1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

2 1. Fundamental Facts

Here Mn(C) is then x n complex matrices and tr is its unique tracial

state. A collection of n x n matrices { ei,j h.::;i,j:s;n is called a system of matrix.

units if ei jest = 5j 8 ei t· It is often convenient to index our matrices over a

finite set F, in whi~h ~ase we write Mp(C) (= JR(.€^2 (F))) and let {ep,q}p,qEF

denote the canonical matrix units.

We reserve A, B, C and D for C* -algebras while M and N will typically

denote von Neumann algebras. We let Asa be the self-adjoint elements, Ai

the closed unit ball and A+ the positive elements in A. The symbols E and

F will denote operator systems (or operator spaces). We usually use I and

J for ideals in C*-algebras (e.g. I <l A), though they are occasionally index

sets too. All ideals are assumed closed and two-sided.

The set of states on A -positive linear functionals of norm one - will be

denoted S(A). If <p E S(A), we let L^2 (A, <p) be the GNS (Gelfand-Naimark-

Segal) Hilbert space and Jr'P: A--+ JIB(L^2 (A,<p)) be the GNS representation.

For an element a EA, we let a E L^2 (A, <p) denote its natural image.

1.2. C*-algebras

Quasicentral approximate units. Quasicentral approximate units are an

indispensable tool. See [53, Theorem I.9.16] for a proof of the following fact.

Theorem 1.2.1. Let I <l A be an ideal. Then I has an approximate unit

{ ei} C I such that lleia - aeill --+ 0, as i --+ oo, for all a E A. In fact,

if {fk} c I is any approximate unit for I, then a quasicentral approximate

unit can always be extracted from its convex hull.

We don't need it too many times, but it is worth mentioning that qua-
sicentral approximate units allow a particular type of approximate decom-
position.

Proposition 1.2.2. Let A be unital and { ei} c I <l A be a quasicentral
approximate unit. For every pair a, b E A such that a - b E I we have

Proof. First notice that for every x EA and polynomialp we have llp(ei)x-
xp(ei)ll --+ 0 (by some standard estimates). Since continuous functions can
1 1
be approximated 1 by polynomials, 1 it follows that II e[ x - xe i ~II --+ 0 and
II (1 - ei) 2 x - x(l - ei) 211 --+ 0.

Next, observe that
1 1 1 1 1 1
II (1 - ei) 2 a(l - ei) 2 - (1 - ei) 2 b(l - ei) 211 = II (1-ei) 2 (a - b) (1 - ei) 211 --+ 0