Realizing the Braided Temperley–Lieb–Jones C*-Tensor Categories as Hilbert C*-Modules

We associate to each Temperley–Lieb–Jones C*-tensor category TLJ(δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )$$\end{document} with parameter δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} in the discrete range {2cos(π/(k+2)):k=1,2,…}∪{2}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{2\cos (\pi /(k+2)):\,k=1,2,\ldots \}\cup \{2\}$$\end{document} a certain C*-algebra B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document} of compact operators. We use the unitary braiding on TLJ(δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )$$\end{document} to equip the category ModB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {Mod}_{{\mathcal {B}}}$$\end{document} of (right) Hilbert B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document}-modules with the structure of a braided C*-tensor category. We show that TLJ(δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {T}}{\mathcal {L}}{\mathcal {J}}(\delta )$$\end{document} is equivalent, as a braided C*-tensor category, to the full subcategory ModBf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {Mod}_{{\mathcal {B}}}^f$$\end{document} of ModB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {Mod}_{{\mathcal {B}}}$$\end{document} whose objects are those modules which admit a finite orthonormal basis. Finally, we indicate how these considerations generalize to arbitrary finitely generated rigid braided C*-tensor categories.


Motivation.
Ultimately, our goal of describing TLJ (δ) in terms of Hilbert C*modules is motivated by a connection with K -theory (cf. e.g. [6,27,49]), namely the theorem of Freed, Hopkins and Teleman (cf. [15][16][17]) describing the fusion ring of the category of level k representations of the loop group LSU (2) in terms of twisted equivariant K -theory. Related to this, we observed in [1] that the K 0group of certain approximately finite-dimensional (AF) C*-algebras has a ring structure that is closely related to the fusion ring of TLJ (δ). For example, the K 0group of the inductive limit TLJ ∞ (δ) = lim n TLJ n (δ) of Temperley-Lieb-Jones C*algebras, whose Bratteli diagram is given in [30], is a localization of the fusion ring of TLJ (δ). The present paper is a result of our efforts to lift such a ring structure in K 0 -theory to a tensor product structure on an underlying category of modules. We found it natural to use the framework of Hilbert C*-modules, which generalize both Hilbert spaces and vector bundles and find uses in diverse areas of mathematics, including K -theory, Kasparov's KK -theory, and quantum groups (cf. e.g. [6,37]).

Related work.
Given a (small) rigid C*-tensor category C, Yuan in [57] constructed a unital C*-algebra A and a fully faithful monoidal *-functor from C into the category A Mod A of finite type Hilbert C*-bimodules over A, the tensor product in A Mod A being given by interior tensor product. A variant of Yuan's construction yields a fully faithful monoidal *-functor from TLJ (δ) into A Mod A , where A is the unital AF-algebra whose Bratteli diagram arises from the fusion graph of f (0) ⊕ f (1) (in the notation of Sect. 2.4.3). For example, when δ = 2 cos(π/5), this diagram is In the present paper, we make use of Yuan's formalism in defining certain Hilbert spaces and bounded operators. In turn, Yuan was influenced by earlier realizations of C*-tensor categories in terms of bimodules over von Neumann algebras (for which we refer to the citations in [57]).
On the other hand, based on the work of Guionnet, Jones and Shlyakhtenko [24], Hartglass and Penneys in [25] associated a C*-algebra B along with a Hilbert C*bimodule X over B to an arbitrary factor planar algebra P • . They then fed this bimodule into a construction due to Pimsner (cf. [42]) in order to associate Cuntz and Toeplitz type algebras to planar algebras. When P • is the Temperley-Lieb-Jones planar algebra with parameter δ, K 0 (B) is isomorphic to the fusion ring of TLJ (δ). This led us to consider modules over a variant of the C*-algebra B.
It should also be mentioned that the tensor product that is defined in the present paper is related to a tensor product of modules over Temperley-Lieb algebras with varying numbers of strands that was introduced in [47,48] and studied further in [19], [3,18]. Moreover, the definition of the modified version of the C*-algebra B of Hartglass and Penneys that we use is influenced by the notion of dilute Temperley-Lieb algebras, which originated in [5,23].
Our contribution starts in Sect. 3. Using the formalism of Yuan and the notion of dilute Temperley-Lieb diagrams (as presented in [5]), we construct a variant of the C*algebra B of Hartglass and Penneys (Sect. 3.1). Next, we explain a way to associate operators in B and its strong closure to certain infinite diagrams (Sect. 3.2). Using an idea of Erlijman and Wenzl (cf. [11]), we then harness the unitary braiding on TLJ (δ) to define a *-homomorphism : B ⊗ B → B by superposition of diagrams (Sect. 3.3) and observe that the product on K 0 (B) induced by recaptures the product in the fusion ring of TLJ (δ) (Remark 3.4).
In Sect. 4, we first use as well as interior and exterior tensor products of Hilbert C*-modules to define a tensor product of Hilbert B-modules (Sect. 4.1). We next use this tensor product to equip the category Mod B of Hilbert B-modules with the structure of a C*-tensor category (Sect. 4.2) and supply it with a unitary braiding (Sect. 4.3).
In Sect. 5, we first define a *-functor F from TLJ (δ) into Mod B and show that it is monoidal and braided (Sect. 5.1). In Sect. 5.2, we then use F to prove Theorem 5.3, which states that TLJ (δ) is equivalent, as a braided C*-tensor category, to the full subcategory Mod f B of Mod B whose objects are those modules which admit a finite orthonormal basis (and which is introduced in Sect. 4.4). Thereafter, we note that the tensor category Mod B "categorifies" the ring K 0 (B) (Remark 5.5) and indicate how one can prove a version of Theorem 5.3 for arbitrary finitely generated rigid braided C*-tensor categories (Remark 5.6).
Finally, in Sect. 6, we pose some questions concerning representability of C*-tensor categories on Hilbert C*-modules and realizability of the representation category of the Virasoro algebra.

