of W), we also have
and hence
(2.2.6) Wyg^wtf^ = WV,W.®^g..
In parallel with our introductory discussion of the case d = 1, let us think of W
as an operator acting on certain formal function spaces. For this purpose, we let
z ( z i , . . . , Zd) and £ = (Ci i Cd) D e two s e ^ s °f noncommuting indeterminants.
If v = gin .. . g^ is a word in Td, we define zv to be zin ... Zix and similarly for (v.
The rules are that z^'s do not commute with each other, Q's do not commute with
each other, but we do have z^j = Qzi for i,j = 1,..., d. We denote by L(Td x T&-, £)
the space of all formal power series f(z, () = J ^
fv,wZv(w m the two sets of
indeterminants (z,() with coefficients fViW G £. The subspace V{Td x Td,£) of
"polynomials" consists of all such formal series f(z,() = J^v ^ e . ^ fv,wZvC,w such
that / y ^ = 0 for all but finitely many v,w G JF^ x ^ . Given a block matrix
W = [Wv,w;a,p] with rows and columns indexed by Td x J ^ as above, we may think
of W as an operator from V(Td x Td-£) to L{Td, xTd,£) by defining
/ = Wp if /„,„, = ^ WVjW]0t,i3Pa,(3 where p(z, C) = ^ P a , / ^ " ^
(where the sum on the right is finite since p is assumed to be a polynomial). If W
arises from a row-unitary as in (2.2.2), then W is positive semidefinite in the sense
(2.2.7) (Wp,p)L2 0 for all p G V(Td x Td, £).
Here we are using that the formal L2 inner product
(/PL2 = Yl (fv,w,Pv,w)£
gives a well-defined pairing between L(Td x Td,£) and V{Td x Td,£), since the
sum defining the inner product for such a pair collapses to a finite sum. We may
therefore define a space Cw a s the completion of WV{Td x Td,£) in the inner
Following the terminology of [BaV04], we say that W is a Cuntz weight if VF is a
positive semidefinite Haplitz operator with the Cuntz property, i.e., if W satisfies
(2.2.3), (2.2.4), (2.2.5) and (2.2.7). So far we have argued that any W arising from
a row-unitary hi as in (2.2.2) is a Cuntz weight; for complete details, we refer to
Since, for any / G WV(Td x Td, £), e G £ and v,w G Td we have
(2.2.8) (/, W[zaC0e])Cw = /, za^e)L2 = (fa0,e)s,
we see that the map / \-^ fv^w of / to a given one of its Fourier coefficients is
bounded in £w-norm, and hence elements of the completion / may still be identified
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