Detailed high-resolution observations of the innermost regions of nearby galaxies have revealed the presence of supermassive black holes. These black holes may interact with their host galaxies by means of 'feedback' in the form of energy and material jets; this feedback affects the evolution of the host and gives rise to observed relations between the black hole and the host. Here we report observations of the ultraviolet emissions of massive early-type galaxies. We derive an empirical relation for a critical black-hole mass (as a function of velocity dispersion) above which the outflows from these black holes suppress star formation in their hosts by heating and expelling all available cold gas. Supermassive black holes are negligible in mass compared to their hosts but nevertheless seem to play a critical role in the star formation history of galaxies.