Timeline for Determine whether the center of a $C^*$-algebra is 0
Current License: CC BY-SA 4.0
18 events
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| Feb 28, 2022 at 1:49 | comment | added | math112358 | @Branimir Ćaćić, excuse me, I have another question. In the example you constructed, $A$ has no tracial states. If $A$ is aasumed to have a tracial state, does there exist other exmples? | |
| Feb 28, 2022 at 1:02 | comment | added | Branimir Ćaćić | Which is $0$ in $K(H) \rtimes_r \mathbb{Z} = K(H) \otimes \mathrm{C}^\ast_r(\mathbb{Z})$. Note that $K(H)$ is non-unital. | |
| Feb 28, 2022 at 0:37 | comment | added | Branimir Ćaćić | Yes, by the above observation, take $A = K(H)$ the $C^\ast$-algebra of compact operators on a separable infinite Hilbert space $H$ together with the trivial action of $G = \mathbb{Z}$. | |
| Feb 27, 2022 at 23:53 | comment | added | math112358 | @Branimir Ćaćić,does there exist a concrete example such that the center of $C_c(G,A)$ is 0? | |
| Feb 27, 2022 at 17:26 | comment | added | Yemon Choi | I think in its current form the question is far too broad and almost asking for people to provide both the hypotheses of a theorem as well as its proof. Consider $G$ discrete ICC and $A$ to be abelian and non-unital, for instance | |
| Feb 27, 2022 at 17:17 | comment | added | Branimir Ćaćić | In the case of $\mathbb{Z}$ with the counting measure, since $C_c(\mathbb{Z},A)$ is spanned by maps with singleton support, you can check that $f \in C_c(\mathbb{Z},A)$ is central with respect to convolution iff $f(n) \cdot \alpha_n(a) = a \cdot \alpha_n(f(n))$ for every $n \in \mathbb{Z}$ and every $a \in A$. I don’t know how much more you can extract without additional information… | |
| Feb 27, 2022 at 15:38 | history | edited | math112358 | CC BY-SA 4.0 |
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| Feb 27, 2022 at 9:30 | history | edited | YCor |
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| Feb 27, 2022 at 8:48 | history | edited | math112358 | CC BY-SA 4.0 |
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| Feb 27, 2022 at 8:16 | history | edited | math112358 | CC BY-SA 4.0 |
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| Feb 26, 2022 at 16:35 | comment | added | Matthew Daws | Could you clarify what algebra product is placed on $C_c(G,A)$. The pointwise product would not seem to use the structure that $G$ is a group in any way... | |
| Feb 26, 2022 at 15:47 | answer | added | Onur Oktay | timeline score: 0 | |
| Feb 26, 2022 at 15:34 | comment | added | math112358 | But not every $f\in C_c(G,A)$ has the form $t\mapsto f(t)x$. | |
| Feb 26, 2022 at 14:06 | comment | added | Onur Oktay | Functions of the form $t\to f(t)x$ where $x\in A$ and $f\in C_c(G)$ suggest that $Z(C_c(G,A) = \{0\}$ iff $Z(A)=\{0\}$. | |
| Feb 24, 2022 at 20:05 | review | Close votes | |||
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| S Feb 24, 2022 at 15:42 | history | edited | LSpice | CC BY-SA 4.0 |
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| Feb 24, 2022 at 15:41 | review | Suggested edits | |||
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| Feb 24, 2022 at 15:30 | history | asked | math112358 | CC BY-SA 4.0 |