Realizing the braided Temperley-Lieb-Jones C*-tensor categories as Hilbert C*-modules

We associate to each Temperley-Lieb-Jones C*-tensor category $\mathcal{T}\!\mathcal{L}\mathcal{J}(\delta)$ with parameter $\delta$ in the discrete range $\{2\cos(\pi/(k+2))\,:\,k=1,2,\ldots\}\cup\{2\}$ a certain C*-algebra $\mathcal{B}$ of compact operators. We use the unitary braiding on $\mathcal{T}\!\mathcal{L}\mathcal{J}(\delta)$ to equip the category $\mathrm{Mod}_{\mathcal{B}}$ of (right) Hilbert $\mathcal{B}$-modules with the structure of a braided C*-tensor category. We show that $\mathcal{T}\!\mathcal{L}\mathcal{J}(\delta)$ is equivalent, as a braided C*-tensor category, to the full subcategory $\mathrm{Mod}_{\mathcal{B}}^f$ of $\mathrm{Mod}_{\mathcal{B}}$ 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.


Background
Temperley-Lieb algebras first appeared in the work of Temperley and Lieb [49] on Potts and ice-type models in statistical mechanics, in which they were defined in terms of generators and relations. These relations reappeared in the work of Jones [29], in which (quotients of) Temperley-Lieb algebras manifested as subalgebras of higher relative commutants of (von Neumann) subfactors (see also [22]). A description of the Temperley-Lieb algebras in terms of what are now known as Temperley-Lieb diagrams first appeared in the work of Kauffman [35] (see also [33]), who was studying a knot invariant introduced by Jones [30]. Later, it was realized that a diagrammatic description could be given for tensor categories (cf. e.g. [50]) and standard invariants of subfactors (cf. Jones' introduction of subfactor planar algebras [31] based on the work of Popa [42]). In particular, diagrammatic Temperley-Lieb-Jones C*-tensor categories were considered (cf. e.g. [55], [9], [13]), which can be viewed as arising from the Temperley-Lieb-Jones factor planar algebras (cf. [31]; see also [38], [7]). When the parameter δ is confined to t2 cospπ{pk`2qq : k " 1, 2, . . .u, the associated Temperley-Lieb-Jones C*-tensor categories TLJ pδq are known to describe (up to equivalence) categories that have appeared in various contexts, including • representations of affine Lie algebras and vertex operator algebras arising from SUp2q Wess-Zumino-Witten models at finite levels k " 1, 2, . . . in 2D conformal field theory (cf. e.g. [27] and the references therein); • representations of the loop group LSUp2q at finite levels k " 1, 2, . . . (cf. [45], [51]); • representations of quantum SUp2q at certain roots of unity (cf. [53]).
We refer the reader to [25] for an overview and further references. It should also be mentioned that TLJ pδq can be recovered as the C*-tensor category of M -bimodules arising from certain subfactors pN Ă M q (cf. [54]; see also Remark 8.2 in [43]). A special feature of the C*-tensor categories TLJ pδq with δ P t2 cospπ{pk`2qq : k " 1, 2, . . .u Y t2u is the presence of a unitary braiding (cf. e.g. [38]), which we will use extensively in the present paper.

Motivation
Ultimately, our goal of describing TLJ pδq in terms of Hilbert C*-modules is motivated by a connection with K-theory (cf. e.g. [6], [48], [26]), 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 LSUp2q in terms of twisted equivariant K-theory. Related to this, we observed in [1] that the K 0 -group of certain approximately finite-dimensional (AF) C*-algebras has a ring structure that is closely related to the fusion ring of TLJ pδq. For example, the K 0 -group of the inductive limit TLJ 8 pδq " lim n TLJ n pδq of Temperley-Lieb-Jones C*-algebras, whose Bratteli diagram is given in [29], is a localization of the fusion ring of TLJ pδq. 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. [36], [6]).

