We next show thatR ⊂M. Let A∈ R be a self-adjoint operator. Define the dense subspaceCA according to the spectral decomposition ofA:
CA:=
∪∞ n=1
ran(EA([−n, n])), where
A=
∫
R
λdEA(λ).
Since allξ∈CAis an entire analytic vector for A, we have eitAξ= lim
n→∞
∑n k=0
(itA)k k! ξ.
LetMA be a von Neumann algebra generated by{EA(J) ; J ∈ B(R)}, where B(R) is the one dimensional Borelσ-field. SinceMAis abelian, it is a finite von Neumann algebra. It is also clear that
Bn:=
∑n k=0
(itA)k
k! ∈(MA)∩R
and eitA ∈ MA. Since CA is completely dense for MA, Bn converges almost everywhere toeitAin (MA). AsMAis finite, we see thatBnconverges toeitAin the strong resolvent topology. On the other hand,Ris SRT-closed and therefore eitA∈R∩B(H) =M, for allt∈R. This impliesA∈M. For a general operator B∈R, using real-imaginary part decompositionB= Re(B) +iIm(B), we have B∈M.
We shall show that M ⊂ R. Let A ∈ M and A = U|A| be its polar decomposition, thenU ∈ M⊂R and |A| ∈ M. Let |A| =:∫∞
0 λdE|A|(λ) be the spectral decomposition of|A|. Put
xn:=
∫ n 0
λdE|A|(λ)∈M⊂R,
then xn converges to |A| in the strong resolvent topology. Thus |A| ∈ R. ThereforeA=U|A| ∈R.
The finiteness ofMfollows immediately from Theorem 2.21.
Note that for each finite von Neumann algebra M on a Hilbert space H, (M,H) is an object infRng.
The main result of this section is the next theorem.
Theorem 5.5. The category fRngis a tensor category. Moreover, fRng and fvNare isomorphic as a tensor category.
To prove this theorem, we need many lemmata. The proof is divided into several steps.
Next, we will define the tensor product R1⊗R2 of objects Ri (i = 1,2) in fRng (cf. Definition 5.9). For this purpose, let us review the notion of the tensor product of closed operators. LetA, Bbe densely defined closed operators on Hilbert spacesH, K, respectively. LetA⊗0B be an operator defined by
dom(A⊗0B) := dom(A)⊗algdom(B),
(A⊗0B)(ξ⊗η) :=Aξ⊗Bη, ξ∈dom(A), η∈dom(B).
It is easy to see thatA⊗0B is closable and denote its closure byA⊗B.
Lemma 5.6. Let M1, M2 be finite von Neumann algebras acting on Hilbert spacesH1,H2, respectively. LetA∈M1 andB∈M2. Then we haveA⊗B∈ M1⊗M2.
Proof. Letxi ∈M′i (i= 1,2). For anyξ∈dom(A) andη∈dom(B), we have (x1⊗x2)(ξ⊗η)∈dom(A⊗0B) and
{(x1⊗x2)(A⊗0B)}(ξ⊗η) ={(A⊗0B)(x1⊗x2)}(ξ⊗η).
Therefore, by the linearity, we have (x1⊗x2)(A⊗0B) ⊂(A⊗0B)(x1⊗x2).
Since (M1⊗M2)′=M′1⊗M′2is the strong closure ofM′1⊗algM′2, we have y(A⊗0B)⊂(A⊗B)y, for ally∈(M1⊗M2)′.
Therefore by the limiting argument, we havey(A⊗B)⊂(A⊗B)y, which implies A⊗B is affiliated withM1⊗M2.
Lemma 5.7. Let A, B be densely defined closed operators on Hilbert spaces H, K with cores DA,DB respectively. Then D := DA⊗alg DB is a core of A⊗B.
Proof. From the definition of A⊗B, for any ζ ∈ dom(A⊗B) and for any ε >0, there exists someζε=∑n
i=1ξi⊗ηi∈dom(A)⊗algdom(B) such that
||ζ−ζε||< ε, ||(A⊗B)ζ−(A⊗B)ζε||< ε.
Put
C:= max
1≤i≤n{||ξi||, ||Aξi||}+ 1>0.
SinceDB is a core ofB, there existsηεi ∈DB such that
||ηi−ηiε||< ε
nC, ||Bηi−Bηεi||< ε nC. Put
C′:= max
1≤i≤n{||ηεi||, ||Bηεi||}+ 1>0.
Similarly, sinceDA is a core ofA, there existsξiε∈ DA such that
||ξi−ξεi||< ε
nC′, ||Aξi−Aξiε||< ε nC′.
