  We study the von Neumann algebra description of the inflationary quasi-de
Sitter (dS) space. Unlike perfect dS space, quasi-dS space allows the nonzero
energy flux across the horizon, which can be identified with the expectation
value of the static time translation generator. Moreover, as a dS isometry
associated with the static time translation is spontaneously broken, the
fluctuation in time is accumulated, which induces the fluctuation in the energy
flux. When the inflationary period is given by $(\epsilon_H H)^{-1}$ where
$\epsilon_H$ is the slow-roll parameter measuring the increasing rate of the
Hubble radius, both the energy flux and its fluctuation diverge in the $G \to
0$ limit. Taking the fluctuation in the energy flux and that in the observer's
energy into account, we argue that the inflationary quasi-dS space is described
by Type II$_\infty$ algebra. As the entropy is not bounded from above, this is
different from Type II$_1$ description of perfect dS space in which the entropy
is maximized by the maximal entanglement. We also show that our result is
consistent with the observation that the von Neumann entropy for the density
matrix reflecting the fluctuations above is interpreted as the generalized
entropy.
