# Bicommutant categories from fusion categories

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## Abstract

Bicommutant categories are higher categorical analogs of von Neumann algebras that were recently introduced by the first author. In this article, we prove that every unitary fusion category gives an example of a bicommutant category. This theorem categorifies the well-known result according to which a finite dimensional \(*\)-algebra that can be faithfully represented on a Hilbert space is in fact a von Neumann algebra.

## 1 Introduction

Bicommutant categories were introduced by the first author in the recent preprint [14], as a categorification of the notion of a von Neumann algebra.

*finite*direct sum of simple objects. In other words, every object is of the form \(\bigoplus _i c_i\otimes V_i\) for some finite dimensional vector spaces \(V_i\in \mathsf {Vec}\) and simple objects \(c_i\in \mathcal {C}\). In order to make \(\mathcal {C}\) into a bicommutant category, we need to allow the \(V_i\) to be arbitrary separable Hilbert spaces. The resulting category is denoted as \(\mathcal {C}\otimes _\mathsf{Vec}\mathsf{Hilb}\) (this is an instance of balanced tensor product of linear categories [33]). Our main result is as follows:

### Theorem A

If \(\mathcal {C}\) is a unitary fusion category, then \(\mathcal {C}\otimes _\mathsf{Vec}\mathsf{Hilb}\) is a bicommutant category.

By a result of Popa [27], every unitary fusion category \(\mathcal {C}\) can be embedded in \({{\mathrm{Bim}}}(R)\) (see Theorem 3.5). We prove that its bicommutant \(\mathcal {C}''\) is equivalent to \(\mathcal {C}\otimes _\mathsf{Vec}\mathsf{Hilb}\), and that the latter is a bicommutant category.

As a special case of the above theorem, if *G* is a finite group and \(\omega \) is a cocycle representing a class \([\omega ]\in H^3(G,U(1))\), then the tensor category \(\mathsf{Hilb}^\omega [G]\) of *G*-graded Hilbert spaces with associator twisted by \(\omega \) is a bicommutant category. That result was conjectured in [14, §6] as part of a bigger conjecture about categories of representations of twisted loop groups.

An algebra | A tensor category \(\mathcal {C}\) |

A finite dimensional algebra | A fusion category |

The center of an algebra | The Drinfeld center \(\mathcal {Z}(\mathcal {C})\) |

The commutant (or centralizer) \(Z_B(A)\) of | The commutant \(\mathcal {Z}_\mathcal {D}(\mathcal {C})\) of \(\mathcal {C}\) in \(\mathcal {D}\) |

The algebra | The category \({{\mathrm{Bim}}}(R)\) of all bimodules |

On a Hilbert space | On a hyperfinite factor |

The commutant \(A':=Z_{B(H)}(A)\) | The commutant \(\mathcal {C}':=\mathcal {Z}_{{{\mathrm{Bim}}}(R)}(\mathcal {C})\) |

A von Neumann algebra \(A=A''\) | A bicommutant category \(\mathcal {C}\cong \mathcal {C}''\) |

\(*\)-algebra | Bi-involutive tensor category \(\mathcal {C}\) |

## 2 Preliminaries

### 2.1 Involutions on tensor categories

A linear *dagger category* is a linear category \(\mathcal {C}\) over the complex numbers, equipped with an anti-linear map \(\mathcal {C}(x,y)\rightarrow \mathcal {C}(y,x):f\mapsto f^*\) for every \(x,y\in \mathcal {C}\) called the *adjoint* of a morphism. It satisfies \(f^{**}=f\) and \((f\circ g)^*=g^*\circ f^*\), from which it follows that \({{\mathrm{id}}}_x^*={{\mathrm{id}}}_x\). An invertible morphism of a dagger category is called *unitary* if \(f^*=f^{-1}\).

A functor \(F:\mathcal {C}\rightarrow \mathcal {D}\) between dagger categories is a *dagger functor* if \(F(f)^*=F(f^*)\).