Defineζε:=
∑n i=1
ξεi ⊗ηiε∈ D. Then we have
||ζ−ζε|| ≤ ||ζ−ζε||+||ζε−ζε||
≤ε+
∑n i=1
||ξi⊗ηi−ξiε⊗ηiε||
≤ε+
∑n i=1
||ξi⊗ηi−ξi⊗ηεi||+
∑n i=1
||ξi⊗ηiε−ξεi ⊗ηiε||
≤ε+
∑n i=1
||ξi||||ηi−ηiε||+
∑n i=1
||ξi−ξεi||||ηεi||
≤ε+
∑n i=1
C· ε nC +
∑n i=1
ε nC′ ·C′
= 3ε. · · ·(∗) Furthermore,
||(A⊗B)ζ−(A⊗B)ζε||
≤ ||(A⊗B)ζ−(A⊗B)ζε||+||(A⊗B)ζε−(A⊗B)ζε||
≤ε+
∑n i=1
||Aξi⊗Bηi−Aξiε⊗Bηεi||
≤ε+
∑n i=1
||Aξi⊗Bηi−Aξi⊗Bηεi||+
∑n i=1
||Aξi⊗Bηiε−Aξiε⊗Bηεi||
≤ε+
∑n i=1
||Aξi||||Bηi−Bηiε||+
∑n i=1
||Aξi−Aξεi||||Bηεi||
≤ε+
∑n i=1
C· ε nC +
∑n i=1
ε nC′ ·C′
= 3ε · · ·(∗∗)
(∗) and (∗∗) impliesDis a core ofA⊗B.
Next lemma says that the tensor product of algebras of affiliated operators has a natural *-algebraic structures.
Lemma 5.8. LetM, Nbe finite von Neumann algebras acting on Hilbert spaces H, K respectively. LetA, C∈M, B, D∈N. Then we have
(1) (A⊗B)(C⊗D) =AC⊗BD.
(2) (A⊗B)∗=A∗⊗B∗.
(3) A+C⊗B+D=A⊗B+A⊗D+C⊗B+C⊗D.
(4) λ(A⊗B) =λA⊗B=A⊗λB (λ∈C).
Proof. (1). From Proposition 2.13, D1 :={ξ ∈dom(C);Cξ ∈ dom(A)} is a core of AC and D2 := {η ∈dom(D);Dη ∈dom(B)} is a core of BD. Define D:=D1⊗algD2, which is a core ofAC⊗BD. Since
dom((A⊗B)(C⊗D))⊃dom((A⊗B)(C⊗D))⊃ D, it holds that for anyζ=
∑n i=1
ξi⊗ηi∈ D, we have
(A⊗B)(C⊗D)ζ=
∑n i=1
ACξi⊗BDηi= (AC⊗BD)
∑n i=1
ξi⊗ηi
= (AC⊗BD)ζ.
Therefore (A⊗B)(C⊗D)⊃(AC⊗BD)|D. SinceDis a core ofAC⊗BD, we have (by taking the closure)
(A⊗B)(C⊗D)⊃AC⊗BD.
Since both operators belong toM⊗N by Lemma 5.6, we have (A⊗B)(C⊗D) =AC⊗BD.
by Proposition 2.14(2).
(2). It is easy to see that (A⊗B)∗⊃A∗⊗B∗. Since (A⊗B)∗ andA∗⊗B∗ are closed operators belonging to M⊗N, we have (A⊗B)∗ = A∗ ⊗B∗ by Proposition 2.14 (2).
(3) and (4) can be easily shown in a similar manner as in (1).
Now we shall define the tensor product R1⊗R2 of (R1,H1) and (R2,H2) in Obj(fRng). Let Mi be finite von Neumann algebras onHi such that Ri = Mi (i= 1,2), respectively (cf. Lemma 5.4). From Lemma 5.8, the linear space R1⊗algR2 spanned by {A1⊗A2 ; Ai ∈ Ri, i = 1,2} is a *-algebra. Since R1⊗algR2 is a subset ofM1⊗M2, it belongs toRC(H1⊗ H2). Therefore:
Definition 5.9. Under the above notations, we defineR1⊗R2to be the SRT-closure (forH1⊗ H2) ofR1⊗algR2.
Lemma 5.10. Let Ri (i= 1,2) be as above. ThenR1⊗R2 is also an object in fRng. More precisely, ifRi =Mi, where Mi is a finite von Neumann algebra (i= 1,2), thenM1⊗M2=M1⊗M2.
Proof. M1⊗M2⊂M1⊗M2: LetTi ∈Mi (i= 1,2). Then we can show that T1⊗T2∈M1⊗M2by Lemma 5.6. Therefore by the linearity, we obtain
M1⊗algM2⊂M1⊗M2.
