Timeline for Injectivity for bimodules and Hochschild cohomology
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 24, 2013 at 22:52 | comment | added | Richard Jennings | I'd heard about (regular?) Hochschild cohomology but not continuous Hochschild cohomology, what's the difference? Anyone know? | |
| Oct 25, 2010 at 1:52 | vote | accept | CommunityBot | ||
| Oct 25, 2010 at 1:25 | answer | added | Yemon Choi | timeline score: 4 | |
| Oct 23, 2010 at 14:24 | comment | added | user2412 | @Yemon: you should definitely post it as an answer. | |
| Oct 21, 2010 at 21:22 | comment | added | Yemon Choi | @Piotr: That is correct (I am assuming that A is unital, which if you are working with algebras arising from discrete groups, is usually the case). In practice, what one needs to show is that the natural embedding from $J: X \to {\mathcal L}(A\hat\otimes A, X)$, defined by $J(x)(a\otimes b) = axb$, has a (continuous, linear) left inverse which is an $A$-bimodule map -- this is equivalent to being A-bi-injective, at least in the unital case. | |
| Oct 21, 2010 at 19:59 | comment | added | user2412 | @Yemon: Thanks! So, to summarize I take an $A$-bimodule $X$ and view it as a left $A^{\epsilon}$-module. Then if this left $A$-module is injective as a left module then $\mathcal{H}^n(A,X)=0$? If so, this answers the question. In that paper you are referring to the point of view was essentially that of bounded cohomology of groups and the translation to Banach algebras is not straightforward at all. At least that is the motivation for my question. | |
| Oct 21, 2010 at 19:23 | comment | added | Yemon Choi | @Matt: Because I seem to remember Piotr has written a paper using (or referring to) this kind of stuff, and it is all in Helemskii's pink book (HBTA). @Piotr: $A^e$ is the enveloping algebra of A, this is A proj. tensor A equipped with product $(a\otimes b)(c\otimes d) = (ac\otimes db)$, and its main feature is that every $A$-bimodule can be regarded as a left $A^e$-bimodule, and conversely. | |
| Oct 21, 2010 at 18:51 | comment | added | user2412 | @Yemon: what do you mean by $A^{\epsilon}$? Is it the unitization of $A$? | |
| Oct 21, 2010 at 18:25 | comment | added | Matthew Daws | @Yemon: why not write that up into an answer?? | |
| Oct 21, 2010 at 18:18 | comment | added | Yemon Choi | I must be misunderstanding something. Doesn't relative $A$-bi-injectivity of $X$ (a.k.a. relative injectivity as an $A^e$-module) do the job? And if $X$ is a dual module then this is the same as asking for its predual to be (relatively) A-biflat? (This just comes out of Helemskii's version of Ext for Banach modules.) | |
| Oct 21, 2010 at 16:13 | history | asked | user2412 | CC BY-SA 2.5 |