# C ⁎-Basic Construction from the Conditional Expectation on the Drinfeld Double

Authors: Qiaoling Xin; Lining Jiang; Tianqing Cao
Journal: Journal of Function Spaces
 (2019)

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## Abstract

LetD(G) be the Drinfeld double of a finite group G and D(G;H) be the crossed product of C(G) and CH, where H is a subgroup of G. Then the sets D(G) and D(G;H) can be made C⁎-algebras naturally. Considering the C⁎-basic construction C⁎〈D(G),e〉 from the conditional expectation E of D(G) onto D(G;H), one can construct a crossed product C⁎-algebra C(G/H×G)⋊CG, such that the C⁎-basic construction C⁎〈D(G),e〉 is C⁎-algebra isomorphic to C(G/H×G)⋊CG.


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## Body

## 1. Introduction

Index theory for subfactors was initiated by Jones ([1]) and has experienced rapid progress beyond the framework of operator algebras. For example, Jones’ index theory has found important applications in knots theory, conformal field theory, algebraic quantum field theory, and so on ([2–8]). For a nontechnical but broad overview of the subject including a lot of important connections with other areas, the readers can refer to [9].LetN⊆M be factors of type II1 with finite Jones index and tr the faithful normal tracial state on M. Denoted by L2(M,tr) the Hilbert space closure of M is with respect to the norm 〈x,y〉=tr(y∗x). Then M acts on L2(M,tr) by the left multiplication. The involution x→x∗ extends to an isometric conjugate linear operator on L2(M,tr) denoted by J. The remarkable discovery of Jones is that the possible values for the index are [M:N]=4cos2⁡π/n, n=3,4,…, or [M:N]≥4. It is rather easy to construct a reducible inclusion of factors with any index value larger or equal to 4. All the values 4cos2⁡π/n in the discrete series are realized by means of the basic construction. To be precise, let EN be a conditional expectation from M onto N associated with the trace, such that tr(EN(x)y)=tr(xy) for x∈M and y∈N. The extension of EN to L2(M,tr), denoted by eN, is the orthogonal projection of L2(M,tr) onto the closure of N regarded as a subspace of L2(M,tr). Then M1≜〈M,eN〉, the von Neumann algebra generated by M and eN on L2(M,tr), is called the basic construction, and 〈M,eN〉=JN′J, which is a perfect result. Subsequently Jones used the basic construction to obtain an increasing sequence of type II1 factors, N⊆M⊆M1⊆M2⊆M3⊆⋯ which is called the Jones tower, iteratively by adding the Jones projections {en:n≥1,e1=eN} which satisfy the Temperley-Lieb relations. Finally, Jones used this structure to construct an example for any possible index value below 4.The Jones index theory for subfactors of typeII1 has been extended to unital C∗-algebras by Watatani ([10]), and many interesting results of C∗-index theory can be found in [11, 12]. Note that in [10] Watatani showed that if α is an outer action of a finite group G on a simple C∗-algebra A and E is the condition expectation from A onto the fixed point subalgebra Aα given by the average over G, then the basic construction is identified with the crossed product A⋊G. It was also shown in [12] that if A is a pure Hopf algebra acting outerly on a factor M, then M⋊A is a factor and therefore the Jones basic construction M1=M⋊A. However, different from the basic construction for type II1 factors, the C∗-basic construction C∗〈B,γA〉 does not have the concrete form in general, where Γ is a conditional expectation of a C∗-algebra B onto a C∗-subalgebra A. The reason is that any factor of type II1 possesses the faithful trace which is a state of this kind for which the Gelfand-Naimark-Segal construction may be performed, while for general C∗-algebras, the existence of this functional is uncertain.LettingG be a finite group and H its subgroup, denoted by H≤G, then D(G;H) is defined as the crossed product of C(G), the algebra of complex functions on G, and group algebra CH with respect to the adjoint action of the latter on the former. In particular, H=G; then D(G;H)≜D(G) is the Drinfeld double of G. It was shown in [13] that there is the conditional expectation E:D(G)→D(G;H) of index-finite type. In this paper, we prove that the C∗-basic construction from the conditional expectation E can be described as a crossed product C(G/H×G)⋊CG.The paper is organized as follows. In Section2, we give a brief description of the C∗-basic construction for C∗-algebras, and we also collect the necessary definitions and facts about the set D(G;H). In Section 3, we define an action of CG on C(G/H×G), and obtain the resulting crossed product C(G/H×G)⋊CG. Theorem 10 is our main result which means that the C∗-basic construction C∗〈D(G),e〉 is C∗-algebra isomorphic to the C∗-algebras C(G/H×G)⋊CG.