As the left hand side is SRT-closed in RC(H1⊗ H2), we have M1⊗M2 ⊂ M1⊗M2.
M1⊗M2⊂M1⊗M2: It is clear thatM1⊗algM2⊂M1⊗M2. By the Kaplansky density theorem and Lemma 3.5, we haveM1⊗M2 ⊂M1⊗M2. By taking the SRT-closure, we obtainM1⊗M2⊂M1⊗M2.
The above Lemma says that (R1⊗R2,H1⊗ H2) is again an object infRng.
Next, we discuss the extension of morphisms in fvN to ones in fRng. It requires some steps.
Lemma 5.11. Let(M1,H1),(M2,H2)be finite von Neumann algebras. Then the mapping
(M1, SRT)×(M2, SRT)−→(M1⊗M2, SRT), (A, B)7−→A⊗B,
is continuous.
Proof. Let {Aα}α ⊂ M1, {Bα}α ⊂ M2 be SRT-converging nets and A ∈ M1, B ∈ M2 be their limits respectively. We should only show that the net {Aα⊗Bα}α converges toA⊗B in the strong resolvent topology.
Step 1. The above claim is true if all Aα, Bα, A and B are self-adjoint.
Indeed, it is easy to see that
eit(Aα⊗1)=eitAα⊗1→eitA⊗1 =eit(A⊗1),
so that, by the limiting argument and Themrem 3.9, the SRT-convergence of Aα⊗1 to A⊗1 follows. Similarly 1⊗Bα converges to 1⊗B in the strong resolvent topology. Therefore, by Lemma 5.8 and the SRT-continuity of the multiplication, we have
Aα⊗Bα= (Aα⊗1) (1⊗Bα)→(A⊗1) (1⊗B) =A⊗B.
Step 2. In a general case, by Lemma 5.8, we obtain Aα⊗Bα=
(
Re(Aα) +iIm(Aα) )⊗(
Re(Bα) +iIm(Bα) )
= Re(Aα)⊗Re(Bα) +iRe(Aα)⊗Im(Bα) +iIm(Aα)⊗Re(Bα)−Im(Aα)⊗Im(Bα)
→Re(A)⊗Re(B) +iRe(A)⊗Im(B) +iIm(A)⊗Re(B)−Im(A)⊗Im(B)
=A⊗B.
Hence the proof of Lemma 5.11 is complete.
Lemma 5.12. Let M be a finite von Neumann algebra on a Hilbert space H andeis a projection in M′, thenMe is also finite.
Proof. Easy.
Lemma 5.13. Let A be a densely defined closed operator on a Hilbert space H, K be a closed subspace of K such that PKA ⊂ APK. Then the operator B:=A|dom(A)∩K is a densely defined closed operator on K.
Proof. This is a straightforward verification.
The next proposition guarantees the existence and the uniqueness of the extension of morphisms infvNto the morphisms infRng. Note that the claim is not trivial, because manyσ-weakly continuous linear mappings between finite von Neumann algebras cannot be extended SRT-continuously to the algebra of affiliated operators. Indeed, we can not extend anyσ-weakly continuous state onMSRT-continuously on MifMis diffuse.
Proposition 5.14. Let M1,M2 be finite von Neumann algebras on Hilbert spacesH1,H2 respectively.
(1) For each SRT-continuous unital *-homomorphism Φ :M1 →M2, the restrictionφofΦontoM1is aσ-weakly continuous unital *-homomorphism fromM1 toM2.
(2) Conversely, for each σ-weakly continuous unital *-homomorphism φ: M1→M2, there exists a unique SRT-continuous unital *-homomorphism Φ :M1→M2 such thatΦ|M1=φ.
Proof. (1). We have to prove that Φ maps all bounded operators to bounded operators. For anyu∈U(M1) andξ∈dom(Φ(u)∗Φ(u)), we have
||Φ(u)ξ||2=⟨ξ,Φ(u)∗Φ(u)ξ⟩=⟨ξ,Φ(u∗u)ξ⟩
=⟨ξ,Φ(1)ξ⟩=||ξ||2.
Since dom(Φ(u)∗Φ(u)) is a (completely) dense subspace, Φ(u)∈M2 and Φ(u) is an isometry. Therefore the finiteness ofM2 implies Φ(u)∈U(M2). Thus, we see that Φ(U(M1))⊂U(M2). Since any element inM1 is a linear combination ofU(M1), Φ maps M1 intoM2. To show thatφ isσ-weakly continuous, it is sufficient to prove the (σ-) strong continuity on the unit ball, because it is a homomorphism. Since the strong resolvent topology coincides with the strong operator topology on the closed unit ball by Lemma 3.5,φis strongly continuous on the closed unit ball. Thereforeφis a σ-weakly continuous homomorphism.
