Definition 0.1
A
C*-algebra

is called a
nuclear C*-algebra if all C*-norms on every algebraic
tensor
product

, of

with any other C*-algebra

, agree with, and also equal the spatial C*-norm (
viz Lance, 1981). Therefore, there is a unique completion of

to a C*-algebra , for any other C*-algebra

.
Definition 0.2
Let

be a matrix space, let

be a general
operator
space, and also let

be a C*-algebra.
A

-algebra

is exact if it is `finitely representable' in

, that is, if for every finite dimensional subspace

in

and quantity

, there exists a subspace

of some

, and
also a linear isomorphism

such that the

-norm