11.2. NF and strong NF algebras 317Showing that A is isomorphic to the induced limit is reasonably straight-
forward - one verifies directly that the an 's give rise to an injective *-
homomorphism from A onto g lim(Am, --+ 'Pn , m)· Just string together the an's
as in the definition of g~(Am, 'Pn,m)· Everything is well-defined and iso-
metric on the span of the Sn's; hence we can extend to an isometry on all of
A and this extension has no choice but to be self-adjoint and multiplicative.
We leave the details to the reader. 0
Definition 11.1.6. A separable C*-algebra is called MF if it satisfies one
of the equivalent conditions from Theorem 11.1.5.^3
Taking the norm-microstate point of view, the following permanence
properties are easy.
Proposition 11.1.7. The following statements hold:
(1) Subalgebras of MF algebras are MF.
(2) Inductive limits (usual or even generalized} of MF algebras are
again MF.Proof. Just think about it.
We close this section with an inductive limit characterization. Let
AC ITnENMs(n)(C)
EBnENMs(n)(C)be given and E C IT Ms(n) (C) be the pullback of A. If
Pn = 0 EEl ... EEl 0 EEl ls(n) EEl ls(n+l) EEl · · · E II Ms(n) (C)
0and if we consider the inductive system E -+ P1E -+ P2E -+ · · ·, it is
readily seen that A is the corresponding inductive limit. Hence we have
Proposition 11.1.8. A C-algebra is MF if and only if it is isomorphic to
an (old-fashioned} inductive limit of RFD C -algebras.
11.2. NF and strong NF algebras
We now restrict to the case of c.c.p. connecting maps. This turns out to
describe an old friend: nuclear QD C* -algebras. Restricting further to con-
necting maps which are complete order embeddings is also interesting.
Definition 11.2.1. A u.c.p. map <p: A -+ B is called a complete order
embedding if it is completely isometric.
3This terminology is due to Blackadar and Kirchberg and abbreviates "matricial field". We
haven't presented the characterization which inspires the terminology, however.