11.1. Generalized inductive limits 315on polynomials provide better and better approximations, to get represent-
ing sequences in the quotient.
Proposition 11.1.4. The C -algebra A admits norm microstates if and
only if there exist a sequence s(n) in N and a -homomorphic embedding
AC TinEN Ms(n) (CC)
ffinENMs(n)(CC).
Note that every QD C -algebra admits norm microstates -this is imme-
diate from the definition -but there are other examples as well (see Sections
13.5 and 17.3).
For a sequence s(n) EN, we still let 7r: IJrMs(n)(CC) ,~:~~be
the quotient map, but we also need the canonical projection maps
00 q
p~: IJMs(n)(CC) , IJMs(n)(CC).
1 p
Theorem 11.1.5. Let A be a separable C -algebra. The following state-
ments are equivalent:
(1) A is isomorphic to a generalized inductive limit of finite-dimensional
C -algebras;
(2) there exist integers s(n) EN and a -homomorphic embedding
AC ITnENMs(n)(CC).
ffinEN Ms(n)(CC)'
(3) A admits norm microstates.
Proof. Since every finite-dimensional C* -algebra embeds into a full matrix
algebra, the implication (1) ==?-(2) is trivial; hence we only have to show the
converse.
So, assume an embedding
AC ITnENMs(n)(CC)
ffinEN Ms(n) (CC)
is given and fix a linear self-adjoint splitting O": A_____, ITnENMs(n)(CC) (i.e.,
7r o O" = idA).^2 Also fix some finite-dimensional self-adjoint subspaces
S1 c S2 c · · · c A
whose union is dense in A and with the property that
{ab : a, b E Sn} C Sn+l2This map comes from linear algebra and isn't necessarily bounded. If er doesn't preserve
adjoints, then replace it with ~ (er + er*).