2.2. FORMAL P O W E R SERIES
Since W W^ (where X^^*^ is the adjoint of W with respect to the formal
L2-inner
product pairing V[Td x Fd,£) with L(Td x JF^,£)), we see from (2.2.18) that we
also have
(2.2.19)
WUpl*]
=
S?l*]W
on V(Td x fd,£).
It then follows from (2.2.18) and (2.2.19) that Cw is invariant under Up and 5-
and that alternate formulas for Wv^,j and W{^ applicable for a general element of
Cw are
Uwjf = UPf
(2.2.20) WWJ =
SfM/
for / G £
w
for j = 1,..., d. A consequence of the formulas (2.2.12) and (2.2.14) combined with
the intertwining relations (2.2.18) and (2.2.19) are the following formulas for the
action ofUwj a n d ^w,j o n a g e n e r a l element / of Cw'-
(2.2.21) Uwj •• f(z,0 ~ /(0,C) C71 + f(z,Q ZJ,
(2.2.22) Ukj:f(z,0 » /(0, C) 0 + /(*, C) zj1.
Conversely, suppose that W is any Cuntz weight. Since W is positive semi-
definite (as in (2.2.7)), we may define a Hilbert space Cw a s the completion of
W - V{Td x J~d,£) with respect to the inner product (•, •)cw given by (2.1.1) and
define operators Uwj by (2.2.12) for j = 1,..., d. Then the Haplitz property of W
guarantees that Uw = (Mw,i, Mw,d) is row-isometric with the adjoint U-wj °f
Uw,j given by (2.2.14) for j = 1,..., d. Furthermore, the Cuntz property (2.2.5) of
W then guarantees that Uw is actually row-unitary. Moreover, the Fourier repre-
sentation operator &w given by (2.1.2) (with Uw in place of U) is the identity on
Cw and we have the concrete model version of (2.2.11):
(2.2.23) W = $w$w
cw
= $w£w.
It is often more convenient to work with the symbol
w(z,o= Yl
wvwzvr
of W, the formal power series in (z, £) defined by
W(z,£)e = (We)(z,C) for e G £
(where e is identified with the polynomial e = ez^C^ G V(Td, x ^ , £ ) ) . As ex-
plained in [BaV04], any [*]-Haplitz operator (i.e., W for which both W and W^
satisfy (2.2.3) and (2.2.4)) is uniquely determined by its symbol W(z,Q according
to the formula
(2 2 24) W
fl
-
JI*W10T.'' i f
M^H,
( V ^ H -
1
)
1 lf \v\
^
\al
and, conversely, any formal power series W(zX) = Y^v werd
Wv,wzv£w
with co-
efficients WViW equal to operators on £ determines a [*]-Haplitz operator W by
(2.2.24). The action of W on any polynomial /(z, C)
c a n
be computed directly
from the symbol W(z,Q according to the formula
(2.2.25) W[f}(z,0 = W(z',C)kPer(z,Of(z,z'-1)\z,=0 + W(z\Of^z'-1)\z,=z.
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