Preliminaries
2.1. Operators on Hilbert space. In this paper, we consider operators on a complex Hilbert space H . We denote by B(H ) the space of all bounded linear operators on H , which comes equipped with a plethora of topologies. In this paper, we will restrict attention to the norm topology, which is induced by the operator norm, and the strong operator topology, which is the topology of pointwise convergence in the norm on H , that is, a n → a strongly if and only if a n (ξ ) − a(ξ ) → 0 for all ξ ∈ H . We will need the following standard fact.
n=0 be a bounded sequence in B(H ) such that a n a * m = a * n a m = 0 whenever n = m. Then n≥0 a n and n≥0 a * n converge strongly in B(H ). The normed space B(H ) is a C*-algebra. It contains the C*-subalgebra K(H ) of compact operators, which is the smallest C*-subalgebra of B(H ) that contains all operators of finite rank. The following standard fact will be useful to us. Fact 2.2. Let (a n ) ∞ n=0 be a sequence in B(H ) that converges strongly to some operator a. For any compact operator x ∈ K(H ), we have that a n x − ax → 0.

Hilbert C*-modules.
A (right) Hilbert C*-module over a C*-algebra B is a (right) B-module M equipped with a B-valued inner product ·, · : M × M → B such that ξ → ξ, ξ 1/2 is a complete norm. The general theory of such modules is laid out very carefully in [37], to which we refer for precise definitions and all the information that the reader will need.
Let us comment on the notation and terminology used in the present paper. We use the symbol for the exterior tensor product of Hilbert C*-modules (so that if M is a Hilbert A-module and N is a Hilbert B-module then M N is a Hilbert (A⊗ B)-module) and the symbol ⊗ φ for the interior tensor product with respect to a *-homomorphism φ. By an orthonormal basis for a Hilbert B-module M, we shall mean a (possibly infinite) family (ξ j ) j∈J of elements in M such that (iii) the Fourier expansion η = j∈J ξ j ξ j , η is valid for all η ∈ M.
[By definition, a (unitary) isomorphism in C is a morphism u such that u * u = id and uu * = id.] (5) There is a distinguished object 1 in C (called the tensor unit) and natural unitary isomorphisms π : 1 ⊗ π → π and r π : π ⊗ 1 → π (called left and right unit constraints) satisfying the triangle identity (see Definition 2.1.1(iv) of [40] or Eq. (4.4) below). (6) ( f ⊗ g) * = f * ⊗ g * for all morphisms f and g. (7) The category C has subobjects and finite direct sums (see Definition 2.1.1(vi), (vii) of [40]). (8) The tensor unit is simple. [An object π in C is said to be simple if End(π ) = Cid π .] A C*-tensor category is said to be strict if the associators and unit constraints are identity morphisms.

Semisimplicity
Briefly speaking, a C*-tensor category C is said to be semisimple if every object in C is isomorphic to a finite direct sum of simple objects. We next explain what this means in detail. Pick a set S of mutually non-isomorphic simple objects such that every simple object in C is isomorphic to some s ∈ S. Given an object ρ in C, there exist non-negative integers N s (with N s = 0 for all but finitely many s) such that ρ ∼ = s∈S s ⊕N s , where s ⊕N s denotes a direct sum of N s copies of s. This means that, for each s with N s > 0, there exist N s morphisms v s,1 , . . . , v s,N s ∈ Hom(s, ρ) such that v * s, j v s, j = id s for all j and id ρ = s∈S N s j=1 v s, j v * s, j . In fact, v s,1 , . . . , v s,N s form an orthonormal basis for Hom(s, ρ) equipped with the inner product ·, · given by ξ, η id s = ξ * η for ξ, η ∈ Hom(s, ρ). The number N s is called the multiplicity of s in ρ and is sometimes denoted by (s, ρ). We write s ≺ ρ if (s, ρ) > 0. Since we mention it in a few places, we also recall that the fusion ring Z[S] of C is the free abelian group generated by S and equipped with the product s · t = r ∈S (r, s ⊗ t)r .

Monoidal functors
A functor F : C → D between C*-tensor categories C and D is called a *-functor if F is linear and satisfies F( f * ) = F( f ) * for all morphisms f . It is said to be monoidal (or to be a tensor functor) if there are natural unitary isomorphisms J π,ρ : F(π ) ⊗ F(ρ) → F(π ⊗ ρ) that are compatible with the associators and unit constraints (see Definition 2.1.3 of [40] or Eqs. (5.1)-(5.3) below). If F is a monoidal *-functor and C and D are both braided then we say that F is braided if the isomorphisms J are compatible with the braiding (see Eq. (5.4) below).

