Let $S$ be a subset of $A_+$ the positive elements of a $C^\ast$-algebra $A$ which is weakly compact. I want to show that $S$ has a minimal and maximal elements.
I know that $S$ has a partial order since its a subset of $A_+$ we say $a\leq b$ for $a,b\in S$ if $b-a$ is a positive element. How can I show that any chain in $S$ has an upper bound?
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adham
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Without other details about $S$, it's impossible; there are subsets of $A_+$ which don't have minimal or maximal elements. – Noah Schweber Dec 09 '16 at 00:29
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@NoahSchweber is weak compactness not enough? – adham Dec 09 '16 at 00:31
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Ah, I read weak compactness as applying to $A$, not $S$. – Noah Schweber Dec 09 '16 at 02:54
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Since $S$ is weakly compact, given a chain $\{a_j\}\subset S$, there is a weakly convergent subnet. Let $a\in S$ be the limit of the subnet. Is is then easy to see that $a$ is an upper bound for $\{a_j\}$.
Martin Argerami
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Since you are dealing with a chain, it is not different than if you were dealing with numbers. – Martin Argerami Dec 10 '16 at 01:48