  Let $\{T_t\}_{t>0}$ be a strongly continuous semigroup of positive
contractions on $L_p(X,\mu)$ with $1<p<\infty$. Let $E$ be a UMD Banach lattice
of measurable functions on another measure space $(\Omega,\nu)$. For $f\in
L_p(X; E)$ define
  $$\mathcal M(f)(x,
\omega)=\sup_{t>0}\frac1t\Big|\int_0^tT_s(f(\cdot,\omega))(x)ds\Big|,\quad
(x,\omega)\in X\times\Omega.$$ Then the following maximal ergodic inequality
holds
  $$\big\|\mathcal M(f)\big\|_{L_p(X; E)}\lesssim \big\|f\big\|_{L_p(X;
E)},\quad f\in L_p(X; E).$$ If the semigroup $\{T_t\}_{t>0}$ is additionally
assumed to be analytic, then $\{T_t\}_{t>0}$ extends to an analytic semigroup
on $L_p(X; E)$ and $\mathcal M(f)$ in the above inequality can be replaced by
the following sectorial maximal function
  $$\mathcal T_\theta(f)(x, \omega)=\sup_{|{\rm
arg}(z)|<\theta}\big|T_z(f(\cdot,\omega))(x)\big|$$ for some $\theta>0$.
  Under the latter analyticity assumption and if $E$ is a complex interpolation
space between a Hilbert space and a UMD Banach space, then $\{T_t\}_{t>0}$
extends to an analytic semigroup on $L_p(X; E)$ and its negative generator has
a bounded $H^\infty(\Sigma_\sigma)$ calculus for some $\sigma<\pi/2$.
