values around $v_w/2c\approx 10^{-1}$ associated with covering factors of order $10^{-1}$ (see Chartas et al. 2003; Pounds et al. 2003). We than compute the fraction $\Delta m$ of cold gas expelled by AGNs; the galactic cold gas expelled by the AGN feedback enriches the hot gas phase which fills the dark matter potential wells of the structure (group or cluster) hosting the galaxy. Expanding out of the galaxy, the blastwave also expels a tiny fraction of the hot gas in the host structure and re-sets the hydrostatic equilibrium of such a hot gas to a new temperature $T+\Delta T$ larger than the initial temperature $T$ by a factor $1/\beta$ (with values in the range 1-1.1 in most cases), also computed after Cavaliere, Lapi & Menci (2002).

RESULTS
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Here we present our results for a $\Lambda$-CDM cosmology with $\Omega_0=0.3$, $\Omega_{\lambda}=0.7$, a baryon fraction $\Omega_b=0.05$, and Hubble constant $h=0.7$ in units of 100 km s$^{-1}$ Mpc$^{-1}$. The star formation and stellar feedback parameters are the same as in Menci et al. (2005), while the AGN feedback efficiency (see sect. 2.2) is set to $f=0.05$.

Testing the model: The AGN evolution and the Intergalactic Gas
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Since our feedback is tightly related 