  Star formation in the universe's most massive galaxies proceeds furiously
early in time but then nearly ceases. Plenty of hot gas remains available but
does not cool and condense into star-forming clouds. Active galactic nuclei
(AGN) release enough energy to inhibit cooling of the hot gas, but energetic
arguments alone do not explain why quenching of star formation is most
effective in high-mass galaxies. In fact, optical observations show that
quenching is more closely related to a galaxy's central stellar velocity
dispersion ($\sigma_v$) than to any other characteristic. Here, we show that
high $\sigma_v$ is critical to quenching because a deep central potential well
maximizes the efficacy of AGN feedback. In order to remain quenched, a galaxy
must continually sweep out the gas ejected from its aging stars. Supernova
heating can accomplish this task as long as the AGN sufficiently reduces the
gas pressure of the surrounding circumgalactic medium (CGM). We find that CGM
pressure acts as the control knob on a valve that regulates AGN feedback and
suggest that feedback power self-adjusts so that it suffices to lift the CGM
out of the galaxy's potential well. Supernova heating then drives a galactic
outflow that remains homogeneous if $\sigma_v \gtrsim 240 \, {\rm km \,
s^{-1}}$. AGN feedback can effectively quench galaxies with a comparable
velocity dispersion, but feedback in galaxies with a much lower velocity
dispersion tends to result in convective circulation and accumulation of
multiphase gas within the galaxy.
