Exercise 9, § 8

Let X be a compact topological space and B a unital Banach subalgebra of $\mathscr{C}(X)$ (for the induced norm). Every element $x$ of X defines a character $\chi_x$ of B. If $x,x'$ are in X, we write $x\sim x'$ when $\|\chi_x-\chi_{x'}\| \not= 2$. a) Let $x$ and $x'$ be in X. If there exists a sequence $(f_n)$ of elements of B such that $\|f_n\|\leqslant 1$, that $|f_n(x)|$ tends to 1 and that lim inf$_{n\rightarrow \infty}|f_n(x)-f_n(x')|>0$, then one does not have $x\sim x'$. (One may suppose that $f_n(x)$ tends to 1. Consider $g_n= (f_n-\lambda_n)(1-\lambda_nf_n)^{-1}$, where $(\lambda_n)$ is a sequence of numbers of $[0,1[$ tending to 1. One has $\|g_n\|\leqslant 1$. If the sequence $(\lambda_n)$ is suitably chosen, $g_n(x)$ tends to 1 and $g_n(x')$ tends to $-1$.)

b) Let R be the set of real parts of the elements of B. If $x\sim x'$, there exists $c\in ]0,1[$ such that $f(x)\geqslant cf(x')$ and $f(x')\geqslant cf(x)$ for every $f\geqslant 0$ in R. (It suffices to obtain the first condition; if it is not true, there exists a sequence $(f_n)$ in R with $f_n\geqslant 0,f_n(x') = 1,f_n(x)$ tending to 0; let $g_n\in B$ with $\mathscr{R}(g_n) =f_n$. Then $e^{-g_n}\in B,\|e^{-g_n}\|\leqslant 1,|e^{-g_n(x')}|= 1/e,|e^{-g_n(x)}|$ tends to 1, which contradicts a).)

c) There exist positive measures $\alpha$ and $\beta$ on X such that $f(x)-cf(x') =\alpha (f)$ and $f(x')-cf(x) =\beta (f)$ for $f\in R$. Setting $\mu_x= (1-c^2)^{-1}(c\beta +\alpha )$, $\mu_{x'}= (1-c^2)^{-1}(c\alpha +\beta )$, one has $\mu_x(f) =f(x)$ and $\mu_{x'}(f) =f(x')$ for every $f\in B$, and $c\mu_x\leqslant \mu_{x'},c\mu_{x'}\leqslant \mu_x$.

d) Suppose again that $x\sim x'$. Prove that if $\mu$ is a positive measure on X such that $\mu(f) =f(x)$ for every $f\in B$, there exists a positive measure $\mu'$ on X such that $\mu'(f) =f(x')$ for every $f\in B$ and $c\mu\leqslant \mu'$ for a $c\in ]0,1[$. (With the notation of c), set $\mu'=\mu_{x'}-c\mu_x+c\mu$.)

e) The relation $x\sim x'$ is an equivalence relation in X. (Use d).)