Timeline for tensor product of the disc algebra with itself
Current License: CC BY-SA 3.0
5 events
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| Apr 26, 2014 at 19:45 | comment | added | Yemon Choi | Ali: 1) yes 2) don't know 3) yes, and for the disc algebra, the injective tensor product also gives you a Banach algebra. See Tomiyama's 1960 paper projecteuclid.org/euclid.tmj/1178244494 for more on tensor products of commutative Banach algebras | |
| Apr 26, 2014 at 19:34 | comment | added | Ali Taghavi | Dear @Yemon thank you very much. Your comment is very interesting. It is also strange because in this question I was (very indirectly), somehow motivated by the following question:mathoverflow.net/questions/164169/…. on the other hand your map is $\phi \otimes Id$ in the above question. First Does your statement a consequence of the fact that the ideal in $C^{*}$ algebras are $*$-closed? second: Do you think that the tube lemma is true forthe disc algebra? Is the projective tensor product a reasonable norm? Thanks again for interesting comment. | |
| Apr 26, 2014 at 19:16 | comment | added | Yemon Choi | The answer to your first question is no, because for any reasonable cross norm, the map $f\otimes g \to f(0)g$ will be a continuous algebra homomorphism of $A\otimes_\alpha A$ onto $A$. Banach algebra quotients of (algebras isomorphic to) abelian $C^*$-algebras are themselves abelian $C^*$-algebras, and $A$ is not isomorphic as a Banach algebra to any $C(X)$, so $A\otimes_\alpha A$ cannot be isomorphic as a Banach algebra to $C(X)$. The second question might be more interesting. | |
| Apr 26, 2014 at 17:25 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Apr 26, 2014 at 17:09 | history | asked | Ali Taghavi | CC BY-SA 3.0 |