The Temperley-Lieb-Jones categories.
In this section, we recall the notion of Temperley-Lieb diagrams and of certain vector spaces, algebras and categories that one can associate to them.

Temperley-Lieb-Jones algebras
We recall first the notion of an (m, n)-Temperley-Lieb diagram (for m, n ≥ 0 of equal parity), which first appeared in [36]. Such a diagram consists of (m +n)/2 non-crossing smooth strands inside a rectangle with m nodes (or marked points) on the left side and n nodes on the right side, each node being connected to a unique strand. In particular, TL 0 n,n (δ) is an associative algebra, which is known as the n'th Temperley-Lieb algebra. One can define a linear trace Tr TL n on TL 0 n,n (δ) as follows. If D is an (n, n)-Temperley-Lieb diagram then Tr TL n (D) is defined by a picture such as the one below (in which n = 3), which is turned into a scalar by removing closed loops as explained above. (This trace is usually called a Markov trace.) D Moreover, one can define an antilinear *-operation TL 0 m,n (δ) → TL 0 n,m (δ) by reflecting diagrams about a vertical axis.

Temperley-Lieb-Jones C*-tensor categories
Let δ ∈ {2 cos(π/(k + 2)) : k = 1, 2, . . .} ∪ [2, ∞) be given. The Temperley-Lieb-Jones (or reduced Temperley-Lieb) C*-tensor category TLJ (δ) with parameter δ is defined as follows. Its objects are all formal finite sums P 1 ⊕ · · · ⊕ P k , where P j is a projection in the C*-algebra TLJ n j (δ) for each j. Given projections P ∈ TLJ n (δ) and Q ∈ TLJ m (δ), the morphism set Hom(P, Q) is QTLJ m,n (δ)P. More generally, given objects ⊕ k j=1 P j and ⊕ r i=1 Q i , the morphism set Hom(⊕ k j=1 P j , ⊕ r i=1 Q i ) consists of all r ×k-matrices whose (i, j)'th entry is in Hom(P j , Q i ). Composition of morphisms is given by multiplication of Temperley-Lieb diagrams combined with matrix multiplication. The tensor product in TLJ (δ) is defined as follows. Given projections P ∈ TLJ n (δ) and Q ∈ TLJ m (δ), the tensor product P ⊗ Q is formed by stacking P on top of Q (or rather by the bilinear extension of this procedure applied to pairs of diagrams) to obtain a projection in TLJ n+m (δ). The tensor product of two objects The tensor product of morphisms is given by vertical stacking combined with tensor multiplication of matrices, i.e., ( j,l) . One can show that TLJ (δ) is a strict C*-tensor category, whose tensor unit is the empty Temperley-Lieb diagram.
In this case, f (1) In either case, the category TLJ (δ) is generated by the object π = f (1) in the sense that every simple object occurs as a direct summand of some tensor power π ⊗n of π . Note in this connection that Hom(π ⊗n , π ⊗m ) = TLJ m,n (δ) for all n, m ≥ 0.

The unitary braiding
is a braided C*-tensor category. Specifically, one defines a unitary braiding σ TL as follows. Consider the unitary Kauffman element We will use the following conventional graphical representation of the Kauffman element as a crossing.
Using it, one can define a unitary element σ TL π ⊗n ,π ⊗m of End(π ⊗(n+m) ) = TLJ n+m (δ) by a braid diagram like the one below (which corresponds to the case n = 2 and m = 3).
, one uses the isotopy invariance of the Temperley-Lieb diagrams along with the following two identities, which follow easily from the definition of the crossing.

= =
Finally, the unitary braiding σ TL is given by the unitary isomorphisms σ TL

Definition of B.
For each s ∈ S and μ ∈ G ∞ , the morphism space Hom(s, o( μ)) is equipped with the inner product ·, · given by ξ, η id s = ξ * η. We denote by H s the orthogonal direct sum of the Hilbert spaces Hom(s, o( μ)) as μ varies through G ∞ . In symbols, Next, we put for ξ ∈ Hom(s, o( μ)). It is a bounded operator whose adjoint operator is Moreover, In particular, The semisimplicity of C implies that φ is injective. Since every injective *homomorphism between C*-algebras is isometric, it follows that For each n ≥ 0, denote by B n the finite-dimensional C*-algebra spanned by the operators of the form L x, y (a), where x k = y k = 1 for all k > n. Each B n admits a positive faithful trace Tr n defined by Tr n (L x, y (a)) = δ x, y Tr TL k (a), where k is the number of entries in x that equal π . Moreover, B n ⊆ B n+1 for all n. Denote by B the smallest C*-subalgebra of B(H ) that contains every B n , i.e., The following result describes the structure of B.