Related work
Given a (small) rigid C*-tensor category C, Yuan in [56] 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 pδq into A Mod A , where A is the unital AF-algebra whose Bratteli diagram arises from the fusion graph of f p0q ' f p1q (in the notation of section 2.4.3). For example, when δ " 2 cospπ{5q, 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 [56]). On the other hand, given a factor planar algebra P ‚ , Hartglass and Penneys in [24] defined a C*-algebra B along with a Hilbert C*-bimodule X over B, which they fed into a construction due to Pimsner (cf. [41]) 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 pBq is isomorphic to the fusion ring of TLJ pδq. 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 [46,47] 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 [23], [5].
Our contribution starts in section 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 (section 3.1). Next, we explain a way to associate operators in B and its strong closure to certain infinite diagrams (section 3.2). Using an idea of Erlijman and Wenzl (cf. [11]), we then harness the unitary braiding on TLJ pδq to define a *-homomorphism Φ : B b B Ñ B by superposition of diagrams (section 3.3) and observe that the product on K 0 pBq induced by Φ recaptures the product in the fusion ring of TLJ pδq (Remark 3.4).
In section 4, we first use Φ as well as interior and exterior tensor products of Hilbert C*-modules to define a tensor product of Hilbert B-modules (section 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 (section 4.2) and supply it with a unitary braiding (section 4.3).
In section 5, we first define a *-functor F from TLJ pδq into Mod B and show that it is monoidal and braided (section 5.1). In section 5.2, we then use F to prove Theorem 5.3, which states that TLJ pδq 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 section 4.4). Thereafter, we note that the tensor category Mod B "categorifies" the ring K 0 pBq (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 section 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.

Operators on Hilbert space
In this paper, we consider operators on a complex Hilbert space H. We denote by BpHq 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 Ñ a strongly if and only if }a n pξq´apξq} Ñ 0 for all ξ P H. We will need the following standard fact.
Fact 2.1. Let pa n q 8 n"0 be a bounded sequence in BpHq such that a n am " ana m " 0 whenever n ‰ m. Then ř ně0 a n and ř ně0 an converge strongly in BpHq.
The normed space BpHq is a C*-algebra. It contains the C*-subalgebra KpHq of compact operators, which is the smallest C*-subalgebra of BpHq that contains all operators of finite rank. The following standard fact will be useful to us. Fact 2.2. Let pa n q 8 n"0 be a sequence in BpHq that converges strongly to some operator a. For any compact operator x P KpHq, we have that }a n x´ax} Ñ 0.

Hilbert C*-modules
A (right) Hilbert C*-module over a C*-algebra B is a (right) B-module M equipped with a Bvalued inner product x¨,¨y : MˆM Ñ B such that ξ Þ Ñ }xξ, ξy} 1{2 is a complete norm. The general theory of such modules is laid out very carefully in [36], 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 b 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 b N is a Hilbert pA b Bq-module) and the symbol b 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 pξ j q jPJ of elements in M such that (i) xξ i , ξ j y " 0 whenever i ‰ j; (ii) xξ j , ξ j y is a projection in B for all j P J; (iii) the Fourier expansion η " ř jPJ ξ j xξ j , ηy is valid for all η P M .

Definition of a C*-tensor category
A category C is called a C*-tensor category if the following conditions are satisfied (where π, ρ and ν denote arbitrary objects in C): (1) Each morphism set Hompπ, ρq is a complex Banach space. Moreover, composition is bilinear and }f g} ď }f }}g} for any pair pf, gq of composable morphisms.
(2) There is an antilinear contravariant functor * : C Ñ C such that π˚" π for all objects π, f˚˚" f for all morphisms f , and the C*-identity }f˚f } " }f } 2 holds for all morphisms f . In particular, each endomorphism space Endpπq :" Hompπ, πq is a unital C*-algebra.
(3) For any f P Hompπ, ρq, the morphism f˚f is a positive element of Endpπq.
(6) pf b gq˚" f˚b g˚for all morphisms f and g.
(8) The tensor unit is simple. [An object π in C is said to be simple if Endpπq " 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 P 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 -À sPS 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 ą 0, there exist N s morphisms v s,1 , . . . , v s,N s P Homps, ρq such that vs ,j v s,j " id s for all j and id ρ " ř sPS ř N s j"1 v s,j vs ,j . In fact, v s,1 , . . . , v s,N s form an orthonormal basis for Homps, ρq equipped with the inner product x¨,¨y given by xξ, ηy id s " ξ˚η for ξ, η P Homps, ρq. The number N s is called the multiplicity of s in ρ and is sometimes denoted by ps, ρq. We write s ă ρ if ps, ρq ą 0. Since we mention it in a few places, we also recall that the fusion ring ZrSs of C is the free abelian group generated by S and equipped with the product s¨t " ř rPS pr, s b tqr.

