  X-ray emission from hot (T = 10^7 K) interstellar gas in massive elliptical
galaxies indicates that 10^{10} M_sun has cooled over a Hubble time, but
optical and radio evidence for this cold gas is lacking. We provide detailed
theoretical support for the hypothesis that this gas has formed into low
luminosity stars. Within several kpc of the galactic center, interstellar gas
first cools to T = 10^4 K where it is heated by stellar UV and emits the
observed diffuse optical line emission. This cooling occurs at a large number
(10^6) of isolated sites. After less than a solar mass of gas has accumulated
(10^{-6} M_sun/yr) at a typical cooling site, a neutral (HI or H_2) core
develops in the HII cloud where gas temperatures drop to T = 15 K and the
ionization level (from thermal X-rays) is very low (x = 10^{-6}). We show that
the maximum mass of cores that become gravitationally unstable is only about 2
M_sun. No star can exceed this mass. Fragmentation of collapsing cores produces
a population of low mass stars with a bottom-heavy IMF and radial orbits.
Gravitational collapse and ambipolar diffusion are rapid. The total mass of
star-forming (dust-free) HI or H_2 cores in a typical bright elliptical is only
10^6 M_sun, below current observational thresholds.
