Can states on commutative Banach algebras be understood as probability measures? Suppose $\mathcal{A}$ is a commutative Banach algebra (over $\mathbb{R}$) or commutative Banach *-algebra (over $\mathbb{C}$). Is there always a measurable space $(\Omega,\mathcal{F})$ such that there is a bijective correspondence between states (positive linear functionals of norm 1) on $\mathcal{A}$ and probability measures on $(\Omega,\mathcal{F})$ together with a map $f:\mathcal{F}\to\mathcal{A}$ such that $$\mu(\Delta)=\phi(f(\Delta))$$ for all $\Delta\in\mathcal{F}$ where $\mu$ is the probability measure corresponding to the state $\phi$?
If $\mathcal{A}$ is a C*-algebra the answer is yes. The measurable space is obtained by using the Gelfand transform establishing an isometry between $\mathcal{A}$ and $C_0(\Omega)$. The correspondence between states and probability measures is obtained by using Riesz' representation theorem. This construction no longer works in the general case though because now the Gelfand representation need not be an isometry. But this of course does not rule out a positive answer to the above question.  
 A: *

*The Gelfand space and transform you mention are defined for any commutative Banach algebra $A$, as $A^\dagger=\{$nonzero linear functionals $\chi:A\to\mathbf C:\chi(ab)=\chi(a)\chi(b)\}$ and $\hat a(\chi)=\chi(a)$.

*On the other hand, the definition of a state $m$ on $A$ asks that $m$ be positive ($m(a^*a)\geqslant 0$) and so requires that your $A$ be at least a $^*$-algebra, right?

*Now, is it a Banach $^*$-algebra, i.e. $\|a^*\|=\|a\|$? If so and $\smash{\hat A := \{\chi\in A^\dagger:\chi(a^*)=\overline{\chi(a)}\}}$ then one has what you want (Fell-Doran, p. 492):

Bochner's Theorem. The formula $m(a) = \int_{\hat A}\chi(a)\,d\mu(\chi)$ defines a bijection between
  
  
*
  
*all bounded regular non-negative Borel measures $\mu$ on $\hat A$,
  
*all extendable positive linear functionals $m$ on $A$. 
  

Here extendable means: $m$ can be extended to a positive linear functional on the $^*$-algebra $A_1$ obtained by adjoining a unit (if one wasn't present) to $A$.

*Bochner's Theorem does not require that $A$ be a C$^*$-algebra. If it is, then $\smash{\hat A = A^\dagger}$ and every positive linear functional $m$ is extendable, but not otherwise (ibid., pp. 391, 475).  