Unitary braidings
A unitary braiding σ on a C*-tensor category C is an assignment of a unitary isomorphism σ π,ρ : π b ρ Ñ ρ b π to every pair pπ, ρq of objects in C, natural in π and ρ, satisfying the hexagon identities (see [34] or equations (4.6) and (4.7) below). As in [11], we call a C*-tensor category with a choice of unitary braiding a braided C*-tensor category.

Monoidal functors
A functor F : C Ñ D between C*-tensor categories C and D is called a *-functor if F is linear and satisfies F pf˚q " F pf q˚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 pπq b F pρq Ñ F pπ b ρq that are compatible with the associators and unit constraints (see Definition 2.1.3 of [39] or equations (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 equation (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 pm, nq-Temperley-Lieb diagram (for m, n ě 0 of equal parity), which first appeared in [35]. Such a diagram consists of pm`nq{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.
(Some examples are shown in the next figure.) Given δ P C, denote by TL 0 m,n pδq the formal complex linear span of all isotopy classes of pm, nq-Temperley-Lieb diagrams and define a product TL 0 m,n pδqˆTL 0 n,k pδq Ñ TL 0 m,k pδq as follows. In order to multiply an pm, nq-Temperley-Lieb diagram by an pn, kq-Temperley-Lieb diagram, start by juxtaposing them, matching up the nodes to form a new diagram. Next, remove each closed loop at the cost of multiplying by the scalar δ. The following figure gives an example of the product of a p2, 4q-Temperley-Lieb diagram and a p4, 0q-Temperley-Lieb diagram.
" δ In particular, TL 0 n,n pδq 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 pδq as follows. If D is an pn, nq-Temperley-Lieb diagram then Tr TL n pDq 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 pδq Ñ TL 0 n,m pδq by reflecting diagrams about a vertical axis.
Jones famously proved (cf. [29]) that the linear trace Tr TL n is positive for all n if and only if δ P t2 cospπ{pk`2qq : k " 1, 2, . . .uY r2, 8q. Given δ in this range, put TLJ m,n pδq " TL 0 m,n pδq{tx P TL 0 m,n pδq : Tr n px˚xq " 0u. Then the product above descends to a product TLJ m,n pδqˆTLJ n,k pδq Ñ TLJ m,k pδq, the above *-operation descends to a *-operation TLJ m,n pδq Ñ TLJ n,m pδq, and the trace Tr TL n descends to a positive faithful trace on TLJ n,n pδq. Thus, TLJ n pδq :" TLJ n,n pδq is a finitedimensional C*-algebra, which is known as the n'th Temperley-Lieb-Jones C*-algebra.

Temperley-Lieb-Jones C*-tensor categories
Let δ P t2 cospπ{pk`2qq : k " 1, 2, . . .u Y r2, 8q be given. The Temperley-Lieb-Jones (or reduced Temperley-Lieb) C*-tensor category TLJ pδq 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 pδq for each j. Given projections P P TLJ n pδq and Q P TLJ m pδq, the morphism set HompP, Qq is QTLJ m,n pδqP . More generally, given objects ' k j"1 P j and ' r i"1 Q i , the morphism set Homp' k j"1 P j , ' r i"1 Q i q consists of all rˆk-matrices whose pi, jq'th entry is in HompP j , Q i q. Composition of morphisms is given by multiplication of Temperley-Lieb diagrams combined with matrix multiplication. The tensor product in TLJ pδq is defined as follows. Given projections P P TLJ n pδq and Q P TLJ m pδq, the tensor product P bQ 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 pδq. The tensor product of two objects ' k i"1 P i and ' r j"1 Q j is simply ' pi,jq pP i b Q j q. The tensor product of morphisms is given by vertical stacking combined with tensor multiplication of matrices, i.e., pa ij q i,j b pb kl q k,l " pa ij b b kl q pi,kq,pj,lq . One can show that TLJ pδq is a strict C*-tensor category, whose tensor unit is the empty Temperley-Lieb diagram.

