Basic Properties

Proposition 2.9. The families of seminorms q_∆and p_∆onLandC, respectively, where the symbols∆denote bounded Borel sets, constitute nets with respect to the inclusion relation. For any∆and∆⁰we have

∆⊆∆⁰ ⇒ q_∆(L)6q_∆0(L), L∈L,

∆⊆∆⁰ ⇒ p_∆(C)6p_∆0(C), C∈C.

Proof. For the q_∆-seminorms on L the assertion follows from the order relation for operators inB(H)⁺. Let L belong to the left idealL, then

Q^(L_∆^∗L)6Q^(L_∆0^∗L), which by Definition2.8has the consequence

q_∆(L)²= Q^(L_∆^∗L)

6

Q^(L_∆0^∗L)

=q_∆⁰(L)². This relation extends by continuity of the seminorms to all ofL.

In case of the p_∆-topologies, note that for any Borel set ∆ the functional φ^E(∆), defined throughφ^E(∆)(.) .

=φ(E(∆).E(∆)), belongs toB(H)_∗,1ifφdoes. From this we infer, since moreover∆⊆∆⁰implies E(∆) =E(∆)E(∆⁰) =E(∆⁰)E(∆), that

nZ

R^s

d^sx

φ E(∆)αx(C)E(∆)

:φ∈B(H)_∗,1o

⊆nZ

R^s

d^sx

φ E(∆⁰)αx(C)E(∆⁰)

:φ∈B(H)_∗,1o for any C∈Cand thus, by (2.17b), that p_∆(C)6p_∆0(C), a relation which by continuity of the seminorms is likewise valid for any operator in the completionC.

The continuous extensions of the seminorms q_∆ and p_∆ toL andC, respectively, can be explicitly computed on the subspacesA_L andA_CofA.

2.3 Characteristics of the Spectral Seminorms 15

Lemma 2.10. Let∆denote an arbitrary bounded Borel subset ofR^s+1. (i) For any L∈A_Lwe have

q_∆(L) =sup nZ

R^s

d^sxω E(∆)αx(L^∗L)E(∆)

:ω∈B(H)⁺_∗,1 o1/2

. (2.18a)

(ii) For any C∈A_Cthere holds the relation p_∆(C) =sup

nZ

R^s

d^sx

φ E(∆)αx(C)E(∆)

:φ∈B(H)_∗,1

o

. (2.18b)

Proof. (i) Note, that we can define a linear subspaceA_q0 ofAconsisting of all those operators L⁰which fulfill

q⁰_∆(L⁰)² .

=sup nZ

R^s

d^sxω E(∆)αx(L^0∗L⁰)E(∆)

:ω∈B(H)⁺_∗,1o

<∞

for any bounded Borel set∆. On this space the mappings q⁰_∆act as seminorms whose restrictions toLcoincide with q_∆ (cf. the Remark following Definition2.8). Now let L∈A_Lbe arbitrary. Given a bounded Borel set∆we can then find a sequence

Ln n∈N

inLsatisfying

nlim→∞q_∆(L−Ln) =0 and lim

n→∞kL−Lnk=0.

The second equation implies

n→∞limkLE(∆)−LnE(∆)k=0,

so that Lebesgue’s Dominated Convergence Theorem can be applied to get for any functionalω∈B(H)⁺_∗,1and any compact K⊆R^s

Z

K

d^sxω E(∆)αx(L^∗L)E(∆)

=lim

n→∞

Z

K

d^sxω E(∆)αx(L_n^∗L_n)E(∆) .

According to (2.17c) each term in the sequence on the right-hand side is majorized by the corresponding q_∆(Ln)² and this sequence in turn converges to q_∆(L)² by assump-tion, so that in passing from K to R^s and to the supremum over all ω∈B(H)⁺_∗,1we get

sup nZ

R^s

d^sxω E(∆)αx(L^∗L)E(∆)

:ω∈B(H)⁺_∗,1o

6q_∆(L)².

This final estimate shows, by arbitrariness of L∈A_L and the selected∆, thatA_L is a subspace ofA_q0 and, from q⁰_∆L=q_∆, it eventually follows that for all these L and∆

q_∆(L) =supnZ

R^s

d^sxω E(∆)αx(L^∗L)E(∆)

:ω∈B(H)⁺_∗,1o1/2

.

