In the reduced free product of C
∗
^{*}
–algebras,
(
A
,
ϕ
)
=
(
A
1
,
ϕ
1
)
∗
(
A
2
,
ϕ
2
)
(A,\phi )=(A_{1},\phi _{1})*(A_{2},\phi _{2})
with respect to faithful states
ϕ
1
\phi _{1}
and
ϕ
2
\phi _{2}
,
A
A
is purely infinite and simple if
A
1
A_{1}
is a reduced crossed product
B
⋊
α
,
r
G
B\rtimes _{\alpha ,r}G
for
G
G
an infinite group, if
ϕ
1
\phi _{1}
is well behaved with respect to this crossed product decomposition, if
A
2
≠
C
A_{2}\ne \mathbf {C}
and if
ϕ
\phi
is not a trace.