Positive functionals on the disc algebra I don't know how to work out 10(d) in Rudin's functional analysis -
Let $U$ be the open unit radius disc in the complex plane and let $A := \{ f \in C\left(\bar{U}\right) : f|_U \text{ is holomorphic} \}$ be equipped with the sup norm as well as the involution $f^*(z):= \overline{f(\bar{z})}$ (Not a $C^*$ algebra).
Given a positive Borel measure $\mu$ on $[-1,1]$, the functional $f\mapsto \int fd\mu$ is clearly positive.
Are there any other positive functionals on $A$?
Attempt: (You should probably ignore unless you have a neurosis about questions lacking effort) My guess would be to connect such positive functionals to functionals on the space $C[-1,1]$. The correspondingly defined involution here does turn the space into a $C^*$ algebra whence the positive functionals are determined by a Bochner-type theorem.
There is the restriction map $r:A \to C[-1,1]$ which is continuous, injective, has dense image (polynomials) and relates both involutions.
Let $\phi$ be a positive linear functional on $A$.
Given a $g\in C[-1,1]$, I'm tempted to take a sequence $f_n \in A$ with $r(f_n) \to g$ and use it to define a functional.
However the $f_n$ need not be Cauchy. E.g. if $g$ is a bump function, any such sequence of holomorphic functions $f_n$ cannot have a convergent sub-sequence? Have I messed up here? Is this approach useless?
Please help me.
 A: Given a positive linear functional $\varphi $ on $A$,  let $(\pi , H, \xi )$ be the associated GNS representation,  which means that
$\pi $ is a *-representation of $A$ on the Hilbert space $H$,  and $\xi $ is a cyclic vector for $\pi $, such that
$$
  \varphi (f) = \langle \pi (f)\xi , \xi \rangle , \quad \forall f \in A.
  $$
Setting $f_1(z)=z$, observe that $f_1^*(z) = \overline {f_1(\bar z)}= z = f_1(z)$, which means that $f_1$ is self-adjoint.  It
follows that the operator
$
  T:= \pi (f_1)
  $
is self-adjoint, and clearly
$$
  \|T\| = \|\pi (f_1)\| \leq  \|f_1\| = 1.
  $$
We then have that $\sigma (T)\subseteq [-1,1]$, while
the functional calculus for $T$ is a *-homomorphism
$$
  \gamma :C(\sigma (T)) \to  B(H),
  $$
sending the function $f_1$ (more precisely its restriction to $\sigma (T)$) to $T$.
The functional
$$
  \psi : g\in   C(\sigma (T)) \mapsto  \langle \gamma (f)\xi , \xi \rangle \in  \mathbb C
  $$
is clearly positive, so by Riesz there exists a positive Borel measure $\mu $ on $\sigma (T)$ such that
$$
  \psi (g) = \int_{\sigma (T)}g(t)\, d\mu (t), \quad \forall g \in C(\sigma (T)).
  $$
On the other hand,
if we let $\rho$ be the *-homomorphism defined by
$$
  \rho :f\in A\mapsto f|_{\sigma (T)}\in  C(\sigma (T)),
  $$
we get  that $\pi (f_1) = T = \gamma \big (\rho (f_1)\big )$.  Observing that $A$ is generated by $f_1$ as  a  unital Banach algebra,
we conclude that $\pi = \gamma \circ \rho $, whence for all $f$ in $A$ we have
$$
  \varphi (f) = \langle \pi (f)\xi , \xi \rangle  = \langle \gamma \rho (f)\xi , \xi \rangle  = \psi (\rho (f)) = \int_{\sigma (T)}f(t)\, d\mu (t).
  $$
Recalling that $\sigma (T)\subseteq [-1, 1]$,  we may view $\mu $ as a measure on $[-1, 1]$ (whose support is contained in $\sigma (T)$), in
which case we may write the above as
$$
  \varphi (f) = \int_{-1}^1f(t)\, d\mu (t).
  $$
