Timeline for Completely equivalent operator norms on $*$-Banach algebras.
Current License: CC BY-SA 2.5
8 events
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| Aug 11, 2010 at 15:15 | vote | accept | Kolya Ivankov | ||
| Aug 11, 2010 at 7:19 | comment | added | Kolya Ivankov | I think my question was not posed well enough. It seems I need to improve my background provided with the new references and to pose the original question, from which this one seemed to arrive. Thank you for the comment! | |
| Aug 10, 2010 at 17:55 | comment | added | Yemon Choi | Could you please clarify what you mean by "an operator norm" and (as per the comments of Matthew Daws and the answer of Andreas Thom below) what operator space structure you are equipping $A_1$ and $A_2$ with? | |
| Aug 10, 2010 at 16:16 | comment | added | Kolya Ivankov | Thanks for the question! Well, presumably it is like this. The algebra $A$ is a dense subspace of a $C^*$-algebra. I have an $A$-valued inner product on $A^n$, defined in usual way, and this inner product defines a Banach norm on $A^n$ as $\|\xi\|_A=\|\langle \xi^*;\xi\rangle\|_A$. This norm, in turn, defines a norm on $M_n(A)$ as $\|(a_{ij})\|_A=\|sup_{\xi\in\mathcal{B}(A^n)}\|(a_{ij})\xi\|_A$ where $\mathcal{B}(A^n)$ is a unit ball. Hope I haven't written a complete nonsence. | |
| Aug 10, 2010 at 15:28 | answer | added | Andreas Thom | timeline score: 4 | |
| Aug 10, 2010 at 15:27 | comment | added | Matthew Daws | What norm do you want to put on the matrix algebras $M_n(A)$ and $M_n(B)$? If A is a C*-algebra than $M_n(A)$ is also a C*-algebra, and so this question doesn't arise. But it seems that for a general Banach *-algebra, there are loads of options for the norm on $M_n(A)$. | |
| Aug 10, 2010 at 13:10 | history | edited | Kolya Ivankov |
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| Aug 10, 2010 at 12:40 | history | asked | Kolya Ivankov | CC BY-SA 2.5 |