Lemma 3.2. We have that
Proof. Note first that B ⊂ K(H ). Indeed, each operator L x, y (a) is compact because it can be written as L x, y (a)P, where P is the orthogonal projection onto the finitedimensional subspace s≺o( y) Hom(s, o( y)). Conversely, if ξ is a unit vector in Hom(s, o( y)) and η is a unit vector in Hom(s, o( x)) then B contains the rank one operator L x, y (ηξ * ) ∈ K(H s ), which maps ξ onto η. Thus, for each s ∈ S, B contains a complete set of matrix units for K(H s ). The result follows.
The next lemma will be used to define certain morphisms between tensor products of B-modules. Proof. Note that v * = n≥0 v * n , where the sum converges in the strong operator topology. Let b ∈ B be given. By Fact 2.2, n≥0 v n b converges to vb in norm because b ∈ K(H ). Similarly, n≥0 v * n b * converges to v * b * in norm. Since B is a C*-subalgebra of B(H ), the lemma follows.

Diagrammatic operators.
In effect, the above construction allows us to associate operators to certain kinds of diagrams. These diagrams all consist of strands inside a rectangle with an infinite sequence of nodes, some empty and some non-empty (or filled-in), attached to each of its (left and right) sides such that every strand connects two distinct non-empty nodes and every non-empty node is the end point of a unique strand. The simplest such diagram is a dilute Temperley-Lieb diagram (cf. e.g. [5]). It has only finitely many non-empty nodes, which are connected by non-crossing strands. The top of such a diagram is depicted below.
The following figure illustrates the product of two diagrammatic operators. Note that the patterns of empty and non-empty nodes have to match in the middle for the product to be non-zero.
The unitary braiding on C allows us to also associate operators to certain diagrams that involve crossings. For instance, we can associate operators to what one might call "finite dilute braid diagrams". Such a diagram has only finitely many non-empty nodes (which is what the term "finite" in the name of the diagrams refers to). Moreover, every strand connects a node on the left side to one on the right side, and any two given strands are only allowed to cross a finite number of times. The top of a sample diagram of this type is shown below. If one such diagram can be obtained from another by a finite sequence of Reidemeister moves of types 2 and 3 then these two diagrams give rise to the same operator. Indeed, the unitary braiding engenders, in a natural way, a group homomorphism from Artin's braid group on n strands into the unitary group of End(π ⊗n ) for every n (see e.g. page 374 in [11]). In particular, every finite dilute braid diagram gives rise to a partial isometry in B.
We will also in a slightly different way associate operators to what might be termed "(possibly) infinite dilute braid diagrams". These diagrams are defined in the same way as their finite cousins, except that they are allowed to have infinitely many non-empty nodes and hence infinitely many strands. Let D be such a diagram and denote by (D) the pattern of empty and non-empty nodes on its left side. Denote by supp(D) the set of patterns that can be obtained from (D) by replacing all but finitely many non-empty nodes by empty ones. Given x ∈ supp(D), we get a finite dilute braid diagram D x by removing from D every strand whose left end point corresponds to an empty node in x and replacing both end points of each removed strand by empty nodes. As mentioned above, this new diagram gives rise to a partial isometry in B, which we denote by v (D, x).
Although v(D) need not belong to B, Fact 2.1 and Lemma 3.

+ + · · ·
We can think of U 3 as v(D), where D is the diagram on the left, all nodes below the displayed part of the diagram being empty. However, in this case it is just a finite sum.
We can now define a *-homomorphism n : B n ⊗ B n → B 2n by . . . , x n , v 1 , . . . , v n , . . .) (and similarly for y w). The faithfulness of the traces Tr n and the fact that Tr 2n • n = Tr n ⊗ Tr n on elements of the form L x, y (a) ⊗ L v, w (b) imply that n is a well-defined isometric *-homomorphism. The purpose of the unitaries U n is to ensure that n+1 • (ι n ⊗ ι n ) = ι 2n+1 • ι 2n • n for all n ≥ 0, where ι n is the inclusion map B n → B n+1 . This allows us to extend the *-homomorphisms n to an isometric *-homomorphism Diagrammatically, the effect of applying to a tensor product L x, y (a) ⊗ L v, w (b) of operators arising from dilute Temperley-Lieb or braid diagrams is to superimpose the one on the left on top of the one on the right in such a way that the nodes are interleaved.
Remark 3.4. By Lemma 3.2, K 0 (B) is isomorphic to the fusion ring Z[S] as an abelian group. It is also easy to check that the induced product map on K 0 (B) agrees with the product on the fusion ring. (This boils down to the fact that L μ, μ (vv * ) is a rank one projection in K(H s ) for any μ ∈ G ∞ and any unit vector v ∈ Hom(s, o( μ)).) Below, we will "categorify" this statement, by using to define a tensor product of right Hilbert B-modules that recaptures the tensor product in C (see also Remark 5.5).

On the Braided C*-Tensor Categories Mod B and Mod f B
In this section, we use the *-homomorphism from the previous section to endow the category Mod B of (right) Hilbert B-modules with the structure of a braided C*-tensor category. We also introduce the full subcategory Mod f B of modules admitting a finite orthonormal basis.