(2). Regardφ as a composition of a surjection φ′ :M1 →φ(M1) and the inclusion mapι: φ(M1),→M2.Note that theσ-weak continuity of φimplies φ(M1) is a von Neumann algebra. Sinceφ′ is surjective, from Theorem IV.5.5 of [21], there exists a Hilbert space K, a projection e′ ∈P(M′1⊗B(K)) and a unitary operatorU : e′(H1⊗ K)→ H∼ 2such that
φ′(x) =U(x⊗1K)e′U∗
for allx∈M1. Now we want to define the extension Φ′ ofφ′ to M1→φ(M1).
Then we define Φ′ as follows:
Φ′(X) =U(X⊗1K)e′U∗, X ∈M1. M1
·⊗1
Φ′_ _ _ _//
_ _ _ _
_ φ(M1)
M1⊗C1K reduction bye//′(M1⊗C1K)e′
U·U∗
OO
More precisely, we define
Z = (X⊗1)e′ :=e′(X⊗1)|ran(e′)∩dom(X⊗1), Φ′(X) :=U ZU∗. We have Z ∈ (M1⊗C1K)e′. Indeed, since e′ commutes with M⊗C1K, it reduces the operatorX⊗1 and therefore by Lemma 5.13, (X⊗1)e′ is a densely defined closed operator on ran(e′). Since (Nf)′= (N′)f for each von Neumann algebra Nand f ∈P(N′), the affiliation property is manifest. In addition, by Lemma 5.12, (M⊗C1K)e′ is a finite von Neumann algebra. Next, we prove the map M∋ X 7→ (X ⊗1)e′ ∈ (M1⊗C1K)e′ is a SRT-continuous unital *-homomorphism. The continuity follows from Lemma 5.11. To prove that it is a
*-homomorphism, we have to show that forX, Y ∈M, ((X+Y)⊗1)e′= (X⊗1)e′+ (Y ⊗1)e′,
(XY ⊗1))e′= (X⊗1)e′(Y ⊗1)e′, ((X⊗1)e′)∗= (X∗⊗1)e′.
To prove the first equality, by Lemma 5.8, we see that ((X+Y)⊗1)
e′ =(
X⊗1 +Y ⊗1)
e′
⊃(X⊗1)e′ + (Y ⊗1)e′. Taking the closure, by Lemma 2.14, we have
((X+Y)⊗1)
e′ = (X⊗1)e′ + (Y ⊗1)e′.
The others are proved in a similar manner. Next, by Lemma 3.19, the correspon-denceM1 ∋ X 7→ U(X⊗1K)e′U∗ ∈ φ(M1) ⊂M2 defines a SRT-continuous unital *-homomorphism Φ′ which is clearly an extension of φ′. Therefore by considering Φ := ι′ ◦ Φ′ : M1 → M2 is the desired extension of φ, where ι′ : Φ′(M1),→M2 is a mere inclusion. Finally, we prove the uniqueness of the extension. Let Ψ be another SRT-continuous unital *-homomorphism such that Ψ|M1 =φ. LetX ∈M1. Then from the SRT-density ofM1inM1, there exists a sequence{xn} ⊂M1such that lim
n→∞xn =X in the strong resolvent topology.
Therefore we have
Ψ(X) = lim
n→∞Ψ(xn) = lim
n→∞φ(xn)
= lim
n→∞Φ(xn) = Φ(X).
The next lemmata, together with Lemma 5.10, implies thatfRngis a tensor category.
Lemma 5.15. LetRi,Si(i= 1,2)be objects in Obj(fRng). IfΨ1:R1→S1, Ψ2 :R2 →S2 are SRT-continuous unital *-homomorphisms, then there exists a unique SRT-continuous unital *-homomorphismΨ :R1⊗R2→S1⊗S2 such that Ψ(A⊗B) = Ψ1(A)⊗Ψ2(B), for all A ∈ R1 and B ∈ R2. We define Ψ1⊗Ψ2 to be the mapΨ.
Proof. Letψibe the restrictions of ΨiontoMi(i= 1,2). Thenψiis aσ-weakly continuous unital *-homomorphism from Mi to Ni, where Ni = Si. Thus there exists aσ-weakly continuous unital *-homomorphismψfrom M1⊗M2 to N1⊗N2 such that
ψ(x⊗y) =ψ1(x)⊗ψ2(y), x∈M1, y∈M2.
By Proposition 5.14, there exists a SRT-continuous unital *-homomorphism Ψ from R1⊗R2 to S1⊗S2 whose restriction to M1⊗M2 is equal to ψ. For all A ∈ R1, B ∈ R2, we can take sequences {xk}∞k=1 ⊂ M1, {yl}∞l=1 ⊂ M2
converging toA,B in the strong resolvent topology, respectively. Therefore, by Proposition 5.11, we have
Ψ(A⊗B) = lim
k→∞Ψ(xk⊗yk) = lim
k→∞ψ1(xk)⊗ψ2(yk)
= lim
k→∞Ψ1(xk)⊗Ψ2(yk) = Ψ1(A)⊗Ψ2(B).
Lemma 5.16. Let (Ri,Hi) (i = 1,2,3) be objects in fRng. Then we have a unique *-isomorphism which is homeomorphic with respect to the strong resol-vent topology:
(R1⊗R2)⊗R3∼=R1⊗(R2⊗R3)
(X1⊗X2)⊗X37→X1⊗(X2⊗X3),for allXi∈Ri
We denote the map asαR1,R2,R3.
Proof. LetMibe a finite von Neumann algebra such thatRi=Mi(i= 1,2,3).
Letα0be the *-isomorphism from (M1⊗M2)⊗M3ontoM1⊗(M2⊗M3) defined by (x1⊗x2)⊗x3 7→ x1⊗(x2⊗x3). By Lemma 5.10, both (M1⊗M2)⊗M3 andM1⊗(M2⊗M3) are generated by (M1⊗M2)⊗M3 andM1⊗(M2⊗M3), re-spectively. Therefore by Proposition 5.14, α0 can be extended to the desired
*-isomorphismαR1,R2,R3.
Proposition 5.17. fRng is a tensor category.
Proof. We define the tensor product⊗:fRng×fRng→fRngby (R1,H1)⊗(R2,H2) := (R1⊗R2,H1⊗ H2)
and for two morphisms Ψi : (Ri,Hi) → (Si,Ki) (i = 1,2), define Ψ1⊗Ψ2
according to Lemma 5.15. The unit object is I := (C1C,C). The associative constraint αR1,R2,R3 is the map defined in Lemma 5.16. The naturality of αR1,R2,R3 follows from Proposition 5.14. The definition of left (resp. right) constraintλ· (resp. ρ·) might be clear. Now it is a routine task to verify that the data (fRng,⊗, I, α, λ, ρ) constitutes a tensor category.
Now we will prove that fvN is isomorphic to fRng as a tensor category.
Define two functorsE:fvN→fRng, F:fRng→fvN.
Definition 5.18. Define two correspondencesE, F as follows:
(1) For each object (M,H) infvN,
E(M,H) := (M,H),
which is an object in fRng. For each morphism φ: M1 →M2 in fvN, E(φ) :M1 →M2 is the unique SRT-continuous extension ofφtoM1, so thatE(φ) is a morphism infRngby Proposition 5.14.
(2) For each object (R,H) infRng,
F(R,H) := (R∩B(H),H).
For each morphism Φ :R1→R2 in fRng,F(Φ) := Φ|R1∩B(H), which is a morphism infvNby Proposition 5.14.
Lemma 5.19. E andF are tensor functors.
Proof. We define the tensor functor (E, h1, h2), where h1: (C1C,C)−→id (C1C,C) =E((C1C,C)), h2((M1,H1),(M2,H2)) :M1⊗M2
−→id M1⊗M2,
can be taken to be identity morphisms thanks to Lemma 5.15. It is clear that E(1M) = 1M, where 1M and 1M are identity map of M and M, respectively.
LetM1 φ1
−→M2 φ2
−→M3 be a sequence of morphisms infvN. Let x∈M1. It holds that
E(φ2◦φ1)(x) = (φ2◦φ1)(x) =E(φ2)(φ1(x))
=E(φ2)(E(φ1)(x)) = (E(φ2)◦ E(φ1)) (x).
By Proposition 5.14 (2), we haveE(φ2◦φ1) =E(φ2)◦ E(φ1). ThereforeE is a functor. The conditions for (E, h1, h2) to be a tensor functor are described as the following three diagrams, the commutativity of which are almost obvious
by Proposition 5.14 and “∼” symbols are followed from Lemma 5.16.
(M1⊗M2)⊗M3 id
∼ //M1⊗(M2⊗M3)
id
(M1⊗M2)⊗M3 id
M1⊗(M2⊗M3)
id
(M1⊗M2)⊗M3 ∼ //M1⊗(M2⊗M3) C⊗M
id
1⊗X7→X //M M⊗C
id
X⊗17→X //M
C⊗M id //C⊗M
OO
M⊗C id //M⊗C
OO
Thus, (E, h1, h2) is a tensor functor. The proof that (F, h′1, h′2) is a tensor functor, including the definitions ofh′1, h′2 are easier.
Now we are able to prove the main theorem easily.
Proof of Theorem 5.5. We will show that E and F are the inverse tensor functor of each other. By Lemma 5.19, they are tensor functors. Let (Mi,Hi) (i= 1,2) be in Obj(fvN). Letφ:M1→M2be a morphism infvN. Proposition 5.14 impliesφ= (F ◦ E)(φ). By Proposition 5.4, we have
(Mi,Hi) = (Mi∩B(Hi),Hi) = (F ◦ E)(Mi,Hi), thereforeF ◦ E = idfvN.
Let (Ri,Hi) (i = 1,2) be objects in fRng, Φ : (R1,H1)→ (R2,H2) be a morphism infRng. By Proposition 5.4, we haveRi=Mifor a unique (Mi,Hi) in Obj(fvN). Similarly, we can prove that
(Ri,Hi) = (E ◦ F)(Ri,Hi), (E ◦ F)(Φ) = Φ, henceE ◦ F = idfRng.
Finally, we remark the correspondence of factors infvNand ones infRng.
Recall that, for a *-algebraA, itscenterZ(A) is defined by Z(A) :={x∈A ; xy=yx, for ally∈A}. Z(A) is also a *-algebra.
Proposition 5.20. LetMbe a finite von Neumann algebra onH. The following conditions are equivalent.
(1) The centerZ(M)of Mis trivial. I.e.,Z(M) =C1H.
(2) The centerZ(M)of Mis trivial.
Proof. (1)⇒(2) is evident.
(2) ⇒ (1). LetA ∈ Mbe a self-adjoint element of the center Z(M). For anyu∈U(M′), we have uAu∗ =A. Therefore from the unitary covariance of the functional calculus, it holds thatu(A−i)−1u∗= (A−i)−1and (A−i)−1∈ M∩M′ =C1.Hence (A−i)−1=α1 for someα∈C. By operatingA−ion both sides, we see thatA∈C1.For a general closed operator A∈Z(M), we know that there is a canonical decompositionA= Re(A) +iIm(A). SinceAbelongs toZ(M), Re(A), Im(A) also belong toZ(M) =C1.ThereforeA∈C1.
A Direct Sums of Operators
We recall the theory of direct sums of operators and show some facts. We do not give proofs for well-known facts. See e.g., [2].
Let{Hα}αbe a family of Hilbert spaces andH=⊕
αHαbe the direct sum Hilbert space of{Hα}α, i.e.,
H:=
{
ξ={ξ(α)}α ; ξ(α)∈ Hα, ∑
α
∥ξα∥2<∞. }
. For a subspaceDαofHα, we set
d⊕
αDα:=
{
ξ={ξ(α)}α∈ H; ξ(α)∈ Dα, ξ(α)= 0 except finitely many α.
} . It is known thatc⊕
αDα is dense inHwhenever eachDα is dense inHα. Next we recall the direct sum of unbounded operators. LetAαbe a (possibly unbounded) linear operator onHα. We define the liner operatorA=⊕αAαon Has follows:
dom(A) :=
{
ξ={ξ(α)}α∈ H; ξ(α)∈dom(Aα), ∑
α
∥Aαξα∥2<∞. }
, (Aξ)(α):=Aαξ(α), ξ∈dom(A).
Ais said to be the direct sum of{Aα}α. It is easy to see that if eachAα is a densely defined closed operator then so isA. In this case,
A∗=⊕αAα∗
holds. The following lemmata are well-known.
Lemma A.1. Assume the above notations.
(1) A ∈B(H) if and only if each Aα is in B(Hα) and supα∥Aα∥ < ∞. In this case,
∥A∥= sup
α ∥Aα∥ holds.
(2) Ais unitary if and only if each Aα is unitary.
(3) Ais projection if and only if each Aα is projection. In this case, ran(A) =⊕
α
ran(Aα)
holds.
Lemma A.2. Assume that eachAα is closed. Let Dα be a core of Aα. Then c⊕
αDα is a core ofA.
Lemma A.3. Assume that eachAα is (possibly unbounded) self-adjpint.
(1) Ais self-adjoint.
(2) For any complex valued Borel functionf on R, f(A) =⊕αf(Aα) holds.
Finally, we study the direct sum of algebras of operators. LetSα be a set of densely defined closed operators onHα. Put
⊕
α
Sα:={⊕αAα; Aα∈Sα}.
