 comment Cuntz comparison of strictly positive elements in finite C*-algebras Ah right, thank you Aaron! Indeed I have considered the compacts but I failed to realise it was a counterexample before your comment. Jul 18 asked Cuntz comparison of strictly positive elements in finite C*-algebras Jun 28 comment Extending GUE to a measure on operators? pre-kidney indeed Jun 27 comment Extending GUE to a measure on operators? I'm not sure but wouldn't that measure pick up only the zero matrix? May 6 comment Do c.p.c. order zero maps between local C*-algebras map C*-subalgebras to C*-subalgebras? Ah that's right, I forgot to try and play with this one-to-one correspondence around! Many thanks! May 5 accepted Do c.p.c. order zero maps between local C*-algebras map C*-subalgebras to C*-subalgebras? May 5 comment Do c.p.c. order zero maps between local C*-algebras map C*-subalgebras to C*-subalgebras? Also, about the second part of your answer, is there something already known in that direction for c.p.c. order zero maps between C*-algebras? For example, if $a\in A^+$, then $\phi(C^*(a))\subset C^*(h_\phi,\pi_\phi(a))\cap B$, but are there elements $g_1,\ldots, g_n\in B$ such that $\phi(C^*(a))\subset C^*(g_1,\ldots,g_n)$? May 4 comment Do c.p.c. order zero maps between local C*-algebras map C*-subalgebras to C*-subalgebras? Thanks for the clarification, I think I can now see why this is the case (just considering real functions on $X$). However your example shows that there might be a suitable choice of exhaustive families for which the property in the OP holds. Of course, for any c.p.c. order zero map $\phi:A\to B$ between C*-algebras one has that $\phi(A)$ is contained in a sub-C*-algebra of $B$ (which is actually the property I'm trying to get to when $A$ and $B$ are local C*-algebras). May 4 comment Do c.p.c. order zero maps between local C*-algebras map C*-subalgebras to C*-subalgebras? Hi Hannes. Many thanks for your answer. In your example I can see how you can get $A$ from the completion of the union of all the finitely generated C*-subalgebras of $A$, but it seems to me that you are somehow implying that one doesn't need to take the norm-completion, and this is not clear to me at the moment. Perhaps what's misleading me is that I'm thinking of $C(X)$ as the infinite tensor product $C(I)^{\otimes\infty}$ with $I=[0,1]$, seen as an inductive limit with the obvious connecting maps. Apr 28 accepted When are countably generated Hilbert modules generated by c.p.c. order zero maps? Apr 20 comment When are countably generated Hilbert modules generated by c.p.c. order zero maps? OK many thanks. That was really helpful! Apr 18 comment When are countably generated Hilbert modules generated by c.p.c. order zero maps? Perhaps I'm overlooking something, but if there are no non-trivial c.p.c. order zero maps between $A\otimes\mathcal K$ and $B$ then the only sequence one can construct out of a countable family of c.p.c. order zero maps is the constant sequence given by the trivial module, which has limit in the trivial module itself. Apr 17 comment When are countably generated Hilbert modules generated by c.p.c. order zero maps? Many thanks for your answer! I was wondering if the case of $A=\mathcal K$ generalises to any stable C*-algebra just by considering $E_A:=\overline{\phi(A\otimes e)B}$, where $e\in\mathcal K$ is any minimal projection; and if there is the possibility of explicitly constructing the c.p.c. order zero map associated to the limit. I was thinking along the lines of a representation of $A$ on $\ell^2(\mathbb N)$ tensor with some positive element in $I$, but I'm not sure to what extent this intuition is correct.