A tensor product of right Hilbert B-modules.
Given two right Hilbert B-modules M 1 and M 2 , we define their tensor product by where : B ⊗ B → B is the *-homomorphism from the previous section. (See Sect. 2.2 for an explanation of the notation.) Given adjointable maps f 1 : M 1 → N 1 and f 2 : M 2 → N 2 between right Hilbert B-modules, we denote by f 1 ⊗ f 2 the adjointable map M 1 ⊗ M 2 → N 1 ⊗ N 2 given by As a simple example, let p and q be projections in B. Then pB and qB are right Hilbert B-modules (with inner product given by (a, b) → a * b) and there exists a surjective B- We next relate the above tensor product to the standard direct sum of Hilbert Bmodules. Given finite families (M i ) i∈I and (N j ) j∈J of right Hilbert B-modules, we have a surjective B-linear isometry

The C*-tensor category Mod B .
We denote by Mod B the category whose objects are all right Hilbert B-modules and whose morphism sets Hom(M, N ) consist of all adjointable (or, equivalently, all bounded B-linear, cf. [14]) maps M → N . Below, we will endow this category with the structure of a C*-tensor category. Note first that conditions (1), (2), (3), (6) and (7) in Sect. 2.3.1 follow from the general theory of Hilbert C*-modules. Thus, our goal in the present section is to define associators, a tensor unit, and unit constraints satisfying conditions (4), (5) and (8).

Associators
We begin by defining associators in Mod B . To do so, we first define a unitary operator V ∈ B(H ) as the operator associated to the following infinite braid diagram D α . (Note that, in notation introduced on page 10, the multi-colored figure on page 10 depicts D α x (= (D α ) x ), where x = (π, π, π, π, 1, π, 1, π, . . .).) First connect the nodes numbered 4, 8, 12, . . . on the left side to those numbered 2, 4, 6, . . . on the right side by strands in order. (These nodes and strands are colored red in the aforementioned figure.) Next connect, by (green) strands that cross over the ones already drawn, the nodes on the left side numbered 2, 6, 10, . . . to those numbered 3, 7, 11, . . . on the right side. Finally, connect, by (blue) strands that cross over the ones already drawn, the nodes on the left side numbered 1, 3, 5, . . . to those numbered 1, 5, 9, . . . on the right side.
We next observe that The following figures illustrate the case when b 1 , b 2 , b 3 ∈ B 2 . In that case, the left hand side of Eq. In general, one of these diagrams can be obtained from the other by a finite sequence of Reidemeister moves of types 2 and 3. Thus, the associated operators are equal.
We can now define associators as follows. Given right Hilbert B-modules M 1 , M 2 and M 3 , consider the formula 3 and a, b, c, d, e ∈ B. Here, ξ 1 ⊗ ξ 2 a ⊗ b ⊗ ξ 3 cd ⊗ e on the left hand side is viewed as an element of e on the right hand side is viewed as an element of On the one hand, we get that On the other hand, we have that Since these two expressions coincide by Eq. (4.1), the above formula defines a B-linear isometry Similarly, we can define a B-linear isometry

Pentagon identity
In order to show that Mod B along with the associators α M 1 ,M 2 ,M 3 and the unit constraints that we define below is a C*-tensor category, we must verify the pentagon identity, which in the present context is the identity for any objects M 1 , M 2 , M 3 and M 4 in Mod B . We verify it by applying both sides to an element of the form Let us first consider the left hand side. First, α M 1 ,M 2 ,M 3 ⊗ id M 4 maps the given element to Next, α M 1 ,M 2 ⊗M 3 ,M 4 maps the above element to We now consider the right hand side. First, α M 1 ⊗M 2 ,M 3 ,M 4 maps the element in Eq. (4.3) to Next, α M 1 ,M 2 ,M 3 ⊗M 4 maps the above element to which is equal to and, in turn, to We now see that the pentagon identity reduces to the identity Since B is generated by operators arising from dilute Temperley-Lieb diagrams, and because V = x v(D α , x) for a certain infinite braid diagram D α (see page 10), it suffices to prove that whenever x, y, z, μ, ν, β, γ ∈ G ∞ are such that the patterns agree. (Recall that p z was defined on page 8.) In this identity, each side is the operator associated to some finite dilute braid diagram. One can easily check that both of these diagrams consist of strands that live on four separate layers, as we next explain. The bottom layer L 1 consists of those strands whose left end point is at one of the non-empty nodes numbered 4, 8, 12, . . ., the next layer L 2 at those numbered 6, 14, 22, . . ., the next layer L 3 at those numbered 2, 10, 18, . . ., and the top layer L 4 at those numbered 1, 3, 5, . . .. This means that, in both diagrams, every crossing is of the following sort: A strand from L j crosses over a strand from L i with j > i. It is easily deduced from this that one of the diagrams can be obtained from the other by a finite sequence of Reidemeister moves of types 2 and 3, from which the identity follows.

Tensor unit and unit constraints
Denote by p * the operator in B that is associated to the empty diagram. We will exhibit p * B as a tensor unit in Mod B by defining explicit unit constraints for each object M in Mod B . First, we define two partial isometries W and W r in B(H ). Namely, W r is the operator associated to the infinite dilute braid diagram which we will call D r , while W is the operator associated to the diagram which we call D . We have that It follows from this that we may define a unitary isomorphism M and b 0 , b 1 , a ∈ B. Note that the adjoint (and inverse) of r M is given by the formula M and b 0 , b 1 , a ∈ B. Again, the assignment M → M is natural in M.