Jones-Wenzl projections
For any δ P t2 cospπ{pk`2qq : k " 1, 2, . . .u Y r2, 8q, the C*-tensor category TLJ pδq is semisimple. Up to unitary isomorphism, the simple objects are the so-called Jones-Wenzl projections (cf. [52]). If δ ě 2 then the Jones-Wenzl projections form an infinite sequence pf pnq q 8 n"0 with f pnq P TLJ n pδq for all n, where f p0q is the empty diagram and f p1q is a single strand. The remaining Jones-Wenzl projections are defined via Wenzl's recursive formula (see e.g. equation (2.1) in [38], in which δ is equal to q`q´1 in their notation). It is a fact that f p1q b f pnqf pn´1q ' f pn`1q in TLJ pδq for all n ě 1. If δ " 2 cospπ{pk`2qq with k ě 1 then the Jones-Wenzl projections form a finite sequence f p0q , f p1q , . . . , f pkq , defined recursively as above. In this case, In either case, the category TLJ pδq is generated by the object π " f p1q in the sense that every simple object occurs as a direct summand of some tensor power π bn of π.

The unitary braiding
If δ P t2 cospπ{pk`2qq : k " 1, 2, . . .u Y t2u then TLJ pδq 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.
σ TL π,π "`σ TL π,π˘´1 " Using it, one can define a unitary element σ TL π bn ,π bm of Endpπ bpn`mq q " TLJ n`m pδq by a braid diagram like the one below (which corresponds to the case n " 2 and m " 3).
Given projections P P Endpπ bn q " TLJ n pδq and Q P Endpπ bm q " TLJ m pδq, one defines a unitary isomorphism σ TL P,Q in HompP bQ, QbP q by σ TL P,Q " σ TL π bn ,π bm˝p P bQq. Finally, the unitary braiding σ TL is given by the unitary isomorphisms σ TL 3 On a C*-algebra B and a *-homomorphism B b B Ñ B In this section, we define a Hilbert space H, a C*-algebra B Ă BpHq and a *-homomorphism Φ : B b B Ñ B, drawing inspiration from [5], [24], [56] and [11]. Our starting point is the braided C*-tensor category C " TLJ pδq with δ P t2 cospπ{pk`2qq : k " 1, 2, . . .u Y t2u, its tensor unit ½, the generating object π, and a set S of simple objects in C chosen as in section 2.3.2. Put G " t½, πu and denote by G 8 the set of infinite sequences µ " pµ 1 , µ 2 , . . .q of elements in G for which there exists n " n µ ě 0 such that µ k " ½ for k ą n. Given such a sequence µ, we put op µq " µ 1 b µ 2 b¨¨¨. As C is a strict C*-tensor category, this infinite tensor product makes sense.

