Timeline for Spectral theorem for unital $C^{*}$-algebras
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 28, 2023 at 15:51 | vote | accept | MathMath | ||
| Jan 26, 2023 at 8:25 | answer | added | Dmitri Pavlov | timeline score: 1 | |
| Jan 25, 2023 at 20:36 | comment | added | MathMath | @DmitriPavlov what I meant is the following. Suppose $A$ is an abelian $C^{*}$-star and $a \in A$ be normal, with $\sigma(a)$ its spectrum. Can we prove formula (\ref{1}) in this case, when $B$ is replaced by $A$? The only difference is that, in my post, $A$ was not abelian, so I have to restrict to $B=C^{*}(a)$ to use Gelfand's Isomorphism Theorem. But when $A$ is abelian, do the same formulas hold? | |
| Jan 25, 2023 at 17:45 | comment | added | Dmitri Pavlov | So what do you want to play the role of σ(a) in the absence of a? The most natural analogue of σ(a) is just  itself, so your formula becomes a tautology. Unless you have a different formula in mind, which is why I asked the above question. | |
| Jan 25, 2023 at 12:35 | comment | added | MathMath | @DmitriPavlov Although I started the analysis with an arbitrary $C^{*}$-algebra and studied the subalgebra $B= C^{*}(a)$, which is abelian. My question is: if $A$ was abelian, does the analysis follow for all $A$ instead of for an abelian subalgebra of it? | |
| Jan 25, 2023 at 4:18 | comment | added | Dmitri Pavlov | The question is a bit vague as stated. Suppose $a=1$ so that $\sigma(a)$ and $\hat B$ are points. What kind of formula do you have in mind instead of (1) in this case? | |
| Jan 24, 2023 at 1:08 | history | asked | MathMath | CC BY-SA 4.0 |