Given a compact Hausdorff space $X$, does there exist a probability $\mu$ on X such that the support of $\mu$ is $X$? This is equivalent to say, for any unital commutative C*-algebra, can we show the existence of a faithful state?
Existence of a completely supported probability measure
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measure-theory
functional-analysis
probability-theory
c-star-algebras
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1Just an observation: If the space is separable, you can choose a countable, dense sequence $\{x_n\}$ and define $\mu$ by $\mu(E)=\sum 2^{-n}\chi_E(x_n)$. I don't know about the general case though. – 2011-10-29
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0$X$ may be not separable, for example if $X=[0,1]^{[0,1]}$ endowed with the topology of uniform convergence. How is the support defined? – 2012-04-22
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1@DavideGiraudo: Note that your $X$ is not compact. – 2012-04-22
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0Sorry, I meant pointwise convergence. – 2012-04-22
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3@DavideGiraudo: Note that your $X$ [is separable](http://planetmath.org/encyclopedia/HewittMarczewskiPondiczeryTheorem.html) :) – 2012-04-22
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0@t.b. You are right, thanks for the link. – 2012-04-22