Triangle identity
In the present context, the triangle identity states that for any objects M 1 and M 2 in Mod B . By applying both sides to an element of the form 4 , we see that the verification reduces to proving the identity Similarly to the case of the pentagon identity, it suffices to prove that whenever x, y, z, β, γ ∈ G ∞ are such that the patterns agree. Note that the operator on the left hand side arises from a finite dilute braid diagram such as while the operator on the right hand side arises from which can be obtained from the top diagram by a finite sequence of Reidemeister moves of type 2.

Simplicity of the tensor unit
To finish the proof that Mod B is a C*-tensor category, we note that p * B is a simple object in Mod B . Indeed, one easily checks that End( p * B) ∼ = p * B p * = p * B 0 p * = C p * (see also the proof of Lemma 5.1 below).

A unitary braiding on
Mod B . We next define a unitary braiding on Mod B and verify the hexagon identities.

Definition of the braiding
Denote by U the unitary operator in B(H ) that is associated to the infinite braid diagram D σ that is formed as follows. First connect the nodes on the left side numbered 2, 4, . . . to those on the right side numbered 1, 3, . . . by red strands (as in the following figure). Next, for each of the remaining nodes on the left numbered 2k − 1, say, draw a blue strand from it to the top of the diagram, crossing over the red strands whose left end point is above it, and then continue this strand to the node numbered 2k on the right side, now crossing under the red strands whose right end point is above that node. The following figure shows one of the associated finite dilute braid diagrams D σ Note that Equation (4.5) allows us, given two objects M 1 and M 2 in Mod B , to define a unitary isomorphism for ξ 1 ∈ M 1 , ξ 2 ∈ M 2 and a ∈ B. The assignment (M 1 , M 2 ) → σ M 1 ,M 2 is clearly natural in M 1 and M 2 and will turn out to be a unitary braiding on Mod B .

Hexagon identities
In the present context, the two hexagon identities are for any objects M 1 , M 2 and M 2 in Mod B . Let us prove the first identity and leave the second one to the reader. The left hand side maps an element of the form while the right hand side maps it to Thus, the first hexagon identity would follow from the identities for a, b, c, d, e ∈ B. As in the case of the pentagon identity, this reduces to showing that whenever x, y, z, μ, ν, β, γ , ∈ G ∞ are such that the patterns agree. In this identity, the operator on each side arises from a certain finite dilute braid diagram. The next figure shows a sample pair of diagrams that can appear. On the left hand side, we could have which would be paired with the following diagram on the right hand side.
Note that, in both diagrams, the blue strands always cross over the green strands. Thus, one can transform both diagrams into the same diagram by pulling the green and blue strands up and pulling the red strands down. In the case of our sample pair of diagrams, the common diagram is Since this only involves Reidemeister moves of types 2 and 3, the associated operators are equal.

The full C*-tensor subcategory Mod
Note that this could also be deduced from the following easily proved fact. Then M is isomorphic to k j=1 p j B, where p j = ξ j , ξ j for all j.

Realizing TLJ (δ) as Right Hilbert B-Modules
In this section, we show that TLJ (δ) is equivalent to Mod f B as a braided C*-tensor category.

A braided monoidal *-functor F
We will now define a functor F : C → Mod B (where C = TLJ (δ), as above). The following notation will be convenient. Setting x n = (π, . . . , π, 1, 1, . . .), with n leading copies of π , we denote L x n , x m (a) by L n,m (a) for any a ∈ Hom(π ⊗m , π ⊗n ) = TLJ n,m (δ). We also put L n (a) = L n,n (a) for any a ∈ End(π ⊗n ) = TLJ n (δ). Finally, we denote by p n the projection p x n (as defined on page 8).
We define F on objects as follows. Given a projection P ∈ TLJ n (δ), we define F(P) by the formula F(P) = L n (P)B, on the right hand side of which we view P as a morphism. Given an object ⊕ j P j in C, we put F ⊕ j P j = ⊕ j F(P j ).
On the right hand side, the symbol ⊕ denotes the standard direct sum of right Hilbert B-modules.
We next define F on morphisms. Given a morphism a ∈ Hom(P, Q), where P ∈ TLJ n (δ) and Q ∈ TLJ m (δ) are projections, we define F(a) to be the adjointable map L n (P)B → L m (Q)B given by left-multiplication by L m,n (a). For any morphism (a i j ) i, j ∈ Hom(⊕ j P j , ⊕ i Q i ), we define F((a i j ) i, j ) to be the adjointable map ⊕ j F(P j ) → ⊕ i F(Q i ) associated to the matrix (F(a i j ) It is clear that F is a *-functor. We will prove that it is in fact a braided monoidal *-functor. Thus, given any two objects ρ and ν in C, we will define a unitary isomorphism J ρ,ν : F(ρ) ⊗ F(ν) → F(ρ ⊗ ν) in such a way that the assignment (ρ, ν) → J ρ,ν is natural in ρ and ν and all four of the following identities hold for all objects ρ, ν and μ in C: The fact that J n,m is a well-defined unitary isomorphism comes down to the easily verified identities U * n,m U n,m = ( p n ⊗ p m ) and U n,m U * n,m = p n+m . Given projections P ∈ TLJ n (δ) and Q ∈ TLJ m (δ), we define J P,Q as the restriction of J n,m . More generally, given two objects ⊕ i P i and ⊕ j Q j in C, we define It is easy to reduce the naturality of J as well as the identities in Eq. (5.1)-(5.4) to the case where ρ, ν and μ are projections in Temperley-Lieb-Jones C*-algebras. Since J P,Q is defined as the restriction of J n,m , it is in fact enough to verify these identities in the case where ρ, ν and μ are identity elements in such algebras. This case can be taken care of by straightforward diagrammatic arguments. For the convenience of the reader, we indicate the proofs of Eqs. (5.1) and (5.4), starting with the latter. In the case under consideration, Eq. (5.4) is just F(σ TL π ⊗n ,π ⊗m ) • J n,m = J m,n • σ F(π ⊗n ),F(π ⊗m ) . The verification of this identity amounts to proving that In the case when n = m = 3, the left hand side arises from the finite braid diagram while the right hand side arises from the diagram As one of these diagrams can in general be obtained from the other by a finite sequence of Reidemeister moves of type 2, we get Eq. (5.5). Similarly, Eq. (5.1) reduces to the identity In the case when n = 3 and m = k = 2, the left hand side arises from the diagram while the right hand side arises from As in the proof of the pentagon identity, Eq. (5.6) is verified in general by noting that the strands live on three separate layers (corresponding to the three colors used in the figures).

