We give a separable Brown-Douglas-Fillmore theorem. Let
A
A
be a separable amenable
C
∗
C^*
-algebra which satisfies the approximate UCT,
B
B
be a unital separable amenable purely infinite simple
C
∗
C^*
-algebra and
h
1
,
h
2
:
A
→
B
h_1, \, h_2: A\to B
be two monomorphisms. We show that
h
1
h_1
and
h
2
h_2
are approximately unitarily equivalent if and only if
[
h
1
]
=
[
h
2
]
in
K
L
(
A
,
B
)
.
[h_1]=[h_2]\,\,\,\,\textrm {in}\,\,\, KL(A,B).
We prove that, for any
ε
>
0
\varepsilon >0
and any finite subset
F
⊂
A
\mathcal {F}\subset A
, there exist
δ
>
0
\delta >0
and a finite subset
G
⊂
A
\mathcal {G}\subset A
satisfying the following: for any amenable purely infinite simple
C
∗
C^*
-algebra
B
B
and for any contractive positive linear map
L
:
A
→
B
L: A\to B
such that
\[
‖
L
(
a
b
)
−
L
(
a
)
L
(
b
)
‖
>
δ
a
n
d
‖
L
(
a
)
‖
≥
(
1
/
2
)
‖
a
‖
\|L(ab)-L(a)L(b)\|>\delta \quad \mathrm {and}\quad \|L(a)\|\ge (1/2)\|a\|
\]
for all
a
∈
G
,
a\in \mathcal {G},
there exists a homomorphism
h
:
A
→
B
h: A\to B
such that
\[
‖
h
(
a
)
−
L
(
a
)
‖
>
ε
f
o
r
a
l
l
a
∈
F
\|h(a)-L(a)\|>\varepsilon \,\,\,\,\,\mathrm {for\,all}\,\,\, a\in \mathcal {F}
\]
provided, in addition, that
K
i
(
A
)
K_i(A)
are finitely generated. We also show that every separable amenable simple
C
∗
C^*
-algebra
A
A
with finitely generated
K
K
-theory which is in the so-called bootstrap class is weakly stable with respect to the class of amenable purely infinite simple
C
∗
C^*
-algebras. As an application, related to perturbations in the rotation
C
∗
C^*
-algebras studied by U. Haagerup and M. Rørdam, we show that for any irrational number
θ
\theta
and any
ε
>
0
\varepsilon >0
there is
δ
>
0
\delta >0
such that in any unital amenable purely infinite simple
C
∗
C^*
-algebra
B
B
if
\[
‖
u
v
−
e
i
θ
π
v
u
‖
>
δ
\|uv-e^{i\theta \pi }vu\|>\delta
\]
for a pair of unitaries, then there exists a pair of unitaries
u
1
u_1
and
v
1
v_1
in
B
B
such that
\[
u
1
v
1
=
e
i
θ
π
v
1
u
1
,
‖
u
1
−
u
‖
>
ε
and
‖
v
1
−
v
‖
>
ε
.
u_1v_1=e^{i\theta \pi }v_1u_1,\,\,\,\,\,\|u_1-u\|>\varepsilon \quad \text {and} \quad \|v_1-v\|>\varepsilon .
\]