## 2. TheC∗-Basic Construction of the Drinfeld Double

In this section, suppose thatB is a C∗-algebra over C and A a C∗-subalgebra with a common unit 1. Let us recall these definitions and facts which can be found in the works of Watatani ([10]) and Jiang ([13]).Definition 1.
A linear mapΓ:B→A is called a conditional expectation if it satisfies the following conditions:(1)
Γ(1)=1;(2)
(bimodular property) fora1,a2∈A, b∈B, (1)Γa1ba2=a1Γba2;(3)
Γ is positive; i.e., Γ(b∗b) is a positive element in A for any b∈B.
Actually, ifΓ is a conditional expectation from B onto A, then Γ is a projection of norm one ([14]).Definition 2.
LetΓ:B→A be a conditional expectation. A finite family {(u1,u1∗),(u2,u2∗),…,(un,un∗)}⊆B×B is called a quasi-basis for Γ if for all x∈B, (2)∑i=1nuiΓui∗x=x=∑i=1nΓxuiui∗. Furthermore, if there exists a quasi-basis for Γ, we call Γ of index-finite type. In this case we define the index of Γ by (3)IndexΓ=∑i=1nuiui∗.
IfΓ is of index-finite type, then IndexΓ is in the center of B and does not depend on the choice of quasi-basis ([10]).In the following we recall theC∗-basic construction from the conditional expectation Γ:B→A.Notice thatB is an A-bimodule algebra; one can consider the module tensor product algebra B⊗AB, the product of which depends on Γ as follows:(4)b1⊗b2b3⊗b4=b1Γb2b3⊗b4 for bi∈B, i=1,2,3,4. Then B⊗AB turns out to be an algebra. And define an involution ∗ by (5)x⊗y∗=y∗⊗x∗ for x,y∈B. The involution ∗ is well defined by considering conjugate operation in C∗-algebras. Thus B⊗AB becomes an ∗-algebra.Recall thatL(X) denotes the algebra of all bounded linear operators on a Hilbert space X.Lemma 3.
LetΓ:B→A be a conditional expectation. Let ϕ:B⊗AB→L(X) be a ∗-representation of B⊗AB on a Hilbert space X. Consider a conditional expectation Γ⊗id:B⊗Mn→A⊗Mn. Then there exists a ∗-representation ϕ′:(B⊗Mn)⊗A⊗Mn(B⊗Mn)→L(X)⊗Mn such that (6)ϕ′xijij⊗yjkjk=∑jϕxij⊗yjkik for (xij)ij,(yjk)jk∈B⊗Mn.
Forc=∑ixi⊗yi∈B⊗AB, set(7)cmax=sup⁡ϕc:ϕis a∗-representation ofB⊗AB, then(8)cmax=Γxi∗xjij1/2Γyiyj∗ij1/2<+∞. Thus ·max is a C∗-seminorm on B⊗AB ([10]).Definition 4.
The completion ofB⊗AB by the norm ·max (after taking quotient by the ideal {c∈B⊗AB:cmax=0} if necessary) is called the C∗-basic construction from the conditional expectation Γ. We denote this C∗-algebra by C∗〈B,γA〉.Now assume thatG is a finite group with a subgroup H and t1=u,t2,…,tk is a left coset representation of H in G, namely, G=⋃i=1ktiH and i≠j induces that tiH∩tjH=∅, where u is the unit of G. Let us begin with the following definitions.Definition 5.
D ( G ; H ) is the crossed product of C(G) and group algebra CH, where C(G) denotes the set of complex functions on G, with respect to the adjoint action of the latter on the former.Using the basis elements(g,h) of D(G;H), the multiplication rule is given as follows:(9)g1,h1g2,h2=δg1h1,h1g2g1,h1h2. The unit of D(G;H) is ∑g∈G(g,u)≜I. D(G;H) becomes a unital ∗-algebra by defining the ∗-operation (g,h)∗=(h-1gh,h-1) on the basis elements and extending antilinearly to D(G;H).The coproduct△, the counit ε, and the antipode S are defined by (10)△g,h=∑t∈Ht,h⊗t-1g,h,εg,h=δg,u,Sg,h=h-1g-1h,h-1, where g∈G,h∈H, and δg,h=1,ifh=g0,ifh≠g.. One can prove D(G;H) becomes a Hopf ∗-algebra ([15]). There exists a unique element zH=1/H∑h∈Hu,h, called a cointegral, satisfying ∀a∈D(G;H), azH=zHa=ε(a)zH, and ε(zH)=1. As a result, D(G;H) is semisimple with a natural ∗-structure. Consequently it can be a C∗-algebra of finite dimension.In particular, ifH=G, then D(G;H)≜D(G) is the Drinfeld double of G. For more information about D(G) one can refer to [16–18]. The main difference between D(G) and D(G;H) is that the former is a quasitriangular Hopf algebra while the latter is not ([19]).Considering a linear map(11)E:DG⟶DG;H∑g,h∈Gkg,hg,h↦∑g∈G,h∈Hkg,hg,h where kg,h∈C, one can show that it is the conditional expectation from D(G) onto D(G;H). Moreover, setting ui=∑g∈G(g,ti), then ui∗=∑g∈G(g,ti-1) and {(ui,ui∗)} is a quasibasis for E. Thus IndexE=[G:H]I. In this case, E is of index-finite type.Definition 6.
The completion ofD(G)⊗D(G;H)D(G) with respect to the norm ·max (after taking quotient by the ideal {c∈D(G)⊗D(G;H)D(G):cmax=0} if necessary) is called the C∗-basic construction from the conditional expectation E of D(G) onto D(G;H). We denote this C∗-algebra by C∗〈D(G),e〉.Note that∑g∈G∑i=1k(g,ti)⊗(g,ti)∗ is the unit of C∗〈D(G),e〉.


