Let A Banach algebra, A is contractible if H1(A,X)=0 for all Banach A-bimodules X
. Now to my question
Let A be Banach algebra and I be closed ideal of A. If I and A/I are both contractible, then is A contractible? My attempt: Let X be a Banach A-bimodules, and let D∈Z1(A,X), then D|I∈Z1(I,X) since I contractible, there is x∈X such that Da=a.x−x.a=adx(a) for all a∈I. Let D¯=D−adx, then D¯|I=0
.
I'm stuck here, Does anyone have a recommendation of how to continue?
I am grateful for anyone to have a good recommendation of books//lectures/resources/etc.? for contractible Banach algebra Thanks.
this is extention closed property. please try to construct exact sequence between homology modules.
See the following interesting references
https://booksc.xyz/book/2323006/397e0c
https://booksc.xyz/book/3686181/26da98
https://booksc.xyz/book/3686588/89ad0f
https://booksc.xyz/book/18735609/c147f8
https://booksc.xyz/book/21579660/8ed261
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