### Definition 2.1

([31, §7]) A *dagger tensor category* is a linear dagger category \(\mathcal {C}\) equipped with a monoidal structure whose associators \(\alpha _{x,y,z}:(x\otimes y)\otimes z\rightarrow x\otimes (y\otimes z)\) and unitors \(\lambda _x:1\otimes x\rightarrow x\) and \(\rho _x:x\otimes 1\rightarrow x\) are unitary, and which satisfies the compatibility condition \((f\otimes g)^* =f^*\otimes g^*\).

The last condition can be rephrased as saying that the monoidal product \(\otimes :\mathcal {C}\otimes _{\mathsf {Vec}} \mathcal {C}\rightarrow \mathcal {C}\) is a dagger functor. From now on, we shall abuse notation, and omit all associators and unitors from our formulas. We trust the reader to insert them wherever needed.

### Definition 2.2

*dagger tensor functor*\(F: \mathcal {C}\rightarrow \mathcal {D}\) is a dagger functor equipped with a unitary natural transformation \(\mu _{x,y}:F(x)\otimes F(y)\rightarrow F(x\otimes y)\) and a unitary isomorphism \(i:1_\mathcal {D}\rightarrow F(1_\mathcal {C})\) such that the following identities hold for all \(x,y,z\in \mathcal {C}\):

We shall be interested in dagger tensor categories which are equipped with a second involution, this time at the level of objects (compare [15, Def. 1.3]):

### Definition 2.3

*bi-involutive tensor category*is a dagger tensor category \(\mathcal {C}\) with a covariant anti-linear dagger functor \(\overline{\,\cdot \,}:\mathcal {C}\rightarrow \mathcal {C}\) called the conjugate. This functor should be involutive, meaning that for every \(x\in \mathcal {C}\), we are given a unitary natural isomorphisms \(\varphi _x:x\rightarrow \overline{\overline{x}}\) satisfying \(\varphi _{\overline{x}}=\overline{\varphi _{x}}\). It should be anti-compatible with the tensor structure, meaning that we have unitary natural isomorphisms

### Remark 2.4

It is interesting to note that the map *j* can be recovered from the other data as \(j= \lambda _{\overline{1}}\circ (\varphi ^{-1}_1\otimes {{\mathrm{id}}}_{\overline{1}})\circ \nu _{\overline{1}1}^{-1}\circ \overline{\lambda _{\overline{1}}}^{-1}\circ \varphi _1\). We believe that the notion of bi-involutive category as presented above is equivalent to its variant without *j* (and without the axioms that involve *j*). Nevertheless, we find it more pleasant to include this piece of data in the definition.

Note that in the category of Hilbert spaces, the isomorphism \(\varphi _H:H\rightarrow \overline{\overline{H}}\) is an identity arrow. Whenever that is the case, we have \(\overline{j}=j^{-1}\) and \(\overline{\nu _{y,x}}=\nu _{\overline{x},\overline{y}}^{-1}\).

### Definition 2.5

Let \(\mathcal {C}\) and \(\mathcal {D}\) be bi-involutive tensor categories. A *bi-involutive tensor functor* is a dagger tensor functor \(F:\mathcal {C}\rightarrow \mathcal {D}\), equipped with a unitary natural transformation \(\upsilon _x:F(\overline{x})\rightarrow \overline{F(x)}\) satisfying the three conditions \(\upsilon _{\overline{x}}= \overline{\upsilon _{x}}^{-1}\circ \varphi _{F(x)}\circ F(\varphi _x)^{-1} \), \(\upsilon _{1_\mathcal {C}}= \overline{i}\circ j_\mathcal {D}\circ i^{-1}\circ F(j_\mathcal {C})^{-1}\), and \(\upsilon _{x\otimes y}= \overline{\mu _{x,y}}\circ \nu _{F(y),F(x)}\circ (\upsilon _y\otimes \upsilon _x)\circ \mu _{\overline{y},\overline{x}}^{-1}\circ F(\nu _{y,x})^{-1}\).

### 2.2 Unitary fusion categories

*rigid*if for every object \(x\in \mathcal {C}\), there exists an object \(x^\vee \in \mathcal {C}\), called the dual of

*x*, and maps \(\mathrm {ev}_x : x^\vee \otimes x \rightarrow 1\) and \(\mathrm {coev}_x : 1 \rightarrow x \otimes x^\vee \) satisfying the zigzag axioms

*semisimple*if it is equivalent to a direct sum of copies of \(\mathsf {Vec}\), possibly infinitely many. Equivalently, it is semisimple if it admits finite direct sums (including the zero sum), and every object is a direct sum of finitely many (possibly zero) simple objects.