## 3. The Construction of a Crossed ProductC∗-Algebra

Let us continue to suppose thatG is a finite group and H≤G. Denoted by G/H the set of all left cosets of H, i.e., G/H={t1,t2,…,[tk]}. Let C(G/H×G) and CG be the algebra of complex functions on G/H×G and the group algebra over C, respectively.The set{χ[ti]:i=1,2,…,k} is a linear basis of C(G/H) where χ[ti][tj]=1,ifj=i,0,ifj≠i, while the set {δg:g∈G} is a linear basis of C(G) where δg(h)=1,ifh=g,0,ifh≠g. Since G is a finite group and C(G/H×G)≅C(G/H)⊗C(G), then {(ti,g):i=1,2,…,k;g∈G} can be regarded as a linear basis of C(G/H×G), where we write (ti,g) instead of (χ[ti],δg) for notational convenience.For anyh∈G, we can define the map σh:C(G/H×G)→C(G/H×G) given on the basis elements (ti,g) of C(G/H×G) as(12)σhti,g=hti,hgh-1, which can be linearly extended in C(G/H×G). One can verify that σh is an automorphic map, and then σ defines an automorphic action of C(G) on C(G/H×G).Assume thatC(G/H×G) acts on a Hilbert space l2(G/H×G), and an action σ of C(G) on C(G/H×G) is given as above. Let H=l2(G/H×G)⊗l2(G). We view H as the Hilbert space of all square summable l2(G/H×G)-valued functions on G, and define: (13)πti,gξh=σh-1ti,gξh,λgξh=ξg-1h, for g,h∈G, ξ∈H, i=1,2,…,k.Lemma 7.
π is a faithful ∗-representation of C(G/H×G) on H, while λ is a unitary representation of G on H. And(14)λhπti,gλh∗=πσhti,g,g,h∈G,i=1,2,…,k.Definition 8.
The associativeC∗-algebra on H generated by {π(ti,g),λ(h):g,h∈G;i=1,2,…,k} is called the crossed product of C(G/H×G) by CG with respect to σ, and we denote it by C(G/H×G)⋊σCG (or simply by C(G/H×G)⋊CG).
Here and from now on, byh(ti,g),(ti,g) and h we always denote σh(ti,g),π(ti,g) and λ(h), respectively.Lemma 9.
(1)
The element∑g∈G(t1,g) in C(G/H×G) is a projection on H, i.e., (15)∑g∈Gt1,g=∑g∈Gt1,g2=∑g∈Gt1,g∗. (2)
Forg,h∈G, we have (16)∑α∈Gt1,α∑i=1kti,gh∑β∈Gt1,β=δt1,h∑i=1kti,gh∑γ∈Gt1,γ. (3)
Leth∈G, then h∈H if and only if (17)∑i=1kti,gh∑γ∈Gt1,γ=∑γ∈Gt1,γ∑i=1kti,gh.Proof.
