Timeline for Spectral theorem for unital $C^{*}$-algebras
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 28, 2023 at 15:51 | vote | accept | MathMath | ||
| Jan 26, 2023 at 18:38 | comment | added | Dmitri Pavlov | @MathMath: No, Â is isomorphic to σ(a) if and only if a generates A as a C*-algebra. Otherwise, you just have a continuous map Â→σ(a) provided by the Gelfand transformation, but it is no longer a homeomorphism. | |
| Jan 26, 2023 at 14:47 | comment | added | MathMath | Dmitri, thank you for your answer. I was puzzled because, in the analysis of my post, I used the fact that $\hat{B} \cong \sigma(a)$. But if $A$ is abelian and $a \in A$ is normal, I believe the same analysis provide an isomorphism $\hat{A}\cong \sigma(a)$, right? And then formula (\ref{1}) follows from my analysis, which justifies your answer. | |
| Jan 26, 2023 at 8:25 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |