Continuity of a positive preserving operator between C(X) and C(Y)

Asked Mar 17 '14 at 16:27

Active Dec 30 '14 at 07:53

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I've been struggling with this question in Reed and Simon while I'm prepping for quals.

Suppose that $T:C(X)\rightarrow C(Y)$ is a positive operator. Prove that T is continuous and $\Vert T\Vert_{C(Y)^{*}}=\Vert T1\Vert_{C(Y)}$.

The proof for positive linear functionals doesn't carry over and I'm unsure how to proceed.

edited Dec 30 '14 at 07:53

Martin Sleziak

asked Mar 17 '14 at 16:27

What is $T_1$? ${}{}$ – copper.hat Mar 17 '14 at 16:32
Use the map with the function $x\mapsto\lVert f\rVert_\infty-f(x)$, $f\in C(X)$. – Davide Giraudo Mar 17 '14 at 16:47
@copper.hat It's the sup norm in $C(Y)$ of T applied to the constant function that evaluates to 1. – SingularDegenerate Mar 17 '14 at 22:40
@DavideGiraudo Applying T to that function works in the case that we're dealing with real-valued continuous functions. What about the complex case? – SingularDegenerate Mar 17 '14 at 22:42
The proof does carry over. http://math.stackexchange.com/questions/426487/why-is-every-positive-linear-map-between-c-algebras-bounded – Tomasz Kania Dec 30 '14 at 07:42

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