Timeline for Complete regularity in C*-algebras
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Oct 29, 2015 at 10:19 | comment | added | Tristan Bice | True, I just thought the other terms should be explained too. Incidentally, one might hope that a truly non-commutative Urysohn lemma would apply to projections that do not commute, leading me to ask this question. | |
| Oct 26, 2015 at 3:13 | comment | added | Nik Weaver | I think you can just say that the pure states correspond to the minimal projections in $A^{**}$. | |
| Oct 25, 2015 at 15:28 | comment | added | Tristan Bice | Yes, there is a relation, specifically the map taking projections p in A** to the set of states Φ with Φ(p)=0 is a bijection between open projections and weak*-closed faces of the state space (each of which is the closed convex hull of its extreme points, i.e. pure states). A projection p in A** is closed iff 1-p is open and compact iff there is also some positive a in A with p<=a (so if A is unital closed=compact). In particular, pure states correspond to minimal compact projections. See Pedersen's "C*-Algebras and their Automorphism Groups" or the papers by Akemann et al mentioned there. | |
| Oct 25, 2015 at 14:57 | comment | added | Onion Dip Carlip | Thanks a lot Nik. This is helpful and I will most definitely look into this. But I was looking more at something that uses pure states. Is there some obvious relation between pure states and projections in $A^{**}$ that I am missing? | |
| Sep 4, 2015 at 18:37 | history | answered | Nik Weaver | CC BY-SA 3.0 |