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We examine the origin of scatter in the relationship between the gas fractionand mass of dark matter haloes hosting present-day $\sim L^\star$ centralgalaxies in the EAGLE simulations. The scatter is uncorrelated with theaccretion rate of the central galaxy's black hole (BH), but correlates stronglyand negatively with the BH's mass, implicating differences in the expulsion ofgas by active galactic nucleus feedback, throughout the assembly of the halo,as the main cause of scatter. Haloes whose central galaxies host undermassiveBHs also tend to retain a higher gas fraction, and exhibit elevated starformation rates (SFRs). Diversity in the mass of central BHs stems primarilyfrom diversity in the dark matter halo binding energy, as these quantities arestrongly and positively correlated at fixed halo mass, such that $\sim L^\star$galaxies hosted by haloes that are more (less) tightly-bound develop centralBHs that are more (less) massive than is typical for their halo mass.Variations in the halo gas fraction at fixed halo mass are reflected in boththe soft X-ray luminosity and thermal Sunyaev-Zel'dovich flux, suggesting thatthe prediction of a strong coupling between the properties of galaxies andtheir halo gas fractions can be tested with measurements of these diagnosticsfor galaxies with diverse SFRs but similar halo masses.

The gas fractions of dark matter haloes hosting simulated $\sim L^\star$ galaxies are governed by the feedback history of their black holes