  This paper gives a systematic study of operator-valued local Hardy spaces.
These spaces are localizations of the Hardy spaces defined by Tao Mei, and
share many properties with Mei's Hardy spaces. We prove the ${\rm h}_1$-$\rm
bmo$ duality, as well as the ${\rm h}_p$-${\rm h}_q$ duality for any conjugate
pair $(p,q)$ when $1<p< \infty$. We show that ${\rm h}_1(\mathbb{R}^d, \mathcal
M)$ and ${\rm bmo}(\mathbb{R}^d, \mathcal M)$ are also good endpoints of
$L_p(L_\infty(\mathbb{R}^d) \overline{\otimes} \mathcal M)$ for interpolation.
We obtain the local version of Calder\'on-Zygmund theory, and then deduce that
the Poisson kernel in our definition of the local Hardy norms can be replaced
by any reasonable test function. Finally, we establish the atomic decomposition
of the local Hardy space ${\rm h}_1^c(\mathbb{R}^d,\mathcal M)$.
