Timeline for Linear independency and compactness of the set of pure states of a $C^*$-algebra
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jul 24, 2016 at 18:54 | comment | added | Masayoshi Kaneda | (Continuation) For any $i\in\{1,\dots,n\}$, by the complete regularity of $\Omega(\mathcal{A})$, $\exists\hat{a}\in C_0(\Omega(\mathcal{A}))$ such that $\hat{a}(\tau_i)=1$ and $\hat{a}(\tau_j)=0$ for $j\ne i$. Thus $0=(\lambda_1\tau_1+\cdots+\lambda_n\tau_n)(a)=\lambda_1\tau_1(a)+\cdots+\lambda_n\tau_n(a)=\lambda_1\hat{a}(\tau_1)+\cdots+\lambda_n\hat{a}(\tau_n)=\lambda_i$. | |
| Jul 24, 2016 at 18:53 | comment | added | Masayoshi Kaneda | Thank you for your comments and a partial answer, Igor and Nate. In the commutative case, the set of pure states is the character space (= maximal ideal space) $\Omega(\mathcal{A})$ which is a locally compact Hausdorff space in the weak* topology, and $\mathcal{A}\cong C_0(\Omega(\mathcal{A}))$ $*$-isomorphically. Let $\lambda_1\tau_1+\cdots+\lambda_n\tau_n=0$, where $\lambda_1,\dots,\lambda_n\in\mathbb{C}$ and $\tau_1,\dots,\tau_n\in\Omega(\mathcal{A})$. (Continued below) | |
| Jul 24, 2016 at 17:55 | vote | accept | Masayoshi Kaneda | ||
| Jul 24, 2016 at 15:47 | answer | added | Nik Weaver | timeline score: 10 | |
| Jul 24, 2016 at 14:46 | comment | added | Nate Eldredge | @IgorKhavkine: I assume we mean "linearly independent as a subset of the dual space $\mathcal{A}^*$." So your example seems to show that this is false. | |
| Jul 24, 2016 at 13:44 | comment | added | Igor Khavkine | In what sense should $PS(\mathcal{A})$ should be "linearly indpendent"? It's not even a linear space. For example, for $\mathcal{A} = M_2(\mathbb{C})$, the pure states constitute a 2-sphere (the Bloch sphere). | |
| Jul 24, 2016 at 13:40 | comment | added | Nate Eldredge | Where is the commutativity of $\mathcal{A}$ used in the proof you already know? | |
| Jul 24, 2016 at 12:44 | history | asked | Masayoshi Kaneda | CC BY-SA 3.0 |