Definition of B
For each s P S and µ P G 8 , the morphism space Homps, op µqq is equipped with the inner product x¨,¨y given by xξ, ηy id s " ξ˚η. We denote by H s the orthogonal direct sum of the Hilbert spaces Homps, op µqq as µ varies through G 8 . In symbols, Homps, op µqq.
Given x, y P G 8 and a P Hompop yq, op xqq, define a linear operator L x, y paq : H Ñ H by the formula L x, y paqξ " δ y, µ pa˝ξq P Homps, op xqq for ξ P Homps, op µqq. It is a bounded operator whose adjoint operator is In particular, Lemma 3.1. We have that }L x, y paq} " }a} for all a P Hompop yq, op xqq.
For each n ě 0, denote by B n the finite-dimensional C*-algebra spanned by the operators of the form L x, y paq, where x k " y k " ½ for all k ą n. Each B n admits a positive faithful trace Tr n defined by Tr n pL x, y paqq " δ x, y Tr TL k paq, where k is the number of entries in x that equal π. Moreover, B n Ď B n`1 for all n. Denote by B the smallest C*-subalgebra of BpHq that contains every B n , i.e., The following result describes the structure of B.
Proof. Note first that B Ă KpHq. Indeed, each operator L x, y paq is compact because it can be written as L x, y paqP , where P is the orthogonal projection onto the finite-dimensional subspace À săop yq Homps, op yqq. Conversely, if ξ is a unit vector in Homps, op yqq and η is a unit vector in Homps, op xqq then B contains the rank one operator L x, y pηξ˚q P KpH s q, which maps ξ onto η. Thus, for each s P S, B contains a complete set of matrix units for KpH s q. The result follows.
The next lemma will be used to define certain morphisms between tensor products of B-modules. Proof. Note that v˚" ř ně0 vn, where the sum converges in the strong operator topology. Let b P B be given. By Fact 2.2, ř ně0 v n b converges to vb in norm because b P KpHq. Similarly, ř ně0 vnbc onverges to v˚b˚in norm. Since B is a C*-subalgebra of BpHq, 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 diagram in the figure gives rise to the operator L x, y paq, where x " pπ, ½, ½, π, ½, . . .q, y " pπ, π, π, ½, π, . . .q, and a is the morphism given by the pictured Temperley-Lieb diagram. By definition, the C*-algebra B is generated by operators arising from dilute Temperley-Lieb diagrams.
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 nonzero.
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 Endpπ bn q 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 ℓpDq the pattern of empty and non-empty nodes on its left side. Denote by supppDq the set of patterns that can be obtained from ℓpDq by replacing all but finitely many non-empty nodes by empty ones. Given x P supppDq, 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 vpD, xq. Since vpD, xq˚vpD, yq " vpD, xqvpD, yq˚" 0 whenever x ‰ y, Fact 2.1 implies that ř xPsupppDq vpD, xq is strongly convergent in BpHq. We put vpDq " ÿ xPsupppDq vpD, xq.
Although vpDq need not belong to B, Fact 2.1 and Lemma 3.3 imply that vpDq¨b P B, b¨vpDq P B for all b P B. If D has no empty nodes then vpDq is a unitary operator in BpHq. This follows from the fact that multiplication in BpHq is jointly strongly continuous on bounded sets. In general, vpDq is a partial isometry in BpHq whose range projection is ř xPsupppDq p x . (Recall that p x was defined on page 8.)

Definition of Φ : B b B Ñ B
Define, for each n ě 0, a unitary element U n P B 2n in terms of the unitary braiding σ TL on C in the same way as on page 374 in [11] (when s there is 2), except that we sum over all patterns x P G 2n . As an example, the following figure shows two of the terms in the definition of U 3 .

+ +¨¨Ẅ
e can think of U 3 as vpDq, 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 Φ n : where x v " px 1 , . . . , x n , v 1 , . . . , v n , . . .q (and similarly for y w). The faithfulness of the traces Tr n and the fact that Tr 2n˝Φn " Tr n b Tr n on elements of the form L x, y paq b L v, w pbq imply that Φ n is a well-defined isometric *-homomorphism. The purpose of the unitaries U n is to ensure that Φ n`1˝p ι n b ι n q " ι 2n`1˝ι2n˝Φn for all n ě 0, where ι n is the inclusion map B n Ñ B n`1 . This allows us to extend the *-homomorphisms Φ n to an isometric *-homomorphism Diagrammatically, the effect of applying Φ to a tensor product L x, y paqbL v, w pbq 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 pBq is isomorphic to the fusion ring ZrSs as an abelian group. It is also easy to check that the induced product map on K 0 pBq agrees with the product on the fusion ring. (This boils down to the fact that L µ, µ pvv˚q is a rank one projection in KpH s q for any µ P G 8 and any unit vector v P Homps, op µqq.) Below, we will "categorify" this statement, by using Φ to define a tensor product of right Hilbert B-modules that recaptures the tensor product in C (see also Remark 5.5).