16 Localizing Operators and Spectral Seminorms

(ii) The proof of the second part follows the same lines of thought. We introduce the subspaceA_p0⊆Aconsisting of operators C⁰satisfying

p⁰_∆(C⁰) .

=supnZ

R^s

d^sx

φ E(∆)αx(C⁰)E(∆)

:φ∈B(H)_∗,1o

<∞

for any bounded Borel set∆and furnish it with the locally convex topology defined by the seminorms p⁰_∆. An arbitrary C∈A_Cis, for given ∆, approximated by a sequence Cn n∈Nwith respect to the norm and the p_∆-topology. As above one has

n→∞limkE(∆)CE(∆)−E(∆)CnE(∆)k=0 and infers

sup nZ

R^s

d^sx

φ E(∆)αx(C⁰)E(∆)

:φ∈B(H)∗,1

o

6p_∆(C).

This establishes, by arbitraryness of C∈A_C and∆, that A_C⊆A_p0, and the equation p⁰_∆C=p_∆implies that for these C and∆

p_∆(C) =sup nZ

R^s

d^sx

φ E(∆)αx(C)E(∆)

:φ∈B(H)_∗,1

o .

An immediate consequence of this result is the subsequent lemma, which in some way reverts the arguments given in the concluding remark of the last section in order to establish the Hausdorff property for(L,T_q)and(C,T_p).

Lemma 2.11. Let∆be a bounded Borel set.

(i) For L∈A_Lwith LE(∆) =0 there holds q_∆(L) =0.

(ii) If C∈A_Csatisfies E(∆)CE(∆) =0 one has p_∆(C) =0.

Next we deal with an implication of the fact, thatLis an ideal of the C^∗-algebraA, and clarify the relationship between the seminorms q_∆and p_∆.

Lemma 2.12. Let∆denote bounded Borel subsets ofR^s+1. (i) A_L is a left ideal of the quasi-local algebraAand satisfies

q_∆(AL)6kAkq_∆(L) (2.19)

for any L∈A_Land A∈A.

(ii) Let Li, i=1,2, be operators inA_L and A∈A, then L1∗AL2 belongs toA_C. If in addition the operators Li have energy-momentum transfer inΓi⊆R^s+1 and∆i are Borel subsets ofR^s+1containing∆+Γi, respectively, then

p_∆(L1∗AL2)6kE(∆1)AE(∆2)kq_∆(L1)q_∆(L2). (2.20) Proof. (i) For any L∈A_L⊆A and arbitrary A∈Athe relation L^∗A^∗AL6kAk²L^∗L leads to the estimate

Z

R^s

d^sxω E(∆)αx(L^∗A^∗AL)E(∆)

6kAk² Z

R^s

d^sxω E(∆)αx(L^∗L)E(∆)

2.3 Characteristics of the Spectral Seminorms 17

for anyω∈B(H)⁺_∗,1and thus, by (2.18a) and the notation of the proof of Lemma2.10, to

q⁰_∆(AL) =supnZ

R^s

d^sxω E(∆)αx(L^∗A^∗AL)E(∆)

:ω∈B(H)⁺_∗,1o1/2

6kAksup nZ

R^s

d^sxω E(∆)αx(L^∗L)E(∆)

:ω∈B(H)⁺_∗,1 o1/2

=kAkq_∆(L).

This shows that AL belongs toA_Land at the same time that the seminorm q⁰_∆(onA_q0) can be replaced by q_∆to yield (2.19).

(ii) Letφbe a normal functional onB(H)withkφk61. By polar decomposition there exist a partial isometry V and a positive normal functional|φ|withk|φ|k61 such that φ(.) =|φ|(.V). Then

φ E(∆)αx(L1∗AL2)E(∆)

=|φ| E(∆)αx(L₁^∗)E(∆1)αx(A)E(∆2)αx(L₂)E(∆)V 6kE(∆1)αx(A)E(∆2)k

q

|φ| E(∆)αx(L1∗L1)E(∆)q

|φ| V^∗E(∆)αx(L2∗L2)E(∆)V for any x∈R^s and the method used in the proof of Proposition2.7 can be applied to get, in analogy to (2.16),

supnZ

R^s

d^sx

φ E(∆)αx(L₁^∗AL₂)E(∆)

