8 A. L. CAREY, V. GAYRAL, A. RENNIE, and F. A. SUKOCHEV
Definition 1.1. For any s 0, we define the weight ϕs on N by
T N+ ϕs(T ) := τ
(
(1 +
D2)−s/4T
(1 +
D2)−s/4
)
[0, +∞].
As usual, we set
dom(ϕs) := span{dom(ϕs)+} = span
(
dom(ϕs)1/2
)∗
dom(ϕs)1/2}
N ,
where
dom(ϕs)+ := {T N+ : ϕs(T ) ∞} ,
dom(ϕs)1/2
:= {T N : T
∗T
dom(ϕs)+}.
In the following, dom(ϕs)+ is called the positive domain and
dom(ϕs)1/2
the
half domain.
Lemma 1.2. The weights ϕs, s 0, are faithful normal and semifinite, with
modular group given by
N T (1 +
D2)−is/2T
(1 +
D2)is/2.
Proof. Normality of ϕs follows directly from the normality of τ. To prove
faithfulness of ϕs, using faithfulness of τ, we also need the fact that the bounded
operator (1 +
D2)−s/4
is injective. Let S
dom(ϕs)1/2
and T :=
S∗S
dom(ϕs)+
with ϕs(T ) = 0. From the trace property, we obtain ϕs(T ) = τ(S(1 + D2)−s/2S∗),
so by the faithfulness of τ, we obtain 0 = S(1 +
D2)−s/2S∗
= |(1 +
D2)−s/4S∗|2,
so (1 +
D2)−s/4S∗
= 0, which by injectivity implies
S∗
= 0 and thus T = 0.
Regarding semifiniteness of ϕs, one uses semifiniteness of τ to obtain that for any
T N+, there exists S N+ of finite trace, with S (1 +
D2)−s/4T
(1 +
D2)−s/4.
Thus S := (1 + D2)s/4S(1 + D2)s/4 T is non-negative, bounded and belongs to
dom(ϕs)+, as needed. The form of the modular group follows from the definition
of the modular group of a weight.
Domains of weights, and, a fortiori, intersections of domains of weights, are ∗-
subalgebras of N . However, dom(ϕs)1/2 is not a ∗-algebra but only a left ideal in N .
To obtain a ∗-algebra structure from the latter, we need to force the ∗-invariance.
Since ϕs is faithful for each s 0, the inclusion of dom(ϕs)1/2 (dom(ϕs)1/2)∗ in its
Hilbert space completion (for the inner product coming from ϕs) is injective. Hence
by [57, Theorem 2.6], dom(ϕs)1/2 (dom(ϕs)1/2)∗ is a full left Hilbert algebra.
Thus we may define a ∗-subalgebra of N for each p 1.
Definition 1.3. Let D be a self-adjoint operator affiliated to a semifinite von
Neumann algebra N with faithful normal semifinite trace τ. Then for each p 1
we define
B2(D,p) :=
sp
dom(ϕs)1/2 (dom(ϕs)1/2)∗
.
The norms
(1.1) Qn(T ) :=
(
T
2
+ ϕp+1/n(|T
|2)
+ ϕp+1/n(|T
∗|2)
)1/2
, n N,
take finite values on B2(D,p) and provide a topology on B2(D,p) stronger than the
norm topology. Unless mentioned otherwise we will always suppose that B2(D,p)
has the topology defined by these norms.
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