We consider group measure space II
1
_{1}
factors
M
=
L
∞
(
X
)
⋊
Γ
M=L^{\infty }(X)\rtimes \Gamma
arising from Bernoulli actions of ICC property (T) groups
Γ
\Gamma
(more generally, of groups
Γ
\Gamma
containing an infinite normal subgroup with the relative property (T)) and prove a rigidity result for
∗
*
–homomorphisms
θ
:
M
→
M
⊗
¯
M
\theta :M\rightarrow M\overline {\otimes }M
.
We deduce that the action
Γ
↷
X
\Gamma \curvearrowright X
is W
∗
^{*}
–superrigid, i.e. if
Λ
↷
Y
\Lambda \curvearrowright Y
is any free, ergodic, measure preserving action such that the factors
M
=
L
∞
(
X
)
⋊
Γ
M=L^{\infty }(X)\rtimes \Gamma
and
L
∞
(
Y
)
⋊
Λ
L^{\infty }(Y)\rtimes \Lambda
are isomorphic, then the actions
Γ
↷
X
\Gamma \curvearrowright X
and
Λ
↷
Y
\Lambda \curvearrowright Y
must be conjugate.
Moreover, we show that if
p
∈
M
∖
{
1
}
p\in M\setminus \{1\}
is a projection, then
p
M
p
pMp
does not admit a group measure space decomposition nor a group von Neumann algebra decomposition (the latter under the additional assumption that
Γ
\Gamma
is torsion free).
We also prove a rigidity result for
∗
*
–homomorphisms
θ
:
M
→
M
\theta :M\rightarrow M
, this time for
Γ
\Gamma
in a larger class of groups than above, now including products of non–amenable groups. For certain groups
Γ
\Gamma
, e.g.
Γ
=
F
2
×
F
2
\Gamma =\mathbb {F}_{2}\times \mathbb {F}_{2}
, we deduce that
M
M
does not embed into
p
M
p
pMp
, for any projection
p
∈
M
∖
{
1
}
p\in M\setminus \{1\}
, and obtain a description of the endomorphism semigroup of
M
M
.