Timeline for Non-unital algebras in geometric algebra, smooth envelopes
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Sep 10, 2021 at 17:07 | answer | added | Dmitri Pavlov | timeline score: 1 | |
| Sep 10, 2021 at 16:54 | comment | added | Dmitri Pavlov | @YemonChoi: “Reduces” was used in the (narrow) context of OP's question, i.e., smooth envelopes of geometric R-algebras. For C*-algebras, I recall a discussion here on MathOverflow about unitization, I am not sure whether any examples given there were convincing or not. | |
| Sep 10, 2021 at 1:33 | comment | added | Yemon Choi | @DmitriPavlov: the existence of unitizations of Cstar algebras does not really reduce the non-unital theory to the unital theory, and indeed there are many instances where experience rather than categorical dictat indicates that one should work with the multiplier algebra not the unitization. | |
| Sep 9, 2021 at 18:42 | comment | added | supergeneric | David, thank you for the suggestion; I've edited the question to ask just about smooth envelopes. YCor, thanks for making the title more specific. Dimitri, my motivation is exactly the intuition that Branimir describes (i.e., that I want to work with an "honest" $C^*$-algebra of globally bounded functions like $C_0 (X)$, or a dense subspace of it containing only smooth functions). | |
| Sep 9, 2021 at 18:31 | history | edited | supergeneric | CC BY-SA 4.0 |
deleted 7 characters in body
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| Sep 9, 2021 at 14:23 | comment | added | Branimir Ćaćić | Very crudely speaking, one can argue that encoding a non-compact LCH space $X$ in terms of the non-unital $C^\ast$-algebra $C_0(X)$ is just the price you pay for working with an honest $C^\ast$-algebra of globally bounded functions without invoking any particular choice of compactification. Nestruev’s commutative-algebraic approach doesn’t require anything to be globally bounded, so the only non-unital algebras they ever need invoke are prime ideals of unital algebras. | |
| Sep 9, 2021 at 13:35 | comment | added | Dmitri Pavlov | What is the motivation behind such a question? A nonunital R-algebra can be turned into a unital R-algebra using the unitization functor. This reduces the study of nonunital algebras to unital algebras. In geometric language, this corresponds to studying pointed spaces instead of spaces. | |
| Sep 9, 2021 at 7:43 | history | edited | YCor | CC BY-SA 4.0 |
made title more specific
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| Sep 9, 2021 at 7:24 | comment | added | David Roberts♦ | This is a pretty wide question. Without looking at the book, it looks like you are asking how much of a whole book can be reproduced by dropping an assumption. I suggest picking one single result you are interested in, asking about that. | |
| S Sep 9, 2021 at 5:51 | review | First questions | |||
| Sep 9, 2021 at 8:58 | |||||
| S Sep 9, 2021 at 5:51 | history | asked | supergeneric | CC BY-SA 4.0 |