:φ∈B(H)_∗,1o

6kE(∆1)AE(∆2)kq_∆(L₁)q_∆(L₂), where we made use of (2.18a). According to the notation introduced in the proof of Lemma2.10this result expressed in terms of the seminorm p⁰_∆ onA_p0 reads

p⁰_∆(L₁^∗AL2)6kE(∆1)AE(∆2)kq_∆(L₁)q_∆(L₂),

from which we infer, as in the first part of the present proof, not only that L1∗AL2is an element ofA_Cbut also that p⁰_∆can be substituted by p_∆ to give (2.20).

The second part of the above lemma means that the product L1∗L2, defined by two operators L₁,L₂∈A_L, is continuous with respect to the locally convex spaces (cf. [44, Chapter Four, § 18, 3.(5)])(A_L,T^u_q)×(A_L,T^u_q)and(A_C,T^u_p).

Corollary 2.13. The sesquilinear mapping on the topological product of the locally convex space(A_L,T^u_q)with itself, defined by

A_L×A_L3(L₁,L₂)7→L₁^∗L₂∈A_C,

is continuous with respect to the respective locally convex topologies.

In the special case of coincidence of both operators (L1 =L2 =L) it turns out that p_∆(L^∗L)equals the square of q_∆(L). Another result involving the operation of adjunc-tion is the fact, that this mapping leaves the p_∆-seminorms invariant.

Lemma 2.14. Let∆denote the bounded Borel sets inR^s+1.

18 Localizing Operators and Spectral Seminorms

(i) For any operator L∈A_L there hold the relations p_∆(L^∗L) =q_∆(L)². whereas the reverse inequality is a consequence of Lemma2.12. This proves the asser-tion.

(ii) Note, thatB(H)_∗,1 is invariant under the operation of taking adjoints defined by ψ7→ψ^∗withψ^∗(A) . for any C∈A_p0 (cf. the proof of Lemma2.10), which is sufficient to establish both of the assertions.

The last statement of this subsection on basic properties of the spectral seminorms establishes their invariance under translations in the s+1-dimensional configuration space.

Lemma 2.15. The subspaces A_L andA_C of the quasi-local algebraAare invariant under translations. In particular, let∆be a bounded Borel set inR^s+1and let x∈R^s+1 be arbitrary, then

2.3 Characteristics of the Spectral Seminorms 19

Therefore the introductory remark in combination with (2.18a) yields for any L∈A_L: q⁰_∆ αx(L)2 which, as in the proof of Lemma2.12, establishes the assertions.

(ii) The same argument applies to the seminorm p⁰_∆, so that for C∈A_C

The assumed strong continuity of the automorphism groupα_(Λ,x):(Λ,x)∈P₊^↑ acting on the C^∗-algebraAcarries over to the locally convex spaces(L,T_q)and(C,Tp); and even the infinite differentiability of (Λ,x)7→ α(Λ,x)(L₀) for L₀∈L₀ is conserved in passing from the uniform topology onL₀to that induced by the seminorms q_∆. Proposition 2.16. (i) For fixed L∈Lthe mapping

ΞL:P₊^↑ →L (Λ,x)7→ΞL(Λ,x) .

=α_(Λ,x)(L) is continuous with respect to the locally convex space(L,Tq).

(ii) For given C∈Cthe mapping

ΞC:P₊^↑ →C (Λ,x)7→ΞC(Λ,x) .

=α_(Λ,x)(C) is continuous with respect to the locally convex space(C,T_p).

(iii) Let idL0 denote the identity mapping

id_L0 :(L₀,k.k)→(L₀,T_q) L₀7→id_L0(L₀) .

=L₀

on the spaceL₀ , once endowed with the norm topology and once withT_q. Consider furthermore the familyXL0 of infinitely often differentiable mappings:

XL0

=. ΞL0: L0∈L₀ , ΞL0 :P₊^↑ →L₀ (Λ,x)7→ΞL0(Λ,x) .

=α_(Λ,x)(L0).

Then the linear operator id_L0 isXL0-differentiable in the sense of DefinitionA.16.

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