Toeplitz spectral inclusion and generalized in modulus property

Let $A \subset {C_b}(X)$ be a function algebra and let $\mu$ be a Borel measure on $X$ and ${H^2}(\mu ) = \bar A$--the closure in ${L^2}(\mu )$. It turns out that the spectral inclusion theorem for Toeplitz operators, defined in the above context, implies the density of finite sums $\sum\nolimits_i {{{\left\vert {{u_i}} \right\vert}^2},{u_i} \in A}$, in

(i) the cone of positive functions in ${L^1}(\mu )$,

(ii) the cone of positive functions in ${L^p}(\mu ),p \leq 2$, if $\mu (X){\text{ < }}\infty$,

(iii) the cone of positive functions in $C(X)$, if $X$ is compact.