We study the mod
p
p
cohomology of the classifying space of the projective unitary group
P
U
(
p
)
PU(p)
. We first prove that conjectures due to J.F. Adams and Kono and Yagita (1993) about the structure of the mod
p
p
cohomology of the classifying space of connected compact Lie groups hold in the case of
P
U
(
p
)
PU(p)
. Finally, we prove that the classifying space of the projective unitary group
P
U
(
p
)
PU(p)
is determined by its mod
p
p
cohomology as an unstable algebra over the Steenrod algebra for
p
>
3
p>3
, completing previous work by Dwyer, Miller and Wilkerson (1992) and Broto and Viruel (1998) for the cases
p
=
2
,
3
p=2,3
.