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Let $X$ be a compact subset of $\mathbb {C^n}$. The polynomial convex hall of $X$ is the set

$\hat{X}=\{z\in \mathbb {C^n}: \left|P(z) \right|\leq \left||P |\right|_\infty , \text{for all polynomial } P \}$

Prove that $\Phi_{P(X)}=\hat{X}$

This is my attempt so far

let $P_0(X)=\mathbb {C}[X_1,\ldots, X_n]$ this implies that $\overline {P_0(X)}=P(X)$ for every $q\in P_0(X) $ we have $$q(z)=\sum\alpha_{k_1k_2\cdots k_n}z_1^{k_1}z_1^{k_2}\cdots z_1^{k_n}=q(z_1, z_2, \ldots, z_n)$$ khow we define $q_{k}(z)=z_k$ and for every $\phi\in \Phi_{P(X)} $ we have $$\phi(q(z))=q(\phi(z_1),\ldots, \phi(z_n) )$$ we consider

$$\lambda: \Phi_{P(X)} \to \hat{X} $$ $$\lambda(\phi)=(\phi(z_1),\ldots, \phi(z_n))$$

Any help on how to continue?

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  • $\begingroup$ I think this question is more suited to math.stackexchange.com but let me just give a hint that to prove two sets are equal you should often try to construct a map in one direction (which you have done), then construct a map in the other direction, and then show that the two maps are mutually inverse. To get you started: you want to start with a point in $\widehat{X}$ and obtain from it a character on $P(X)$. What examples of characters do you know? $\endgroup$
    – Yemon Choi
    Jan 28 '19 at 9:36
  • $\begingroup$ @ Yemon Choi, did you mean I have to construct a homeomorphism? $\endgroup$
    – user62498
    Jan 28 '19 at 9:44
  • $\begingroup$ Well, yes, I thought that was obvious from context. Recall however that a continuous bijection from one compact Hausdorff space K to another compact Hausdorff space L is automatically a homeomorphism. So you just have to show that the map $\widehat{X} \to \Phi_{P(X)}$ is continuous, which you can do using the description of the Gelfand topology, and then show it is bijective as I hinted above. $\endgroup$
    – Yemon Choi
    Jan 28 '19 at 10:52

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