TLJ (δ) and
Mod f B are equivalent. Finally, we will prove that F is fully faithful (i.e., restricts to a bijection on each morphism space) and that if we restrict its codomain to the subcategory Mod f B then it is essentially surjective (i.e., hits every isomorphism class of objects).

Lemma 5.1. The functor F is fully faithful.
Proof. It suffices to prove that F restricts to a bijective map Hom(P, Q) → Hom(F(P), F(Q)) for any given pair of projections P ∈ TLJ n (δ) and Q ∈ TLJ m (δ).
We first prove injectivity. Let a ∈ Hom(P, Q) be such that F(a) = 0. Since F is linear on morphisms, we need only show that a = 0. Since F(a) is left-multiplication by L m,n (a), we get that 0 = L m,n (a) • L n (P) = L m,n (a P) = L m,n (a), whereby a = L m,n (a) = 0 by Lemma 3.1.
We next prove surjectivity. Let f ∈ Hom(F(P), F(Q)) be given. Then f performs left-multiplication by b = f (L n (P)) ∈ L m (Q)BL n (P). Set K = max{m, n} so that L n (P), L m (Q) ∈ B K . Since B K is finite-dimensional and L m (Q)B k L n (P) ⊆ L m (Q)B K L n (P) for all k ≥ 0, it follows that b ∈ L m (Q)B K L n (P). Thus, b = L m,n (a) for some morphism a ∈ Hom(P, Q). Writing such a projection p as a finite sum of minimal projections, we get that pB is isomorphic to a direct sum of modules of the form qB, where q is a minimal projection in B. Thus, we may assume that M = qB, where q is a rank one projection in, say, the summand K(H s ). Pick n ≥ 0 such that s ≺ π ⊗n , and let v be a unit vector in Hom(s, π ⊗n ). Then L n (vv * ) is a rank one projection in the summand K(H s ), and therefore Murray-von Neumann equivalent to q in B. It follows that qB ∼ = L n (vv * )B = F(vv * ), which yields the stated result.
In conclusion, we have the following theorem. We end this section with a couple of remarks.
Remark 5.4. Although we have not defined the conjugate of an arbitrary object in Mod f B , we have shown that every such object is isomorphic to F(P) for some object P in TLJ (δ). Since F is a monoidal *-functor, F(P) has a conjugate (namely F(P) = F(P)). Thus, every object in Mod  ( p 1 , . . . , p n ) is any diagonal projection in M n (B) for which X ∼ = n j=1 p j B. Since B-linear maps n j=1 p j B → m i=1 q i B may be identified with m × n-matrices whose (i, j)'th entry belongs to q i B p j in such a way that composition corresponds to matrix multiplication and adjoints correspond to matrix adjoints, we get that φ is well-defined. (In the module picture of K 0 (B), [p] corresponds to [X ⊗B], whereB is the unitalization of B.) The map φ is injective because B is an AF-algebra and hence admits cancellation. Since φ([q s B]) = [q s ] for all s ∈ S, where q s is any minimal projection in the summand K(H s ) of B, and the classes [q s ] generate K 0 (B), it follows that φ is surjective. (q 1 , . . . , q m )]) is the class of the diagonal mn × mnmatrix whose (i, j)'th diagonal entry is ( p i ⊗ q j ). Thus, we may now conclude that φ is an isomorphism of rings. In this sense, the above equivalence of categories TLJ (δ) ∼ = Mod Remark 5.6. Theorem 5.3 can in fact be proved in greater generality, as we next indicate. Let C be a finitely generated rigid (see Definition 2.2.1 of [40]) braided C*-tensor category.
The assumption that C is rigid implies that C is semisimple (see Sect. 2.3.2) and that each End C (ρ) is a finite-dimensional C*-algebra equipped with a canonical positive faithful trace (cf. [38]; see also [40]). The assumption that C is finitely generated means that there exists a finite set L of objects such that every simple object in C occurs as a direct summand of a tensor product of objects in L.
By a version of the Mac Lane Coherence Theorem (that can e.g. be deduced from the proof of Theorem XI.5.3 in [35]), we may assume that C is strict. Denote by π a direct sum of the objects in L. By Theorem 2.17 in [7], for example, the category C is equivalent, as a C*-tensor category, to the category D whose objects are formal finite sums P 1 ⊕ · · · ⊕ P k of projections P j ∈ End C (π ⊗n j ) and whose morphisms ⊕ j P j → ⊕ i Q i are matrices whose (i, j)'th entry belongs to Q i Hom C (π ⊗n j , π ⊗m i )P j (when Q i ∈ End C (π ⊗m i )). One can use the unitary braiding σ on C to define a unitary braiding σ on D byσ P,Q = σ π ⊗n ,π ⊗m • (P ⊗ Q) (when P ∈ End(π ⊗n ) and Q ∈ End(π ⊗m )). Then C and D are equivalent as braided C*-tensor categories.
Next, put G = {1, π} and choose S as in Sect. 2.3.2. Then, as in Sect. 3.1, we construct a Hilbert space H = ⊕ s∈S H s [where H s = ⊕ x∈G ∞ Hom C (s, o( x))], operators L x, y (a), and a C*-algebra B that is *-isomorphic to ⊕ s∈S K(H s ). We can also define a *-homomorphism : B⊗B → B and equip Mod B with associators, unit constraints and a unitary braiding as in Sects. 3.3 and 4 by using the well-known graphical calculus for braided tensor categories (cf. e.g. [51]). Hence, Mod B obtains the structure of a braided C*-tensor category. Finally, we can define a braided monoidal *-functor F : D → Mod f B as in Sect. 5.1 and show, as in Sect. 5.2, that F is an equivalence of categories. Thus, the initial category C is equivalent to Mod f B as a braided C*-tensor category. Let us finally mention some examples of categories to which this generalization of Theorem 5.3 applies. Firstly, C could be the representation category of a compact group. Secondly, and more interestingly for us, C could be a further example of the Verlinde fusion category in conformal field theory e.g. arising from the finite-level, positiveenergy representation theory of the loop group of a compact, simple, connected, simplyconnected Lie group (cf. [46,52]). (The Temperley-Lieb-Jones category is the Verlinde fusion category arising from SU(2).) These latter categories can also be constructed from certain quantum groups at roots of unity (cf. [54]; see also section 6A of [11]). Thirdly, there are examples arising from the quantum double construction applied to not necessarily braided categories, which yields braided C*-tensor categories. The most prominent of these is the quantum double of the Haagerup subfactor, which has attracted much attention recently due to evidence that this system should arise from a conformal field theory (cf. [12]).