B
In this section, we use the *-homomorphism Φ 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 Φ : B b B Ñ B is the *-homomorphism from the previous section. (See section 2.2 for an explanation of the notation.) Given adjointable maps f 1 : M 1 Ñ N 1 and f 2 : M 2 Ñ N 2 between right Hilbert B-modules, we denote by f 1 b f 2 the adjointable map M 1 b M 2 Ñ N 1 b N 2 given by for ξ 1 P M 1 , ξ 2 P M 2 and b P B. As a simple example, let p and q be projections in B. Then pB and qB are right Hilbert Bmodules (with inner product given by pa, bq Þ Ñ a˚b) and there exists a surjective B-linear isometry We next relate the above tensor product to the standard direct sum of Hilbert B-modules. Given finite families pM i q iPI and pN j q jPJ of right Hilbert B-modules, we have a surjective B-linear isometry φ : p' i M i q b p' j N j q ÝÑ ' pi,jq pM i b N j q defined by pξ i q i b pη j q j b b Þ Ñ pξ i b η j b bq pi,jq for ξ i P M i , η j P N j and b P B.

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 HompM, N q consist of all adjointable (or, equivalently, all bounded B-linear, cf. [14]) maps M Ñ 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 section 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 P BpHq 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 9 depicts D α x (" pD α q x ), where x " pπ, π, π, π, ½, π, ½, π, . . .q.) 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 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 P B. Here, ξ 1 b ξ 2 a b b b ξ 3 cd b e on the left hand side is viewed as an element of dqe 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 " e˚Φprb˚Φpxξ 1 , η 1 y b xξ 2 a, η 2 a 1 yqb 1 s b d˚xξ 3 c, η 3 c 1 yd 1 qe 1 " e˚Φpb˚b d˚qΦ pΦpxξ 1 , η 1 y b xξ 2 a, η 2 a 1 yq b xξ 3 c, η 3 c 1 yq Φpb 1 b d 1 qe 1 .
Since these two expressions coincide by equation (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 in the quadruple tensor product`pM 1 b M 2 q b M 3˘b M 4 . Let us first consider the left hand side. First, α M 1 ,M 2 ,M 3 b id M 4 maps the given element to Next, α M 1 ,M 2 bM 3 ,M 4 maps the above element to We now consider the right hand side. First, α M 1 bM 2 ,M 3 ,M 4 maps the element in equation (4.3) to Next, α M 1 ,M 2 ,M 3 bM 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 " ř x vpD α , xq for a certain infinite braid diagram D α (see page 10), it suffices to prove that vpD α , xqΦpvpD α , yq b p z q " Φpp µ b vpD α , νq˚qvpD α , βqvpD α , γq whenever x, y, z, µ, ν, β, γ P G 8 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 ą 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 ℓ and W r in BpHq. Namely, W r is the operator associated to the infinite dilute braid diagram which we will call D r , while W ℓ is the operator associated to the diagram which we call D ℓ . We have that

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 ξb , we see that the verification reduces to proving the identity Similarly to the case of the pentagon identity, it suffices to prove that Φpp x b vpD ℓ , yqqvpD α , zq " ΦpvpD r , βq b p γ q whenever x, y, z, β, γ P G 8 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 Endpp˚Bq -p˚Bp˚" p˚B 0 p˚" Cp( 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 BpHq 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 x (" pD σ q x ).

Note that
for all b 1 , b 2 P B. Equation (4.5) allows us, given two objects M 1 and M 2 in Mod B , to define a unitary isomorphism for ξ 1 P M 1 , ξ 2 P M 2 and a P B. The assignment pM 1 , M 2 q Þ Ñ 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 P B. As in the case of the pentagon identity, this reduces to showing that vpD α , xqvpD σ , yqvpD α , zq " Φpp µ b vpD σ , νqqvpD α , βqΦpvpD σ , γq b p ǫ q whenever x, y, z, µ, ν, β, γ, ǫ P G 8 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 f B
Denote by Mod f B the full subcategory of Mod B whose objects are those right Hilbert B-modules which admit a finite orthonormal basis in the sense of section 2.2. (Note that, by [2], every module in Mod B admits a possibly infinite orthonormal basis.) Clearly, Mod f B contains the tensor unit in Mod B . In order to check that Mod f B is a C*-tensor subcategory of Mod B , we must show that Mod f B is closed under tensor products. To do this, let M and N be objects in Mod f B . Choose finite orthonormal bases pξ i q i and pη j q j for M and N , respectively. Using the identities ξ i xξ i , ξ i y " ξ i and η j xη j , η j y " η j , one verifies that the elements ξ i b η j b Φpxξ i , ξ i y b xη j , η j yq form a finite orthonormal basis for M b N , showing that M b N is an object in Mod f B . Note that this could also be deduced from the following easily proved fact.  Let tξ 1 , . . . , ξ k u be a finite orthonormal basis for a right Hilbert B-module M . Then M is isomorphic to À k j"1 p j B, where p j " xξ j , ξ j y for all j.

