JONES TYPE C∗-BASIC CONSTRUCTION IN NON-EQUILIBRIUM HOPF SPIN MODELS∗

(魏晓敏)

School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing 211816, China

E-mail: wxiaomin@amss.ac.cn

Lining JIANG (蒋立宁)†

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

E-mail: jianglining@bit.edu.cn

Abstract Let H be a finite dimensional Hopf C∗-algebra,and let K be a Hopf ∗-subalgebra of H.Considering that the field algebra FK of a non-equilibrium Hopf spin model carries a D(H,K)-invariant subalgebra AK,this paper shows that the C∗-basic construction for the inclusion AK ⊆FK can be expressed as the crossed product C∗-algebra FK ⋊D(H,K).Here, D(H,K) is a bicrossed product of the opposite dual and K.Furthermore,the natural action of on D(H,K) gives rise to the iterated crossed product FK ⋊D(H,K)⋊,which coincides with the C∗-basic construction for the inclusion FK ⊆FK ⋊D(H,K).In the end,the Jones type tower of field algebra FK is obtained,and the new field algebra emerges exactly as the iterated crossed product.

Key words field algebra;conditional expectation;basic construction;C∗-tower

1 Introduction

Jones’index theory for(sub)factors has visited many homes of mathematics and physics,including manifold topology,statistical mechanics,quantum field theory,and dynamical systems,etc..All of these approaches imply that subfactors could be regarded as fixed points generalizing the group-like algebraic structures,and the symmetries generalizing ordinary group actions are also exhibited on the way.The concept of Jones’ index was proposed to measure the size of a subfactor.For all possible values of index for a subfactorN ⊆M,Jones showed in [5]that the size is｛4 cos2(π/n):n ≥3｝∪[4,∞].In the process of quantizing the index,the basic construction toolkit plays an important role.Since there is always a faithful normal normalized trace,which provides a Hilbert space denoted byL2(M)through the GNS construction,one can get the extended von Neumann algebra〈M,eN〉,called the basic construction for the inclusionN ⊆M.Here,eNis the projection induced by the conditional expectationEN:M →N.

The chain ofN ⊆M ⊆〈M,eN〉⊆〈〈M,eN〉,eM〉⊆··· leads to the Temperley-Lieb algebra([9,24]) for corresponding projections{ei:e1=eN,e2=eM,···}:

Similar relations occur in objects such as Hecke algebras of type A,braid groups ([6]) and so on.

The method concerning the geometric,combinational and discrete nature of II1factors has been developed by many mathematicians.In 1986,Pimsner and Popa ([18]) established the concept of a probability index,then Kosaki ([10]) generalized this to the situation of arbitrary factors,and pointed out that the set of values of an index coincides with the set obtained by Jones,and that a Pimsner-Popa type basis exists when the index is finite.Other attempts have also been made in this regard,e.g.Longo ([12,13]) demonstrated that the square of the index of a quasi-local C∗-algebra represents the statistical dimension of the Doplicher-Haag-Roberts superselection sector.This correspondence shows that index theory creates a strong bond between mathematics and physics.Furthermore,building on the achievements of Jones and Kosaki,Watatani([27])proposed an index theory and its basic construction in the context of C∗-algebras.In their algebraic approach,the Pimsner-Popa basis was generalized to the notion of a quasi-basis for a conditional expectation on a C∗-algebra.Moreover,using a similar tower of Jones’ iterated basic extensions,they showed that the spectrum of their index could be included in the Jones’ index set under a certain condition.Subsequently,Kajiwara,Pinzari and Watatani ([8]) introduced Jones’ index theory for general Hilbert bimodules over pairs of C∗-algebras,which provided a positive answer to what had been an open problem in the theory of conjugation in abstract tensor C∗-categories that appeared in the algebraic formulation of quantum field theory ([7]).

Quantum chains considered as models of 1+1-dimensional quantum field theory reveal many interesting features,such as braid group statistics and quantum symmetry.G-spin models of a finite groupGprovide the simplest examples of lattice field theory exhibiting quantum symmetry,and implement a Doplicher-Haag-Roberts program for exploring the internal symmetries of the model merely from observable data.This data was interpreted as the observable algebra in the models investigated by Szlachányi and Vecsernyés ([20]).Moreover,a large class of models–Hopf spin models generalizingG-spin models–was established in [17].Given a finite dimensional Hopf C∗-algebraHand its dualere is a copy ofH-order on lattice sites and-disorder on links,together with non-trivial commutation relationships between neighboringHand,such that the observable algebraAof Hopf spin models was obtained,so the field algebraFwas realized by the coaction of the Drinfel’d doubleD(H) onA.Furthermore,the symmetry of the superselection sectors ofAwas revealed byD(H).

The above spin models were carried out by setting up order-disorder operators that appear in pairs,which is related to the equilibrium situation.The non-equilibrium case of Hopf spin models is the subject of our interest.In fact,the number of disordered states is much greater than that of ordered states for a macrophysical system,which always tends to a disordered state.From now on,we denote a finite dimensional Hopf C∗-algebra byH,and a Hopf∗-subalgebra ofHbyK.The article [28]constructed non-equilibrium Hopf spin models,copies ofKon lattice sites andon links with non-trivial commutation relations between only on neighboring links and sites.This relationship yields the observable algebraAKwhich carries a coaction of the relative quantum doubleD(H,K).The field algebrawas obtained([29])such thatAKis exactly theD(H,K)-invariant subalgebra ofFK,and thus a conditional expectation of index-finite type fromFKontoAKwas formed.In particular,the non-equilibriumG-spin models determined by a normal group was achieved in [32].Based on these observations,this paper focuses on the Jones type C∗-basic construction for the field algebraFKin non-equilibrium Hopf spin models.It is worth mentioning that a connection between C∗-basic constructions and coactions was also found in[16];this established the Morita equivalences between fixed point algebras and crossed products for a special coaction of compact quantum groups.

The rest of this paper is organized as follows: Section 2 collects some necessary conceptions and facts on non-equilibrium Hopf spin models.Section 3 shows that the C∗-algebra C∗〈FK,e1〉established from a C∗-basic construction for the inclusionAK ⊆FKis characterized by the crossed product C∗-algebraFK⋊D(H,K).Here,the field algebraFKbecomes aD(H,K)-module algebra in terms of the natural action ofD(H,K) on.Furthermore,the C∗-algebraFK⋊D(H,K) is a-module algebra,and this guarantees that the mapEFK:FK⋊D(H,K)→FKdefined by the Haar integral ofis a conditional expectation,which coincides with the dual conditional expectation ofEAK:FK →AK.These results are illustrated in the first part of Section 4.Moreover,Section 4 shows that the C∗-algebra C∗〈FK⋊D(H,K),e2〉 for the inclusionFK ⊆FK⋊D(H,K) is∗-isomorphic to the iterated crossed productIterated as needed,we can obtain the Jones type tower of the field algebraFKin non-equilibrium Hopf spin models,implying the emergence of the higher dimensional non-equilibrium Hopf spin models.We end this paper with a description of the new observable algebra in terms of the Takai duality theorem ([22]).

2 Auxiliaries in Non-equilibrium Hopf Spin Models

We will follow Sweedler’s notations regarding the comultiplication of a Hopf algebraHover a fixed complex field C:,a ∈H.For full textbook treatments see [1].We begin by reviewing the definition of Hopf C∗-algebras.

Definition 2.1([2]) A finite dimensional Hopf C∗-algebra is a C∗-algebraHtogether with a unital∗-homomorphism ∆:H →H ⊗Hsuch that (∆⊗id)◦∆=(id⊗∆)◦∆and the linear spans which are denoted by“[·]”satisfy that[∆(H)(H ⊗1)]=[∆(H)(1⊗H)]=H ⊗H.

Note that for a finite dimensional Hopf C∗-algebra,there exists a counitε:H →C and an antipodeS:H →Hobeying that (ε ⊗id)◦∆=(id⊗ε)◦∆=id and (m ◦(S ⊗id)◦∆)(a)=(m ◦(id⊗S)◦∆)(a)=(ι ◦ε)(a),a ∈H,respectively.Here,m:H ⊗H →Handι: C→Hdenote the multiplication and the unit.The dualofHis also a Hopf C∗-algebra of finite dimension with∗-operation defined byϕ ∈,a ∈H.There is a unique one dimensional central projection inH,i.e.,h=h2=h∗=S(h),which is called the Haar integral.For more details about Hopf C∗-algebras one can refer to [15,25,26].

Throughout the paper,Halways denotes a Hopf C∗-algebra of finite dimension equipped with a counitεand an antipodeS,andKdenotes a Hopf∗-subalgebra ofH.

We now give some necessary conceptions and results which will be used throughout this paper,starting with the observable algebra in non-equilibrium Hopf spin models.We associate to each even integer 2ia copyA2iofKand to each odd integer 2i+1 a copyA2i+1of.Denote byA2i(x) andA2i+1(ϕ) the elements ofA2iandA2i+1,respectively,and thati ∈Z.

Definition 2.2([28]) The unital∗-algebraAK,locis generated byA2i(x),A2i+1(ϕ),x ∈K,ϕ ∈subject to

where〈·,·〉 means the canonical pairing betweenKand.

Define the setJto be

This allows us to define the local field algebra of a finite interval as the crossed product C∗-algebraThe chain of finite dimensional C∗-algebras is⊆⊆···wheneverI ⊆J ⊆···,and their C∗-inductive limit leads us to the field algebraFK([29]) inin other words,taking a unital C∗-algebra generated bynon-equilibrium Hopf spin models.One can interpret its multiplication and∗-operation are given by

Notice that the field algebraFKis a leftD(H,K)-module algebra under the natural action ofD(H,K) on,formulated explicitly by

It is obvious to see that the observable algebraAKis precisely theD(H,K)-invariant subalgebra of the field algebraFK,which is announced by the following proposition:

Proposition 2.3([29])AK=｛F ∈FK:X.F=εD(H,K)(X)F,X ∈D(H,K)｝,whereεD(H,K)is the counit ofD(H,K).

In particular,the Haar integralhof the relative quantum doubleD(H,K)provides an onto linear mapFK →AK,

(3)EAKis positive.

A linear map from a unital C∗-algebra onto its C∗-subalgebra with a common unit,satisfying the above relations(1)–(3),is called a conditional expectation([23]).We say thatEAKis faithful ifEAK(F∗F)=0 implies thatF=0.

Definition 2.4([27]) LetA ⊆Bbe a unital inclusion of unital C∗-algebras.A conditional expectationE:B →Ais of index-finite type if there exists a quasi-basis forE,which is a finite family{(u1,v1),···,(un,vn)}⊆B×Bsatisfying that,for allb ∈B,

The index ofEis defined as

Remark 2.5This C∗-algebra index does not depend on the choice of quasi-basis,and lies in the center ofB([27]).

By Proposition 2.3,the symmetry ([29]) in the field algebraFKrevealed by the relative quantum doubleD(H,K) can be measured;this is shown as follows:

Proposition 2.6([30]) The mappingEAK:FK →AKgiven by (2.1) is a faithful conditional expectation,and is of index-finite type with IndexEAK=dimD(H,K)·1FK.

This will enable us to consider the extension of the field algebraFKthrough the C∗-basic construction procedure in what follows.

3 The C∗-basic Construction for the Inclusion AK ⊆FK

In this section,we will prove that the crossed product C∗-algebraFK⋊D(H,K) arising from the action of the relative quantum doubleD(H,K) is∗-isomorphic to the C∗-basic construction for the inclusionAK ⊆FK.

We now provide some ingredients for the basic construction.

Definition 3.1([11,14]) LetAbe a C∗-algebra.An inner-productA-module is a linear spaceMwhich is a rightA-module equipped with a sesquilinear form〈·,·〉:M×M →Awith the following properties:

(1)〈x,x〉≥0 for anyx ∈M,and〈x,x〉=0 implies thatx=0;

(2)〈y,x〉=〈x,y〉∗for anyx,y ∈M;

(3)〈x,ya〉=〈x,y〉a.

The map〈·,·〉 is called anA-valued inner product.

LetE:B →Abe a faithful conditional expectation.ThenBcould be viewed as a rightA-module via its multiplication,denoted by.The mapdefined by

DenoteLA(BA) as the set of all rightA-module homomorphismsT:BA →BAwith an adjointA-module homomorphismT∗:BA →BAsuch that

ThenLA(BA)is a C∗-algebra with the usual operator norm.On the other hand,the C∗-algebraBcan be embedded as a C∗-subalgebra ofLA(BA) by the injective∗-homomorphism

Definition 3.2([27]) The C∗-subalgebra ofLA(BA) generated by{λ(b):b ∈B} andeAis called the C∗-basic construction,and is denoted by C∗〈B,eA〉.

Remark 3.3The above statement actually gives the reduced version of a C∗-basic construction.The reason that we do not distinguish between reduced and unreduced concepts is that there is a canonical∗-isomorphism between them ([27]).

Lemma 3.4([27]) Letx ∈FK.Thene1λ(x)e1=λ(EAK(x))e1.Moreover,x ∈AKif and only ife1λ(x)=λ(x)e1.