(1) For anyξ∈H and h∈G, observe that (18)∑g∈Gt1,gξh=h-1∑g∈Gt1,gξh=∑g∈Gh-1,h-1ghξh,and (19)∑g∈Gt1,g2ξh=h-1∑f∈Gt1,f∑g∈Gt1,gξh=h-1∑f∈Gt1,fh-1∑g∈Gt1,gξh=h-1∑f∈Gt1,fh-1∑g∈Gt1,gξh=∑f∈Gh-1,h-1fh∑g∈Gh-1,h-1ghξh=∑g∈Gh-1,h-1ghξh, From the above equalities, we conclude that ∑g∈G(t1,g)=(∑g∈G(t1,g))2. And(20)∑g∈Gt1,gξ,η=∑x∈G∑g∈Gt1,gξx,ηx=∑x∈G∑g∈G∑m∈G/H∑n∈Gt1,gξxm,nηxm,n¯=∑x∈G∑g∈G∑m∈G/H∑n∈Gx-1t1,gξxm,nηxm,n¯=∑x∈G∑g∈G∑m∈G/H∑n∈Gx-1,x-1gxm,nξxm,nηxm,n¯=∑x∈G∑g∈G∑m∈G/H∑n∈Gξxm,nx-1,x-1gxm,nηxm,n¯=∑x∈G∑g∈G∑m∈G/H∑n∈Gξxm,nx-1t1,gηxm,n¯=∑x∈G∑g∈Gξx,t1,gηx=ξ,∑g∈Gt1,gη,which shows that ∑g∈G(t1,g)=(∑g∈G(t1,g))∗.
(2) Letg,h∈G, we can compute that (21)∑α∈Gt1,α∑i=1kti,gh∑β∈Gt1,β=∑α∈G∑β∈G∑i=1kt1,αti,ght1,β=∑α∈G∑β∈G∑i=1kt1,αti,ght1,β=∑α∈G∑β∈G∑i=1kt1,αti,gh,hβh-1h=∑β∈Gt1,gh,hβh-1h=δt1,ht1,gh,and (22)∑i=1kti,gh∑γ∈Gt1,γ=∑γ∈G∑i=1kti,ght1,γ=∑γ∈G∑i=1kti,gh,hγh-1h=∑γ∈G∑i=1kδti,hδgh,hγti,gh=t1,gh.
(3) Ifh∈H, then (∑α∈G(t1,α))(∑i=1k(ti,g)h)=∑α∈G∑i=1k(t1,α)(ti,g)h=(t1,g)h and (23)∑i=1kti,gh∑α∈Gt1,α=∑i=1k∑α∈Gti,ght1,α=∑i=1k∑α∈Gti,gh,hαh-1h=t1,gh. Thus ∑i=1k(ti,g)h commutes with ∑α∈G(t1,α).
Conversely, ifh∈G and (∑i=1k(ti,g)h)∑γ∈G(t1,γ)=∑γ∈G(t1,γ)(∑i=1k(ti,g)h), then (h,g)h=(t1,g)h which implies that (h,g)=(t1,g). This shows that h∈H.From the proof of Lemma9, we can obtain (ti,g)∗=(ti,g) for any g∈G, i=1,2,…,k.Now we give the main theorem of this paper.Theorem 10.
There exists an isometric∗-isomorphism of C∗〈D(G),e〉 onto C(G/H×G)⋊CG.Proof.
By the definition of·max, it suffices to show that there exists an ∗-isomorphism Φ of D(G)⊗D(G;H)D(G) on H.
The mapΦ:D(G)⊗D(G;H)D(G)→C(G/H×G)⋊CG is defined on the generators of D(G)⊗D(G;H)D(G) by (24)Φg,α⊗h,β=∑i=1kti,gα∑f∈Gt1,f∑j=1ktj,hβ=δgα,αhα,gαβ, for (g,α), (h,β) in D(G). Since ∑i=1k(ti,g)h and ∑γ∈G(t1,γ) commute, Φ is a well defined map. Then it can be linearly extended in D(G)⊗D(G;H)D(G). Again since (25)Φg,α⊗h,β∗=Φh,β∗⊗g,α∗=Φβ-1hβ,β-1⊗α-1gα,α-1=δgα,αhβ-1,β-1hββ-1α-1, and (26)Φg,α⊗h,β∗=δgα,αhα,gαβ∗=δgα,αhβ-1α-1α,g∗=δgα,αhβ-1α-1α,β-1α-1gαβ=δgα,αhβ-1,β-1hββ-1α-1. Then (27)Φg,α⊗h,β∗=Φg,α⊗h,β∗. Hence, combining with Lemma 9, we have that the map Φ is a ∗-homomorphism.
For(ti,g)h∈C(G/H×G)⋊CG, choose (g,ti)⊗(ti-1gti,ti-1h)∈D(G)⊗D(G;H)D(G) such that Φ((g,ti)⊗(ti-1gti,ti-1h))=(ti,g)h. This shows that Φ is surjective. Finally, we will verify that Φ is isometric.