Realizing TLJ pδq as right Hilbert B-modules
In this section, we show that TLJ pδq is equivalent to Mod f B as a braided C*-tensor category.
5.1 A braided monoidal *-functor F : TLJ pδq Ñ Mod B We will now define a functor F : C Ñ Mod B (where C " TLJ pδq, as above). The following notation will be convenient. Setting x n " pπ, . . . , π, ½, ½, . . .q, with n leading copies of π, we denote L xn, xm paq by L n,m paq for any a P Hompπ bm , π bn q " TLJ n,m pδq. We also put L n paq " L n,n paq for any a P Endpπ bn q " TLJ n pδq. Finally, we denote by p n the projection p xn (as defined on page 8).
We define F on objects as follows. Given a projection P P TLJ n pδq, we define F pP q by the formula F pP q " L n pP qB, on the right hand side of which we view P as a morphism. Given an object ' j P j in C, we put 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 P HompP, Qq, where P P TLJ n pδq and Q P TLJ m pδq are projections, we define F paq to be the adjointable map L n pP qB Ñ L m pQqB given by left-multiplication by L m,n paq. For any morphism pa ij q i,j P Homp' j P j , ' i Q i q, we define F ppa ij q i,j q to be the adjointable map ' j F pP j q Ñ ' i F pQ i q associated to the matrix pF pa ij qq 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 pρqbF pνq Ñ F pρ b νq in such a way that the assignment pρ, νq Þ Ñ 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: F pσ TL ρ,ν q˝J ρ,ν " J ν,ρ˝σF pρq,F pνq . The fact that J n,m is a well-defined unitary isomorphism comes down to the easily verified identities Un ,m U n,m " Φpp n b p m q and U n,m Un ,m " p n`m . Given projections P P TLJ n pδq and Q P TLJ m pδq, 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 J ' i P i ,' j Q j as the composition`' pi,jq J P i ,Q j˘˝φ , where φ is the unitary isomorphism`' i F pP i q˘b`' j F pQ j q˘Ñ ' pi,jq pF pP i q b F pQ j qq from section 4.1. Note that the domain of It is easy to reduce the naturality of J as well as the identities in equations (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 equations (5.1) and (5.4), starting with the latter. In the case under consideration, equation (5.4) is just F pσ TL π bn ,π bm q˝J n,m " J m,n˝σF pπ bn q,F pπ bm q .
The verification of this identity amounts to proving that L n`m pσ TL π bn ,π bm q˝U n,m˝Φ pp n b p m q " U m,n˝U˝Φ pp n b p m q.
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 equation (5.5). Similarly, equation (5.1) reduces to the identity U n,m`k˝Φ pp n b U m,k q˝V " U n`m,k˝Φ pU n,m b p k q.
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, equation (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 pδq 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 HompP, Qq Ñ HompF pP q, F pQqq for any given pair of projections P P TLJ n pδq and Q P TLJ m pδq.
We first prove injectivity. Let a P HompP, Qq be such that F paq " 0. Since F is linear on morphisms, we need only show that a " 0. Since F paq is left-multiplication by L m,n paq, we get that 0 " L m,n paq˝L n pP q " L m,n paP q " L m,n paq, whereby }a} " }L m,n paq} " 0 by Lemma 3.1.
We next prove surjectivity. Let f P HompF pP q, F pQqq be given. Then f performs left-multiplication by b " f pL n pP qq P L m pQqBL n pP q. Set K " maxtm, nu so that L n pP q, L m pQq P B K . Since B K is finite-dimensional and L m pQqB k L n pP q Ď L m pQqB K L n pP q for all k ě 0, it follows that b P L m pQqB K L n pP q. Thus, b " L m,n paq for some morphism a P HompP, Qq. 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 KpH s q. Pick n ě 0 such that s ă π bn , and let v be a unit vector in Homps, π bn q. Then L n pvv˚q is a rank one projection in the summand KpH s q, and therefore Murray-von Neumann equivalent to q in B. It follows that qB -L n pvv˚qB " F pvv˚q, which yields the stated result.