By Lemma 3.4,the C∗-algebra C∗〈FK,e1〉 can be expressed as

Recalling that the field algebraFKis a leftD(H,K)-module algebra,one can construct the∗-algebraFK ⊗D(H,K),which is a linear space equipped with multiplication and∗-operation given by

for (A,ξ),(B,η)∈FK,X,Y ∈D(H,K).

Given an intervalI ∈J,denote by⋊D(H,K) the∗-subalgebra ofFK ⊗D(H,K).In the finite dimension situation,the crossed product for a Hopf C∗-algebra acting on a C∗-algebra is still a C∗-algebra,implying that⋊D(H,K)is a finite dimensional C∗-algebra.ForI ⊆J,⋊D(H,K)→⋊D(H,K) is a unital injective∗-homomorphism,and this information allows us to complete the union of these C∗-algebras via C∗-inductive limit procedure to get the C∗-algebra:

The next observations will provide some materials for the characterization of the C∗-basic construction C∗〈FK,e1〉.

Proposition 3.5(1)The element(1FK,h)inFK⋊D(H,K)is a self-adjoint idempotent element,wherehis the Haar integral inD(H,K).

ProofIt is easy to draw the conclusion that (1FK,h)∗=(1FK,h∗)=(1FK,h)=(1FK,h)2from the relationshiph=h∗=h2.ForF ∈FK,one has that

The proof is finished.

Remark 3.6Since the mapEAKfrom the field algebraFKonto itsD(H,K)-invariant subalgebraAKis a conditional expectation of index-finite type,with IndexEAK=dimD(H,K)·1FK(see Proposition 2.6),the C∗-algebraFK⋊D(H,K) can be expressed as follows ([4,21]):

Theorem 3.7The C∗-algebraFK⋊D(H,K)is∗-isomorphic to the C∗-algebra C∗〈FK,e1〉.

ProofForI ∈J,define a linear map(H,K) given by

First,the mapΦIis well-defined and injective.Indeed,assuming that=0,for anyF ∈,

Therefore,

Here,the second and fifth equalities hold in terms of the covariant relations in the C∗-algebrasFK⋊D(H,K) (Proposition 3.5) and C∗〈,e1〉 (cf.Lemma 3.4).

Noting thatΦI(e1)=(1FK,h) is a self-adjoint element inFK⋊D(H,K),the mapΦIpreserves the∗-operation on generators of C∗〈,e1〉 with

Remark 3.8The C∗-basic construction does not depend on the choice of the conditional expectation up to an isomorphism([27,Proposition 2.10.11]).Namely,ifΓ1andΓ2:B →Aare both conditional expectations of index-finite type,letting C∗〈B,e1〉 and C∗〈B,e2〉 be the corresponding C∗-basic constructions,respectively,then there is a∗-isomorphismθ:C∗〈B,e1〉→C∗〈B,e2〉 such thatθ(b)=b,b ∈B.

4 The C∗-basic Construction for the Inclusion FK ⊆FK ⋊D(H,K)

In this section,we will demonstrate that the Haar integral ofyields a faithful conditional expectationEFKfrom the crossed product C∗-algebraFK⋊D(H,K) onto the field algebraFK,which is also of index-finite type.Furthermore,we show that the C∗-basic construction forFK ⊆FK⋊D(H,K) is precisely consistent with the iterated crossed product,following from the natural-module algebra structure onFK⋊D(H,K).

4.1 The Conditional Expectation from FK ⋊D(H,K) onto FK

which will be verified to be a conditional expectation of index-finite type.For this purpose,we single out a system of elements in the C∗-subalgebra(isomorphic to the matrix algebraMn(C))of.Such a system,which is called a matrix unit([23]),is a family{wij:i,j=1,···,n}satisfying that

With the help of (4.3),we now arrive at

Proposition 4.1The mapEFKgiven by (4.1) is a faithful conditional expectation.

(2) (bimodular property) LettingF1,F2∈FK,T ∈FK⋊D(H,K),

(3) (positive) Noticing that (ξ.T)∗=S(ξ∗).T∗forξ ∈andT ∈FK⋊D(H,K),one has that

Moreover,EFK(TT∗)=0 implies that.T=0 for anyr,i,j,and thus one hasT=0,and the mapEFKis faithful.

by virtue of [27],Proposition 2.3.2,wheren=dimD(H,K).Theis called the dual conditional expectation ofEAK:FK →AK.

Here,“1” denotes the unit of the observable algebraAKand C∗〈FK,e1〉.

