CHAPTER 1

General representations of Od on a separable

Hilbert space

Representations of the Cuntz algebras Od play a role in several recent papers;

see, e.g., [37], [16], [21], [22], [23], [31], [93], [53]. Since Od is purely infinite, there

are few results that cover all representations. The following result does just that,

and serves as a "noncommutative spectral resolution". We will use the convention

that Si denotes the representative of the Cuntz algebra generator Si in any given

representation.

Let ft = Y\^° Zd, and let a be the right shift on Vt\

(1.1) (j(xi,x

2

,...) = (x

2

,x

3

,...)

Define sections Oi of a by

(1.2) 7 i ( x i , x

2

, . . . ) = (hXuX2,--.),

for i = 1,..., d. Note that a is a d-to-1 map and that the sections are injective.

The sets

(1.3) fii - Oi (fi)

form a partition of the Cantor set SI into clopen sets. The c^ are right inverses of

a:

(1.4) GGi = id

for i = 1,..., d. If/i is a probability measure on O, we say that ji is cr-quasi-invariant

if

(1.5)

fl

(E) = 0^fi(o~1(E))=0

for all Borel sets E C tt, and we say that /i is cr^-quasi-invariant if

(1.6) ^(E)=0=^^(a-1(E))=0

for all Borel sets E C fi, where

(y~x{E) = {x | Gi(x) = (i,xi,x2,...) eE} ,

a'1 (E) = {x\ *(x) = (x2,x3,...) e E}.

The set of d conditions (1.6) is equivalent to the set of d conditions

(1.8) VL(Ti(F))=*»(F)=0

for i — 1,..., d. The condition (1.5) is implied by, but does not imply, the condition

(1.9) fi(a(F))=0=n(F) = 0.

3