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,x 2 ,...) 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
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