Timeline for Stabilization in Banach algebras
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 31, 2014 at 18:00 | answer | added | Eusebio Gardella | timeline score: 2 | |
| Jan 25, 2013 at 20:58 | comment | added | user23860 | @Yemon: You are welcome. Following Vincent Lafforgue's works, Walter Paravicini has studied Morita equivalence of Banach algebras too. | |
| Jan 25, 2013 at 19:34 | comment | added | Yemon Choi | Thanks Vahid - the extra information is helpful. I am rather busy right now but will have a look in Gronbaek's papers to see if they give any indication of how to approach 2. | |
| Jan 17, 2013 at 5:58 | comment | added | user23860 | @Yemon: I list some instances that clarify the importance of stabilization in $C^*$-algebras: 1. Both K-theory and KK-theory are stable functors meaning $K(A)\simeq K(A\otimes K(H))$. 2. Two separable $C^*$-algebras $A$ and $B$ are Morita equivalent if and only if they are stably isomorphic, i.e. $A\otimes K(H)\simeq B\otimes K(H)$. 3. Tensoring by $K(L^2(G))$ also appears in some theorems too, for example see Takai-Takesaki duality. So, it is nice to have a similar notion in Banach algebras, for instance proving item 2 for Banach algebras would be a good start. | |
| Jan 16, 2013 at 19:37 | comment | added | Yemon Choi | @Vahid: not being a C-star algebraist, nor a (NC) geometer, I'm afraid I still don't understand what is meant to be special about tensoring with $K(H)$. I mean, why not tensor with $C$? What is it one gains in the C-star world by tensoring with $K(H)$? Is it important to you that the stabilization of the stabilization of A is the stabilization of A? It would help Banach-algebra people like myself if you could add some more specific sub-questions or requirements to your original question. | |
| Jan 16, 2013 at 5:04 | comment | added | user23860 | @Yemon: I explained my ideas in the above. | |
| Jan 15, 2013 at 18:59 | comment | added | Yemon Choi | Regarding Morita equivalence for Banach algebras, I suggest you look up work by Niels Gronbaek (JPAA I think) | |
| Jan 15, 2013 at 18:58 | comment | added | Yemon Choi | Vahid, you still haven't told us what you want stabilization to do. I mean I can always tensor a given Banach algebra A with a fixed Banach algebra B. What is meant to be special or useful about the result? | |
| Jan 15, 2013 at 18:05 | comment | added | user23860 | @Alain: It is a good point. I guess choosing the type of the tensor product should be part of the stabilization process too. Do you have any suggestion? | |
| Jan 15, 2013 at 17:58 | comment | added | Alain Valette | Is it clear which tensor product should be used for Banach algebras? (For $C^*$-algebras, one secretly enjoys nuclearity of $K(H)$...) | |
| Jan 15, 2013 at 12:36 | answer | added | Ulrich Pennig | timeline score: 2 | |
| Jan 15, 2013 at 0:14 | comment | added | user23860 | After being able to define (an appropriate notion for) the stabilization of a Banach algebra, say $A$, I'd like to see if it is Morita equivalent with $A$. Of course, the equality of $K$-groups is the next. And so on. Stabilization of $C^\ast$-algebras is a elementary notion, so I thought, maybe there is a similar notion for Banach algebras too. That's why I asked this question. | |
| Jan 14, 2013 at 23:58 | comment | added | Yemon Choi | @Z254R, are you pointing out deficiencies in a question? ;-) Well in any case I agree completely with you. @Vahid, what do you want a stabilization functor to do? Do you have in mind K-theory? | |
| Jan 14, 2013 at 23:41 | history | asked | user23860 | CC BY-SA 3.0 |