Timeline for Stabilization in Banach algebras
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 1, 2014 at 12:53 | comment | added | Rasmus | With regard to the first paragraph it is perhaps worthwile to point out that strongly self-absorbing $C^*$-algebras need not have the same $K$-theory as $\mathbb C$. | |
| Jan 17, 2013 at 18:26 | comment | added | Ulrich Pennig | Oops, I just realized my last comment was mentioned above already. | |
| Jan 17, 2013 at 18:22 | comment | added | Ulrich Pennig | Stabilization with K is also interesting from the point of view of Hilbert bimodules, since the Brown-Green-Rieffel theorem says that two C*-algebras with countable approximate identities are strongly Morita equivalent if and only if they are stably isomorphic. Apart from that we have an exact sequence $0 \to Inn(A \otimes \mathbb{K}) \to Aut(A \otimes \mathbb{K}) \to Pic(A \otimes \mathbb{K}) \to 0$. | |
| Jan 17, 2013 at 17:05 | comment | added | Leonel Robert | Yemon, one gains, for example idempotents. The group $K_0(A)$ can be constructed looking at the projections of $A\otimes K$ (looking at $\bigcup_n M_n(A)$ also works here, but passing to the completion may be more appealing at times). One also looses after tensoring with $K$ (which can be a good thing). E.g., Brown's theorem says that if $a,b\in A$ are positive then thr C*-algebras $\overline{aAa}$ and $\overline{bAb}$ become isomorphic after tensoring with $K$ if and only if $a$ and $b$ generate the same closed two-sided ideal. | |
| Jan 16, 2013 at 19:38 | comment | added | Yemon Choi | Out of curiosity, Ulrich: *what is it that one gains by tensoring a given $C^\ast$-algebra with $K(H)$? (not a rhetorical question; I am genuinely ignorant of, or have forgotten, the reason one does this). | |
| Jan 15, 2013 at 12:44 | comment | added | user23860 | Thanks Ulrich. I guess one can choose is $K(H)$ for stabilization of Banach algebras too. But I am not sure if this gives rise to similar theorems about Morita equivalence and $K$-theory of Banach algebras. So, it seems there is no obvious choose "known yet" and it is open for more investigations. | |
| Jan 15, 2013 at 12:36 | history | answered | Ulrich Pennig | CC BY-SA 3.0 |