Note that each element in⊕
αSα is a densely defined closed operator on H=
⊕
αHα. If each Sαconsists only of bounded operators, we also define
⊕b α
Sα:=
{
⊕αxα; xα∈Sα, sup
α ∥xα∥<∞. }
.
By Lemma A.1, each element in ⊕b
αSα is bounded. The following is also well-known.
Lemma A.4. Let Mα be a von Neumann algebra acting onHα, and put
M:=
⊕b α
Mα.
ThenMis von Neumann algebra acting on H=⊕
αHα. The sum, the scalar multiplication, the multiplication and the involution are given by
(⊕αxα) + (⊕αyα) =⊕α(xα+yα),
λ(⊕αxα) =⊕α(λxα), f or all λ∈C, (⊕αxα) (⊕αyα) =⊕α(xαyα),
(⊕αxα)∗=⊕α(xα∗). Furthermore the followings hold.
(1) M′=⊕b αM′α.
(2) Mis a finite von Neumann algebra if and only if eachMαis finite von Neumann algebra.
We call⊕b
αMαthedirect sum von Neumann algebraof{Mα}α.
B Fundamental Results of SRT
LetHbe a Hilbert space. The following lemmata are well-known [18]:
Lemma B.1. Let {Aλ}λ∈Λ be a net of self-adjoint operators on H, A be a self-adjoint operator onH, and D be a dense subspace of Hwhich is a core of AandD ⊂∩
λ∈Λdom(Aλ)∩dom(A). Suppose for allξ∈ D,limλ∈ΛAλξ=Aξ, thenAλ converges to Ain the strong resolvent topology.
Lemma B.2. Let {An}∞n=1 be a sequence of self-adjoint operators on H,A be a self-adjoint operator onH. Then An converges to A in the strong resolvent topology if and only ifeitAnconverges strongly toeitAfor allt∈R. In this case, the strong convergence ofeitAn toeitA is uniform on every finite interval oft.
Lemma B.3. Let {An}∞n=1 be a sequence of self-adjoint operators on H,A be a self-adjoint operator onH. SupposeAnconverges toAin the strong resolvent topology, thenEAn((a, b))converges strongly toEA((a, b))for eacha, b∈Rwith a < banda, b /∈σp(A), whereσp(A) is the set of point spectra ofA.
Lemma B.4. Let {An}∞n=1 be a sequence of self-adjoint operators on H,A be a self-adjoint operator onH. SupposeAnconverges toAin the strong resolvent topology, then for all complex valued bounded continuous functionf onR,f(An) converges strongly tof(A).
Lemma B.5. Let {xλ}λ∈Λ be a net of bounded self-adjoint operators on H, x be a bounded self-adjoint operator onH. Suppose that
sup
λ∈Λ
∥xλ∥<∞,
andxλconverges toxin the strong resolvent topology, thenxλconverges strongly tox.
C Tensor Categories
We briefly review the definition of tensor categories. For more details about category theory, see MacLane [9] (we follow the style in Kassel [8], Chapter XI).
Definition C.1. LetC,C′ be categories, F,G be functors from C to C′. A natural transformation θ : F → G is a function which assigns to each object A in C a morphism θ(A) :F(A)→ G(A) of C′ in such a way that for every morphismf :A→B in C, the following diagram commutes:
F(A)
F(f)
θ(A) //G(A)
G(f)
F(B) θ(B) //G(B)
Ifθ(A) is an invertible morphism for everyA, we callθ anatural isomorphism.
Definition C.2. A tensor category(C,⊗, I, α, λ, ρ) is a category C equipped with
(1) a bifunctor⊗:C ×C →C called atensor product2, (2) an objectI inC called aunit object,
(3) a natural isomorphismα:⊗(⊗ ×1C)3→ ⊗(1C × ⊗) called an asso-ciativity constraint.
(3) means for any objectsA, B, C in C, there is an isomorphismαA,B,C : (A⊗ B)⊗C→A⊗(B⊗C) such that the diagram
(A⊗B)⊗C
(f⊗g)⊗h
αA,B,C //A⊗(B⊗C)
f⊗(g⊗h)
(A′⊗B′)⊗C′
αA′,B′,C′
//A′⊗(B′⊗C′)
commutes for all morphismsf, g, hinC.
(4) a natural isomorphismλ:⊗(I×1C)4→1C (resp. ρ:⊗(1C×I)→1C) called aleft(resp. right)unit constraintwith respect to I.
2This implies (f′⊗g′)◦(f⊗g) = (f′◦f)⊗(g′◦g) for all morphisms inC, and 1A⊗1B= 1A⊗B
for all objects inC.
