Embedding of exact C*-algebras and continuous fields in the Cuntz algebra O_2
Abstract
We prove that any separable exact C*-algebra is isomorphic to a subalgebra of the Cuntz algebra
O
2
.
We further prove that if
A
is a simple separable unital nuclear C*-algebra, then
O
2
⊗A≅
O
2
,
and if, in addition,
A
is purely infinite, then
O
∞
⊗A≅A.
The embedding of exact C*-algebras in $\OA{2}$ is continuous in the following sense. If
A
is a continuous field of C*-algebras over a compact manifold or finite CW complex
X
with fiber
A(x)
over
x∈X,
such that the algebra of continuous sections of
A
is separable and exact, then there is a family of injective homomorphisms
ϕ
x
:A(x)→
O
2
such that for every continuous section
a
of
A
the function
x↦
ϕ
x
(a(x))
is continuous. Moreover, one can say something about the modulus of continuity of the functions
x↦
ϕ
x
(a(x))
in terms of the structure of the continuous field. In particular, we show that the continuous field
θ↦
A
θ
of rotation algebras posesses unital embeddings
ϕ
θ
in
O
2
such that the standard generators
u(θ)
and
v(θ)
are mapped to
Lip
1/2
functions.