  We analyse the K-band Hubble diagram for a sample of brightest cluster
galaxies (BCGs) in the redshift range 0<z<1. In good agreement with earlier
studies, we confirm that the scatter in the absolute magnitudes of the galaxies
is small (0.3 magnitudes). The BCGs exhibit very little luminosity evolution in
this redshift range: if q_0=0.0 we detect no luminosity evolution; for q_0=0.5
we measure a small negative evolution (i.e., BCGs were about 0.5 magnitudes
fainter at z=1 than today). If the mass in stars of these galaxies had remained
constant over this period of time, substantial positive luminosity evolution
would be expected: BCGs should have been brighter in the past since their stars
were younger. A likely explanation for the observed zero or negative evolution
is that the stellar mass of the BCGs has been assembled over time through
merging and accretion, as expected in hierarchical models of galaxy formation.
The colour evolution of the BCGs is consistent with that of an old stellar
population (z_form >2) that is evolving passively. We can thus use evolutionary
population synthesis models to estimate the rate of growth in stellar mass for
these systems. We find that the stellar mass in a typical BCG has grown by a
factor ~ 2 since z=1 if q_0=0.0 or by factor ~ 4 if q_0=0.5. These results are
in good agreement with the predictions of semi-analytic models of galaxy
formation and evolution set in the context of a hierarchical scenario for
structure formation. The models predict a scatter in the luminosities of the
BCGs that is somewhat larger than the observed one, but that depends on the
criterion used to select the model clusters.
