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
Let ft = Y\^° Zd, and let a be the right shift on Vt\
(1.1) (j(xi,x
,...) = (x
Define sections Oi of a by
(1.2) 7 i ( x i , x
, . . . ) = (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
(1.4) GGi = id
for i = 1,..., d. If/i is a probability measure on O, we say that ji is cr-quasi-invariant
(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.
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