Concluding Remarks and Outlook
In the present paper, we have shown how to realize certain braided C*-tensor categories as categories of (right) Hilbert C*-modules with a natural tensor product structure (see Theorem 5.3 and Remark 5.6) or, phrased differently, how certain braided C*-tensor categories act faithfully on certain C*-algebras via Hilbert C*-modules. In light of this, it is natural to ask on which C*-algebras a given C*-tensor category (possibly without a unitary braiding) can act (faithfully) in this sense. In this context, it may be noted that, starting from TLJ (δ), for example, one can define a variant of the Hilbert C*-bimodule X of Hartglass and Penneys (cf. [25]) and use Pimsner's construction from [42] to construct from it a Toeplitz type C*-algebra T that is KKequivalent (by Theorem 4.4 of [42]) to the C*-algebra B that appeared in the present paper. Perhaps this allows one to realize TLJ (δ) as a C*-tensor category of Hilbert T -modules.
It is a long standing open problem to rigorously construct a conformal field theory (CFT) from a continuum scaling limit of a statistical mechanical model at criticality-or to construct a CFT from a modular tensor category (cf. e.g. [4,8,12,18,29,33,41,45]). One aspect of this is to derive the category of representations of the Virasoro algebra from representations of Temperley-Lieb algebras TL 0 N ,N (δ) in the N → ∞ limit in a mathematically rigorous way. The representation theory of the Virasoro algebra at central charge c = 1 − 6/(k + 2)(k + 3), where k = 0, 1, 2, . . ., can be realized from the diagonal embedding su(2) k+1 ⊂ su(2) k ⊕ su(2) 1 via a coset construction (cf. [21]). Here, through the Sugawara construction, the affine Lie algebra su(2) k has central charge c k = 3k/(k + 2). It is then intriguing to ask whether there is a parallel coset construction starting from an embedding B (k) ⊗ B (1) ⊂ B (k+1) , where B (k) is the algebra constructed as above from the Temperley-Lieb category with parameter δ = 2 cos(π/(k + 2)), that yields the representation category of the Virasoro algebra at central charge c = 1 − 6/(k + 2)(k + 3). regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.