Letψ:D(G;H)→C(G/H×G)⋊CG⊆L(H) be a map given as ψ(g,α)=(t1,g)α. It is easy to see that ψ is a injective ∗-homomorphism. Define ψ′:D(G;H)⊗Mn→L(H)⊗Mn by ψ′=ψ⊗id. Then ψ′ is also injective. Again, by Lemma 3, there exists (28)Φ′:DG⊗Mn⊗DG;H⊗MnDG⊗Mn→LH⊗Mnsuch that Φ′((xij)ij⊗(yjk)jk)=(∑jΦ(xij⊗yjk))ik. Since ψ′((xij)ij)=Φ′((xij)ij⊗1), (29)Φc2=∑iΦxi⊗yi2=∑iΦxi⊗yi00⋯0000⋯0···⋯····⋯·000⋯02=Φ′x1x2x3⋯xn000⋯0···⋯····⋯·000⋯0⊗y100⋯0y200⋯0···⋯····⋯·yn00⋯02=Φ′X⊗Y2=Φ′X⊗Y∗Φ′X⊗Y=Φ′Y∗⊗X∗Φ′X⊗Y=Φ′Y∗FX∗X⊗Y=Φ′Y∗FX∗X1/2⊗1Φ′1⊗FX∗X1/2Y=Φ′1⊗FX∗X1/2Y∗Φ′1⊗FX∗X1/2Y=Φ′1⊗FX∗X1/2YΦ′1⊗FX∗X1/2Y∗=Φ′FX∗X1/2FYY∗FX∗X1/2⊗1=FX∗X1/2FYY∗FX∗X1/2⊗1=FX∗X1/2FYY∗1/22=Exi∗xjij1/2Eyiyj∗ij1/2, for c=∑ixi⊗yi∈D(G)⊗D(G;H)D(G), where F=E⊗id:D(G)⊗Mn→D(G;H)⊗Mn is a conditional expectation and X=x1x2x3⋯xn000⋯0···⋯····⋯·000⋯0, Y=y100⋯0y200⋯0···⋯····⋯·yn00⋯0. Thus, we have that Φ(c)=c. Now we have shown Φ is an isometric ∗-isomorphism.Example 11.
(1) LetC∗(G) (or C∗(H)) be the group C∗-algebra of G (or H), namely, the C∗-subalgebra of L(l2(G)) generated by the left regular representation of G (or H). Consider the basic construction from the conditional expectation E:C∗(G)→C∗(H) defined by (30)E∑g∈Gkgλg=∑h∈Hkhλh where kg∈C. Let τ:CG→AutC(G/H) be the action induced by translation from left, where AutC(G/H) stands for the group of all automorphism of C(G/H). In [10], Watatani showed that the C∗-basic construction C∗〈C∗(G),e〉 is C∗-algebra isomorphic to C(G/H)⋊τCG. This result is a special case of Theorem 10. In fact, there is an inclusion i:C∗(G)→D(G) given by i(λg)=∑f∈G(f,g).
(2) IfG is a finite abelian group, then D(G) reduces to a symmetry group G^×G, where G^ denotes the Pontryagin dual of G. Let E:G^×G→G^×H be a conditional expectation defined by (31)E∑g,h∈Gkg,hg,h=∑g∈G,h∈Hkg,hg,h. We can define the map ς:CG×C(G/H×G)→C(G/H×G) by(32)ςh×ti,g=hti,g for all (ti,g)∈C(G/H×G). Then ς is an automorphic action of CG on C(G/H×G). Then the C∗-basic construction C∗〈G^×G,e〉 is C∗-algebra isomorphic to C(G/H×G)⋊ςCG. Furthermore, the C∗-basic construction C∗〈G^×G,e〉 can be described as G^×(C(G/H)⋊τCG), where τ is the action induced by translation from left.



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