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.