### Definition 2.6

A fusion category is a tensor category which is rigid, semisimple, with simple unit, and finitely many isomorphism classes of simple objects.

^{1}and such that the norms

### Definition 2.7

A unitary fusion category is a dagger tensor category whose underlying dagger category is a C*-category, and whose underlying tensor category is a fusion category.

*canonical bi-involutive structure*. The conjugation \(\overline{\,\cdot \,}\) is characterized at the level of objects (up to unique unitary isomorphisms) by the data of structure morphisms \(\mathrm {ev}_x : \overline{x} \otimes x \rightarrow 1\) and \(\mathrm {coev}_x : 1 \rightarrow x \otimes \overline{x}\), subject to the two zigzag axioms (1) and the balancing condition

Note that a unitary fusion category is a fusion category with an additional structure. A fusion category could therefore, in principle, have more than one unitary structures. The question of uniqueness is best formulated in the following way (see [9, §5] for related work).

### Question 2.8

Let \(F:\mathcal {C}\mathop {\rightarrow }\limits ^{\simeq }\mathcal {D}\) be a tensor equivalence between two unitary fusion categories. Is any such *F* naturally equivalent to a dagger tensor functor?

*j*are inherited from those of \(\mathcal {C}\) and of \(\mathsf{Hilb}\).

### 2.3 The commutant of a category

*B*and a subalgebra \(A\subset B\), the commutant of

*A*inside

*B*, also called the centralizer, is the algebra

*A*and

*B*are replaced by tensor categories, dagger tensor categories, and finally bi-involutive tensor categories.

### Definition 2.9

### Remark 2.10

The Drinfeld center \(\mathcal {Z}(\mathcal {C})\) is the commutant of \(\mathcal {C}\) in itself.

*unitary commutant*of \(\mathcal {C}\) in \(\mathcal {D}\) (compare [21, Def. 6.1]). Unlike \(\mathcal {Z}_\mathcal {D}(\mathcal {C})\), the unitary commutant is a dagger category, and its \(*\)-operation is inherited from \(\mathcal {D}\).

### Remark 2.11

The inclusion \(\mathcal {Z}_\mathcal {D}^*(\mathcal {C})\hookrightarrow \mathcal {Z}_\mathcal {D}(\mathcal {C})\) is in general not an equivalence. The easiest counterexample is given by \(\mathcal {C}=\mathsf {Vec}[G]\) for *G* some infinite group, and \(\mathcal {D}=\mathsf {Vec}\). Then, \(\mathcal {Z}^*_\mathcal {D}(\mathcal {C})\) is the category of unitary representations of *G*, whereas \(\mathcal {Z}_\mathcal {D}(\mathcal {C})\) is the category of all representations of *G*. See [22, Thm. 6.4] and [9, Proposition 5.24] for some positive results when \(\mathcal {C}\) is a fusion category.

*j*, and \(\nu \) are inherited from \(\mathcal {D}\).

We will be especially interested in the case when \(\mathcal {D}={{\mathrm{Bim}}}(R)\), the tensor category of bimodules over some hyperfinite von Neumann factor *R*. The monoidal product on that category is based on the operation of Connes fusion, which we describe next.

### 2.4 \(L^2\)-spaces and Connes fusion

*R*be a von Neumann algebra, with predual \(R_*\) and positive part \(R_*^+\subset R_*\). The \(L^2\)-space of

*R*(also known as standard form of

*R*), denoted as \(L^2R\), is the Hilbert space generated by symbols \(\sqrt{\phi }\) for \(\phi \in R_*^+\), under the inner product

^{2}The Hilbert space \(L^2R\) is an

*R*–

*R*-bimodule, with the two actions of

*R*are determined by the formula

*H*and a left module

*K*, their fusion \(H\boxtimes _R K\) is the Hilbert space generated by symbols \(\alpha [\xi ]\beta \), for \(\alpha :L^2R\rightarrow H\) a right

*R*-linear map, \(\xi \in L^2R\), and \(\beta :L^2R\rightarrow K\) a left

*R*-linear map, under the inner product

*r*denote the left and right actions of

*R*on its \(L^2\) space, defined by \(\ell (a)(\xi )=a\xi \) and \(r(a)(\xi )=\xi a\), respectively.

