Timeline for Is it true that $A$ is Morita equivalent with $M_I(A)$ [closed]
Current License: CC BY-SA 3.0
24 events
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| Jan 16, 2018 at 4:57 | comment | added | fereidoun | I am so sorry about that. you are right, but as your first comment, to prove Morita equivalence of two Banach algebras, we must first prove that they are self-induced, so I guessed that the above identification must first be prove. Anyway, thank you so much for your remarks. | |
| Jan 15, 2018 at 17:57 | history | rollback | Yemon Choi |
Rollback to Revision 4
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| Jan 15, 2018 at 17:57 | comment | added | Yemon Choi | You have now changed the question completely, this is not fair on the previous commenters. Please ask your new question as a separate question. I am reverting this to the original version, since your original question about Morita equivalence does not seem to be directly related to your second question about being self-induced | |
| Jan 15, 2018 at 12:48 | review | Reopen votes | |||
| Jan 15, 2018 at 17:59 | |||||
| Jan 15, 2018 at 12:28 | history | edited | fereidoun | CC BY-SA 3.0 |
added 558 characters in body; edited title
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| Jan 15, 2018 at 9:22 | history | closed |
abx Jeremy Rickard Michael Albanese David Handelman coudy |
Needs details or clarity | |
| Jan 14, 2018 at 21:37 | comment | added | Yemon Choi | @fereidoun also, please explain to us what you meant when you said "Morita equivalence", because it was not clear from the formulation of your question where you had seen this definition and what you think it means | |
| Jan 14, 2018 at 21:36 | comment | added | Yemon Choi | @fereidoun, please edit your question so that it contains the correct definition. Incidentally "$\ell_1$-Munn Banach space" is bad terminology in my view, and I wish people would not use it. | |
| Jan 14, 2018 at 10:36 | comment | added | fereidoun | Let A be a unital Banach algebra, and $I$ be an arbitrary index set, and let $M_ I (A)$ be the vector space of all $I × I$-matrices $A=(A_i j)$ over $A$ such that $\norm A =\Sigma_i,j\in I \norm A_i j $. Then, it is easy to check that $M_I (A)$ with scaler multiplication, matrix addition, and the above norm is a Banach space. This Banach space is called $\ell_1-Munn$ Banach space over A. If $M_I (A)$ is a Banach algebra, by product as above ($P$ be the $I*I$ matrix with $a_i j=0$ and $a_i i=1_A$), then $M_I (A)$ is called the $\ell_1-Munn$ Banach algebra over $A$ with index set I. | |
| Jan 13, 2018 at 18:40 | review | Close votes | |||
| Jan 15, 2018 at 9:22 | |||||
| Jan 13, 2018 at 18:35 | comment | added | Yemon Choi | @Mare I think it is up to the OP to clarify what he means, since the question seems currently ill-posed | |
| Jan 13, 2018 at 17:31 | comment | added | Benjamin Steinberg | If P is invertible then your Munn algebra is isomorphic to a usual matrix algebras on I. | |
| Jan 13, 2018 at 17:15 | history | edited | YCor |
edited tags
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| Jan 13, 2018 at 14:28 | comment | added | Mare | I did not know about other notions of Morita equivalence or the definition of the multiplication when I answered. So if necessary I will delete my answer in case it was off topic. | |
| Jan 13, 2018 at 14:24 | comment | added | Yemon Choi | As @QiaochuYuan has pointed out, in order for $aPb$ to be well-defined, you must specify some extra conditions on $M_I(A)$, such as convergence with respect to some norm. I have downvoted the question until this is clarified. | |
| Jan 13, 2018 at 14:23 | history | edited | Yemon Choi |
added the BA tag since I expect the OP is thinking about a BA notion of Morita equivalence
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| Jan 13, 2018 at 14:22 | comment | added | Yemon Choi | If you are thinking of what some people like to call Munn algebras, which in turn are motivated by convolution algebras of Rees semigroups, then there was some discussion in a paper of Gourdeau, Gronbaek and White, Studia Mathematica, published between 2005 to 2010 I think? | |
| Jan 13, 2018 at 14:20 | comment | added | Yemon Choi | It is not entirely clear, in the world of Banach algebras, what the "correct" notion of Morita equivalence should be. One possibility was investigated by Niels Gronbaek in a paper in the 1990s in JPAA (I don't have the details to hand right now) | |
| S Jan 13, 2018 at 13:49 | history | suggested | fereidoun | CC BY-SA 3.0 |
the product of $M_I(A)$
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| Jan 13, 2018 at 13:12 | review | Suggested edits | |||
| S Jan 13, 2018 at 13:49 | |||||
| Jan 13, 2018 at 11:47 | comment | added | Qiaochu Yuan | How is $M_I(A)$ a Banach algebra if $I$ is infinite? How do you define multiplication? | |
| Jan 13, 2018 at 9:51 | answer | added | Mare | timeline score: 7 | |
| Jan 13, 2018 at 9:41 | review | First posts | |||
| Jan 13, 2018 at 9:49 | |||||
| Jan 13, 2018 at 9:41 | history | asked | fereidoun | CC BY-SA 3.0 |