Timeline for Is the algebra of compact operators flat?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Nov 3, 2019 at 2:29 | comment | added | Ruy | If $A\hookrightarrow B$ is an inclusion of C*-algebras, and $C$ is any other C*-algebra, then $A⊗_{min}C\hookrightarrow B⊗_{min}C$ is injective. The same is also true for $⊗_{max}$ in the following special cases: (1) $C$ is nuclear, (2) $A$ is nuclear, (3) $A$ is a hereditary subalgebra of $B$, or (4) there exists a conditional expectation from $B$ to $A$. Beyond these cases the statement for $⊗_{max}$ may fail. (Ref: Section (3.6) in Brown-Ozawa book). | |
| Oct 7, 2019 at 20:05 | history | edited | YCor | CC BY-SA 4.0 |
removed many typos
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| Oct 7, 2019 at 10:39 | history | became hot network question | |||
| Oct 7, 2019 at 3:19 | history | edited | Yemon Choi |
added a top-level tag
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| Oct 7, 2019 at 3:16 | answer | added | Yemon Choi | timeline score: 9 | |
| Oct 7, 2019 at 3:13 | answer | added | Nik Weaver | timeline score: 8 | |
| Oct 7, 2019 at 2:57 | comment | added | Yemon Choi | Thirdly, if you want to talk about objects of a category being flat, you need to define your terms rather more precisely, especially when your category is not additive. In particular, "tensoring with this object preserves monomorphisms" might not be enough to categorize "flatness" - if one wants to use a definition based on short exact sequences, then tensoring $0\to J \to A \to A/J$ with a fixed ${\rm C}^*$-algebra need not preserve "exactness in the middle" | |
| Oct 7, 2019 at 2:53 | comment | added | Yemon Choi | Firstly, there are some typos in your question to fix. Secondly, can you clarify whether $\otimes$ denotes ${\rm C}^*$-tensor product, or just the uncompleted algebraic tensor product? | |
| Oct 7, 2019 at 2:38 | history | asked | Less | CC BY-SA 4.0 |