ProofIt suffices to prove that∀T ∈C∗〈FK,e1〉,

where the third equality holds by virtue of Lemma 3.4.

Since the index of a conditional expectation does not depend on the choice of the quasi-basis([27]),one has that

wheren=dimD(H,K).

Remark 4.4The conditional expectationEFK:FK⋊D(H,K)→FKin Proposition 4.1 coincides with the above dual conditional expectationIndeed,there is a one-to-one correspondence between the linear bases

The argument is now completed in terms of the linearity and continuity ofThe above correspondence is equal to the commutative diagram below.

4.2 The ∗-isomorphism Between C∗〈FK ⋊D(H,K),e2〉 and FK ⋊D(H,K)⋊

We continue to investigate the C∗-algebra C∗〈FK⋊D(H,K),e2〉 from the conditional expectationEFK:FK⋊D(H,K)→FK,wheree2is the Jones’ projection ofEFK.More precisely,one has that

Therefore,the iterated crossed product C∗-algebra denoted byFK⋊D(H,K)⋊is obtained.

Similar observations as to those of Proposition 3.5 can be found in the C∗-algebraFK⋊

Proposition 4.5(1)（1FK⋊D(H,K),ζ）=(1FK⋊D(H,K),ζ)∗=(1FK⋊D(H,K),ζ)2,whereζis the Haar integral of

(2) For anyT ∈FK⋊D(H,K),we have the covariant relation

It is now time to arrive at another main conclusion regarding the description of the C∗-algebra C∗〈FK⋊D(H,K),e2〉.

Theorem 4.6There exists a∗-isomorphism between the C∗-algebra C∗〈FK⋊D(H,K),e2〉andFK⋊D(H,K)⋊

ProofConsider the linear mapΨ:C∗〈FK⋊D(H,K),e2〉→FK⋊D(H,K)⋊satisfying that

and similarly to that of Theorem 3.7,the mapΨcan be extended to an isometric isomorphism between C∗〈FK⋊D(H,K),e2〉 andFK⋊D(H,K)⋊

Remark 4.7In particular,lettingHbe the group algebra CGof a finite groupG,K=H,the C∗-basic construction in theG-spin models ([20]) is obtained.

Since the C∗-basic construction does not depend on the choice of the conditional expectation up to an isomorphism,we say the above chain of C∗-algebras is a Jones type tower of the field algebraFKin non-equilibrium Hopf spin models.

Remark 4.8The Jones type tower of the field algebraFKin non-equilibrium Hopf spin models has a periodicity of order two,and the structure it possesses is an invariant for the initial inclusion.Indeed,by the Takai duality ([22]),the C∗-algebraFK⋊D(H,K)=is canonically∗-isomorphic toAK ⊗Mn(C)∼=Mn(AK),wheren=dimD(H,K).Moreover,the concrete construction ofMn(AK) is performed as follows:

Considering Z as a one-dimensional lattice,each even number representing a lattice site,and odd number representing a link,set that

The elementsA2i(x),A2i+1(ϕ) lie inA2i,A2i+1,respectively,andx ∈K,ϕ ∈.

The local observable algebraMn(AK)locis a unital∗-algebra generated by｛A2i(x)⊗wkl,A2i+1(ϕ)⊗wst:x ∈K,ϕ ∈,wkl,wst ∈Mn(C)｝subject to the following relations:

ForI ∈J,letMn(AK)(I)be the∗-subalgebra ofMn(AK)locwith generators,which is then a finite dimensional C∗-algebra through a faithful∗-representation similar to that of the equilibrium Hopf spin models.Furthermore,Mn(AK)(I)⊆Mn(AK)(J) is a unital inclusion for anyI ⊆J,and this allows us to get a C∗-algebraMn(AK)via C∗-inductive limit processThis C∗-algebraMn(AK) is called the observable algebra in the emergent higher dimensional Hopf spin models,which will be called the dual non-equilibrium Hopf spin models.The parameters,x ∈Kand,ϕ ∈act as order and disorder operators in the models.

Natural questions now arise: what is the field algebra in the dual non-equilibrium Hopf spin models? Moreover,what are the corresponding lattice models? The constructions of the field algebra in non-equilibrium Hopf spin models ([29]) suggest further that the coaction of a C∗-algebraMn(AK)(I) makesMn(AK)(I) a left-module algebra.Hence,the field algebra in the dual non-equilibrium Hopf spin models could be constructed asMn(FK) :=,which is exactly(see (4.4)).It would be of great interest to explore these things more in the future.

Conflict of InterestThe authors declare no conflict of interest.