  We study a coarse-grained model for a water monolayer that cannot crystallize
due to the presence of confining interfaces, such as protein powders or
inorganic surfaces. Using both Monte Carlo simulations and mean field
calculations, we calculate three response functions: the isobaric specific heat
$C_P$, the isothermal compressibility $K_T$, and the isobaric thermal
expansivity $\alpha_P$. At low temperature $T$, we find two distinct maxima in
$C_P$, $K_T$ and $|\alpha_P|$, all converging toward a liquid-liquid critical
point (LLCP) with increasing pressure $P$. We show that the maximum in $C_P$ at
higher $T$ is due to the fluctuations of hydrogen (H) bond formation and that
the second maximum at lower $T$ is due to the cooperativity among the H bonds.
We discuss a similar effect in $K_T$ and $|\alpha_P|$. If this cooperativity
were not taken into account, both the lower-$T$ maximum and the LLCP would
disappear. However, comparison with recent experiments on water hydrating
protein powders provides evidence for the existence of the lower-$T$ maximum,
supporting the hypothesized LLCP at positive $P$ and finite $T$. The model also
predicts that when $P$ moves closer to the critical $P$ the $C_P$ maxima move
closer in $T$ until they merge at the LLCP. Considering that other scenarios
for water are thermodynamically possible, we discuss how an experimental
measurement of the changing separation in $T$ between the two maxima of $C_P$
as $P$ increases could determine the best scenario for describing water.
