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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. |