1. PSEUDODIFFERENTIAL CALCULUS AND SUMMABILITY 9

Notation. Given a semifinite von Neumann algebra N with faithful normal

semifinite trace τ, we let

˜p(N

L , τ), 1 ≤ p ∞, denote the set of τ-measurable

operators T aﬃliated to N with τ(|T

|p)

∞. We do not often use this no-

tion of p-integrable elements, preferring to use the bounded analogue,

Lp(N

, τ) :=

˜p(N

L , τ) ∩ N , normed with T → τ(|T |p)1/p + T .

Remarks. (1) If

(1+D2)−s/2

∈

L1(N

, τ) for all (s) p ≥ 1, then B2(D,p) =

N , since then the weights ϕs, s p, are bounded and the norms Qn are all

equivalent to the operator norm.

(2) The triangle inequality for Qn follows from the Cauchy-Schwarz inequal-

ity applied to the inner product T, S

n

= ϕp+1/n(T

∗S),

along with the equality

Qn(T )2 = T 2 + T, T

n

+ T ∗,T ∗

n

. In concrete terms, an element T ∈ N belongs

to B2(D,p) if and only if for all s p, both T (1 +

D2)−s/4

and T

∗(1

+

D2)−s/4

belong to

L2(N

, τ), the ideal of τ-Hilbert-Schmidt operators.

(3) The norms Qn are increasing, in the sense that for n ≤ m we have Qn ≤ Qm.

We leave this as an exercise, but observe that this requires the cyclicity of the trace.

The following result of Brown and Kosaki gives the strongest statement on this

cyclicity. By the preceding Remark (2), we do not need the full power of this result

here, but record it for future use.

Proposition 1.4. [8, Theorem 17] Let τ be a faithful normal semifinite trace

on a von Neumann algebra N , and let A, B be τ-measurable operators aﬃliated to

N . If AB, BA ∈

˜1(N

L , τ) then τ(AB) = τ(BA).

Another important result that we will frequently use comes from Bikchentaev’s

work.

Proposition 1.5. [6, Theorem 3] Let N be a semifinite von Neumann algebra

with faithful normal semfinite trace τ. If A, B ∈ N satisfy A ≥ 0, B ≥ 0, and are

such that AB is trace class, then

B1/2AB1/2

and

A1/2BA1/2

are also trace class,

with τ(AB) =

τ(B1/2AB1/2)

=

τ(A1/2BA1/2).

Next we show that the topological algebra B2(D,p) is complete and thus is a

Fr´ echet algebra. The completeness argument relies on the Fatou property for the

trace τ, [26].

Proposition 1.6. The ∗-algebra B2(D,p) ⊂ N is a Fr´ echet algebra.

Proof. Showing that B2(D,p) is a ∗-algebra is routine with the aid of the

following argument. For T, S ∈ B2(D,p), the operator inequality

S∗T ∗TS

≤

T

∗T S∗S

shows that

ϕp+1/n(|TS|2)

=

ϕp+1/n(S∗T ∗TS)

≤ T

2ϕp+1/n(|S|2),

and, therefore, Qn(TS) ≤ Qn(T ) Qn(S).

For the completeness, let (Tk)k≥1 be a Cauchy sequence in B2(D,p). Then

(Tk)k≥1 converges in norm, and so there exists T ∈ N such that Tk → T in N .

For each norm Qn we have | Qn(Tk) − Qn(Tl) | ≤ Qn(Tk − Tl), so we see that the

numerical sequence (Qn(Tk))k≥1 possesses a limit. Now since

(1+D2)−p/4−1/4nTk ∗ Tk(1+D2)−p/4−1/4n

→

(1+D2)−p/4−1/4nT ∗

T

(1+D2)−p/4−1/4n,