3⊗(⊗ ×1C) is the composition of the functors⊗ ×1C : (C ×C)×C → C×C and
⊗:C×C→C.
4I×1C is the functor fromC to C ×C given by A7→ (I, A) for all objects inC and f7→(1I, f) for all morphisms inC.
(4) means for any objectAinC, there is an isomorphismλA:I⊗A→A(resp.
ρA:A⊗I→A) such that the following two diagrams commute:
I⊗A
1I⊗f
λA //A
f
A⊗I
f⊗1I
ρA //A
f
I⊗A′ λA′ //A′ A′⊗I ρA′ //A′
for each morphismf :A→A′ in C. These functors and natural isomorphisms satisfy the Pentagon Axiom and the Triangle Axiom. Namely, for all objects A, B, C andD, the following diagrams commute:
(A⊗(B⊗C))⊗D
αA,B⊗C,D
))T
TT TT TT TT TT TT TT
((A⊗B)⊗C)⊗D
αA⊗B,C,D
αA,B,C⊗1D
33h
hh hh hh hh hh hh hh hh h
A⊗((B⊗C)⊗D)
1A⊗αB,C,D
(A⊗B)⊗(C⊗D) α
A,B,C⊗D //A⊗(B⊗(C⊗D))
(A⊗I)⊗B
ρA⊗1B
&&
MM MM MM MM MM
αA,I,B
//A⊗(I⊗B)
1A⊗λB
xxqqqqqqqqqq
A⊗B
Definition C.3. Let (C,⊗, I, α, λ, ρ), (C′,⊗, I′, α′, λ′, ρ′) be tensor categories.
(1) A triple (F, h1, h2) is called a tensor functor from C to C′ if F : C →C′ is a functor,h1is an isomorphismI′ ∼→ F(I) andh2 is a natural isomorphism⊗(F × F)5→ F⊗∼ , and they satisfy
(F(A)⊗ F(B))⊗ F(C)αF(A),F(B),F(C)//
h2(A,B)⊗1F(C)
F(A)⊗(F(B)⊗ F(C))
1F(A)⊗h2(B,C)
F(A⊗B)⊗ F(C)
h2(A⊗B,C)
F(A)⊗ F(B⊗C)
h2(A,B⊗C)
F((A⊗B)⊗C)
F(αA,B,C) //F(A⊗(B⊗C))
5⊗(F × F) is a functorC×C→C which assingsF(A)⊗ F(B) for each object (A, B) in C×C andF(f)⊗ F(g) for each morphism (f, g) inC×C
I′⊗ F(A)
h1⊗1F(A)
λ′F(A)
//F(A) F(A)⊗I′
1F(A)⊗h1
ρ′F(A)
//F(A)
F(I)⊗ F(A)
h2(I,A)
//F(I⊗A)
F(λA)
OO
F(A)⊗ F(I)
h2(A,I)
//F(A⊗I)
F(ρA)
OO
for all objectsA, B, C in C.
(2) Anatural tensor transformationη : (F, h1, h2)→(F′, h′1, h′2) between tensor functors from C to C′ is a natural transformation F → F′ such that the following diagrams commute:
F(I)
η(I)
F(A)⊗ F(B)
η(A)⊗η(B)
h2(A,B)//F(A⊗B)
η(A⊗B)
I
h1
=={
{{ {{ {{ {
h′1
!!C
CC CC CC C
F′(I) F′(A)⊗ F′(B)h
′
2(A,B)//F′(A⊗B)
for all objectsA, Bin C. Ifη is also a natural isomorphism, it is called a natural tensor isomorphism.
(3) Atensor equivalencebetween tensor categoriesC,C′is a tensor functor F : C → C′ such that there exists a tensor functor F′ : C′ → C and natural tensor isomorphisms η : 1C′ → F ◦ F∼ ′ and θ : F′◦ F →∼ 1C. If η and θ can be taken to be identity transformations, then F is called a tensor isomorphismand we sayC is isomorphic toC′ as a tensor category.
Acknowledgement
The authors would like to express their sincere thanks to Professor Asao Arai at Hokkaido University, Professor Izumi Ojima at Kyoto University for the fruitful discussions, insightful comments and encouragements. H.A. also thanks to his colleagues: Mr. Ryo Harada, Mr. Takahiro Hasebe, Mr. Kazuya Okamura and Mr. Hayato Saigo for their useful comments and discussions during the seminar. Y.M. also thanks to Mr. Yutaka Shikano for his professional advice about LaTeX. Finally, the authors thank to Professor Izumi Ojima for his careful proofreading and suggestions again.
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