In conclusion, we have the following theorem. Theorem 5.3. If δ P t2 cospπ{pk`2qq : k " 1, 2, . . .u Y t2u then the Temperley-Lieb-Jones C*tensor category TLJ pδq and the category Mod f B are equivalent as braided C*-tensor categories. 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 pP q for some object P in TLJ pδq. Since F is a monoidal *-functor, F pP q has a conjugate (namely F pP q " F pP q). Thus, every object in Mod f B does have a conjugate and Mod f B is in fact a rigid braided C*-tensor category. (See [37] and e.g. section 2.2 of [39] for the concepts of conjugates and rigidity in C*-tensor categories.) Remark 5.5. Denote by R the fusion ring of Mod f B consisting of formal differences rXs´rY s of isomorphism classes of modules in Mod f B . Define a group homomorphism φ : R Ñ K 0 pBq by φprXsq " rps, where p " diagpp 1 , . . . , p n q is any diagonal projection in M n pBq for which X -À n j"1 p j B. Since B-linear maps À n j"1 p j B Ñ À m i"1 q i B may be identified with mˆn-matrices whose pi, jq'th entry belongs to q i Bp 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 pBq, rps corresponds to rX bBs, whereB is the unitalization of B.) The map φ is injective because B is an AF-algebra and hence admits cancellation. Since φprq s Bsq " rq s s for all s P S, where q s is any minimal projection in the summand KpH s q of B, and the classes rq s s generate K 0 pBq, it follows that φ is surjective. If X " À n i"1 p i B and Y " À m j"1 q j B then X b Y -À pi,jq Φpp i b q j qB, and K 0 pΦqprdiagpp 1 , . . . , p n qs b rdiagpq 1 , . . . , q m qsq is the class of the diagonal mnˆmn-matrix whose pi, jq'th diagonal entry is Φpp i b q j q. Thus, we may now conclude that φ is an isomorphism of rings. In this sense, the above equivalence of categories TLJ pδq -Mod f B "categorifies" the isomorphism ZrSs -K 0 pBq of rings that was exhibited in Remark 3.4.
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 [39]) braided C*-tensor category. The assumption that C is rigid implies that C is semisimple (see section 2.3.2) and that each End C pρq is a finitedimensional C*-algebra equipped with a canonical positive faithful trace (cf. [37]; see also [39]).
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 [34]), 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 P End C pπ bn j q and whose morphisms ' j P j Ñ ' i Q i are matrices whose pi, jq'th entry belongs to Q i Hom C pπ bn j , π bm i qP j (when Q i P End C pπ bm i q). One can use the unitary braiding σ on C to define a unitary braidingσ on D byσ P,Q " σ π bn ,π bm˝pP b Qq (when P P Endpπ bn q and Q P Endpπ bm q). Then C and D are equivalent as braided C*-tensor categories.
Next, put G " t½, πu and choose S as in section 2.3.2. Then, as in section 3.1, we construct a Hilbert space H " ' sPS H s [where H s " ' xPG 8 Hom C ps, op xqq], operators L x, y paq, and a C*-algebra B that is *-isomorphic to ' sPS KpH s q. We can also define a *-homomorphism Φ : B b B Ñ B and equip Mod B with associators, unit constraints and a unitary braiding as in sections 3.3 and 4 by using the well-known graphical calculus for braided tensor categories (cf. e.g. [50]). Hence, Mod B obtains the structure of a braided C*-tensor category. Finally, we can define a braided monoidal *-functor F : D Ñ Mod f B as in section 5.1 and show, as in section 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. In particular, this shows that a version of Theorem 5.3 is also valid for representation categories of compact groups as well as for C*-tensor categories arising from certain quantum groups at roots of unity (cf. [53]; see also section 6A of [11]).

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