Timeline for Ultrafilters as a double dual
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 16, 2019 at 21:13 | history | edited | Nik Weaver | CC BY-SA 4.0 |
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| Mar 16, 2019 at 21:13 | comment | added | Nik Weaver | @RobertFurber: yes, of course you're right. It's the multiplier algebra, not the second dual. | |
| Mar 16, 2019 at 19:24 | comment | added | Robert Furber | @NikWeaver I think in the last paragraph you meant to put that the spectrum of $C_b(X)$, the bounded continuous functions, is homeomorphic to $\beta X$. In general $C_0(X)^{**}$ is just an enveloping W*-algebra with no other characterization, and it is not isomorphic to $C(\beta X)$ in cases such as $X = [0,1]$. | |
| Mar 8, 2019 at 16:25 | comment | added | Nik Weaver | Yes, that's right. The first duals generally don't even have a preferred product, so there's nothing like a spectrum. | |
| Mar 8, 2019 at 14:57 | comment | added | Adam P. Goucher | @NikWeaver Elegant! This essentially answers both of my questions, namely why the analogy exists, and also why the 'half-iterate' $\delta X$ cannot be defined: when you take the first dual of $c_0(X)$, the result is not a C*-algebra, so you can't take its spectrum. But the double dual $l^{\infty}(X)$ is a C*-algebra, so you can indeed take its spectrum, and you get $\beta X$. | |
| Mar 8, 2019 at 14:38 | vote | accept | Adam P. Goucher | ||
| Mar 7, 2019 at 22:31 | history | edited | Nik Weaver | CC BY-SA 4.0 |
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| Mar 7, 2019 at 19:38 | comment | added | Nik Weaver | Equivalently, the extension of $f$ to the one point compactification of $X$ (with the discrete topology) which sets $f(\infty) = 0$ is continuous. Hence "goes to zero at infinity". | |
| Mar 7, 2019 at 19:36 | comment | added | Nik Weaver | For any $\epsilon > 0$, there is a finite subset of $X$ off of which $|f(x)| \leq \epsilon$. | |
| Mar 7, 2019 at 19:28 | comment | added | Alex Kruckman | What does it mean for a function from a set $X$ to $\mathbb{C}$ to "go to zero at infinity"? | |
| Mar 7, 2019 at 16:59 | history | answered | Nik Weaver | CC BY-SA 4.0 |