1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

7r o <I>m (A m ) C ffi finENAn A.

WnEN n

Note that the norm closure of the union Um 7r o <I>m(Am) is a C* -algebra

since the definition of a generalized inductive system implies that for each

a, b E Am, the product 7r(<I>m(a))7r(<I>m(b)) belongs to this closure (though

it's not close to 7r(<I>m(ab)), in general).

Definition 11.1.2. If (Am, 'Pn,m) is a generalized inductive system, the

generalized inductive limit, A= g lim(Am, 'Pn m), is the norm closure of

---> '

U (A )

finEN An

7r o <I>m m C ffi A.

mEN WnEN n

Typically we forget about the map 7r and let <I>m: Am --+ A denote the

natural adjoint-preserving linear map.

We will primarily be concerned with the case where each Am is a finite-

dimensional C* -algebra. In this setting there is a related notion which arose

from important work in the context of Voiculescu's free probability theory.

Definition 11.1.3. Let A be a C*-algebra with a finite generating set

{xk}k= 1 C A of self-adjoint elements. We say that A admits norm mi-

crostates if for every finite set ~ of noncommutative polynomials in s vari-

ables and r:; > 0 there exist n E N and self-adjoint matrices Xi, ... , X 8 E

Mn ( C) such that for every P E ~ we have

I llP(x1, · · ·, Xs) II - llP(X1, · · ·, Xs) 11 I < €.

The definition in the nonfinitely generated case is similar, except that one

has to accommodate all noncommutative polynomials in any number of vari-

ables.

The proof of the following proposition is rather technical to write out

precisely, though conceptually it is very easy. We leave the details to the

reader, but the idea is to string together sequences of matrices, whose norms