*R*-linear map and \(\xi \in K\) a vector, and generated by symbols \(\xi ]\beta \) for \(\beta :L^2R\rightarrow K\) a left

*R*-linear map and \(\xi \in H\) a vector. The isomorphisms between the above models are given by

The two actions of *R* on \(L^2R\) are each other’s commutants. That property characterizes the bimodules which are invertible with respect to Connes fusion:

### Lemma 2.12

([30, Prop. 3.1]) Let *A* and *B* be von Neumann algebras, and let *H* be an *A*–*B*-bimodules such that *A* and *B* are each other’s commutants on *H* (in particular, they act faithfully on *H*). Then, *H* is an invertible *A*–*B*-bimodule.

Connes fusion has the following useful *faithfulness* property:

### Lemma 2.13

*R*be a von Neumann algebra, and let

*H*be a faithful right module. Then, for any left modules \(K_1\) and \(K_2\), the map

### Proof

*R*on

*H*. By Lemma 2.12,

*H*is an invertible \(R'\)–

*R*-bimodule. The map (2) can then be factored as the composite of the bijection \({{\mathrm{Hom}}}_R(K_1,K_2)\cong {{\mathrm{Hom}}}_{R'}(H\boxtimes K_1,H\boxtimes K_2)\) with the inclusion

*R*–

*R*-bimodules

^{3}into a tensor category, with unit object \(L^2R\). The associator is given by

*R*-linear map, \(\xi \in K\), and \(\beta :L^2R\rightarrow L\) a left

*R*-linear map, and the two unitors are given by

*H*(with scalar multiplication \(\lambda \overline{\xi }=\overline{\overline{\lambda }\xi }\)), and the two actions of

*R*are given by \(a\overline{\xi }b=\overline{b^*\xi a^*}\). The transformation \(\varphi \) is the identity. The map \(j:L^2R\rightarrow \overline{L^2R}\) is given by \(j(\xi )=\overline{J(\xi )}\), with

*J*the modular conjugation (note that

*j*is linear, and

*J*is anti-linear), and the coherence \(\nu :\overline{H} \boxtimes _R \overline{K} \rightarrow \overline{K \boxtimes _R H}\) is given by

### Remark 2.14

Let \({{\mathrm{Bim}}}^\circ (R)\subset {{\mathrm{Bim}}}(R)\) be the full subcategory of dualizable bimodules (equivalently, the bimodules with finite statistical dimension [2, § 5 and Cor. 7.14]). Then, by [2, Cor. 6.12] , the canonical conjugation on \({{\mathrm{Bim}}}^\circ (R)\) (described in Sect. 2.2) is the restriction of the conjugation on \({{\mathrm{Bim}}}(R)\) described above.

### 2.5 Graphical calculus

*j*, \(\nu \), and \(\varphi \), can be conveniently abbreviated

*dimension*of a dualizable object \(x\in \mathcal {C}\) is given by

### Remark 2.15

The element Open image in new window.

The following lemma lists the most important relations satisfied in the above graphical calculus. To our knowledge, the following relations have not appeared in this exact form in the literature, but they are certainly well known to experts:

### Lemma 2.16

### Proof

Let us now assume that \(\mathcal {C}\) is furthermore a fusion category, and let \( \dim (\mathcal {C}):=\sum _{x\in {{\mathrm{Irr}}}(\mathcal {C})}\, d_x^2 \) be its global dimension. We then have the following result.

### Lemma 2.17

### Proof

### 2.6 Cyclic fusion

*n*), we may define the cyclic tensor product

*cyclic Connes fusion*, first introduced in [1, Appendix A], is the analog of the above construction for Connes fusion.

*n*), then the above trace is not always defined. It is however defined if

*at least two of the maps are Hilbert–Schmidt*.

For bimodules between von Neumann algebras, we propose the following as a categorification of the Hilbert–Schmidt condition:

### Definition 2.18

*coarse*if the action of the algebraic tensor product \(A\odot B^{\text {op}}\) extends to the spatial tensor product \(A\,{\bar{\otimes }}\, B^{\text {op}}\). Equivalently, a bimodule is coarse if it is a direct summand of a bimodule of the form

*A*or

*B*are factors, then any coarse bimodule is of the form (8)).

Coarse bimodules form an ideal in the sense that if \({}_AH_B\) is coarse and \({}_BK_C\) is any bimodule, then \({}_AH\boxtimes _BK_C\) is coarse.

### Definition 2.19

*n*). Assume that

*at least two*of the \(H_i\) are coarse. Then, we define the

*cyclic fusion*by:

*a*and

*b*are chosen so that at least one of the \(\{H_{a+1},\ldots ,H_b\}\) is coarse, and at least one of the \(\{H_{b+1},\ldots ,H_a\}\) is coarse.

### Remark 2.20

*a*and

*b*used to “cut the circle”:In [1, Appendix A], it was shown that when all the \(H_i\) are coarse (and as long as there are at least two of them), the cyclic fusion is well defined up to canonical unitary isomorphism. It is also well defined in the presence of non-coarse bimodules: Let the \(H_{i_1},\ldots ,H_{i_k}\) be coarse, and let the other bimodules be non-coarse. Then, we may define the cyclic fusion in terms of the operation described in [1, Appendix A] as

*one*thick strand).

Later on in this paper, we will combine the above cylinder graphical calculus with the colored dots notation from (3).

## 3 Bicommutant categories

Let *R* be a hyperfinite factor, and let \({{\mathrm{Bim}}}(R)\) be the category of *R*–*R*-bimodules whose underlying Hilbert space is separable. The latter is a bi-involutive tensor category under the operation of Connes fusion, as discussed in Sect. 2.4.

### Notation 3.1

There is an obvious bi-involutive tensor functor \(\mathcal {C}'\rightarrow {{\mathrm{Bim}}}(R)\) given by forgetting the half-braiding. It therefore makes sense to consider the commutant of the commutant. There is also an “inclusion” functor \(\iota :\mathcal {C}\rightarrow \mathcal {C}''\) from the category to its bicommutant. It sends an object \(X\in \mathcal {C}\) to the object \((X,e'_X)\in \mathcal {C}''\) with half-braiding given by \(e'_{X,(Y,e_Y)}:=e_{Y,X}^{-1}\) for \((Y,e_Y)\in \mathcal {C}'\). The coherence data \(\mu \), *i*, \(\upsilon \) for \(\iota \) are all identity morphisms.

### Definition 3.2

A *bicommutant category* is a bi-involutive tensor category \(\mathcal {C}\) for which there exists a hyperfinite factor *R* and a bi-involutive tensor functor \(\mathcal {C}\rightarrow {{\mathrm{Bim}}}(R)\), such that the “inclusion” functor \(\iota :\mathcal {C}\rightarrow \mathcal {C}''\) is an equivalence.

If a bi-involutive tensor functor \(\alpha :\mathcal {C}\rightarrow {{\mathrm{Bim}}}(R)\) is such that the corresponding “inclusion” functor \(\iota \) is an equivalence, then we say that \(\alpha \)*exhibits*\(\mathcal {C}\)*as a bicommutant category*.

### 3.1 Representing tensor categories in \({{\mathrm{Bim}}}(R)\)

A representation of a \(*\)-algebra *A* on a Hilbert space *H* is a \(*\)-algebra homomorphism \(A\rightarrow B(H)\). By analogy, we define a *representation* of a bi-involutive tensor category \(\mathcal {C}\) to be a bi-involutive tensor functor \(\mathcal {C}\rightarrow {{\mathrm{Bim}}}(R)\), for some von Neumann algebra *R*. One can alternatively describe this as an action of \(\mathcal {C}\) on the category \(\mathrm {Mod}(R)\) of left *R*-modules.

### Definition 3.3

A representation \(\mathcal {C}\rightarrow {{\mathrm{Bim}}}(R)\) is called *fully faithful* if non-isomorphic objects of \(\mathcal {C}\) remain non-isomorphic in \({{\mathrm{Bim}}}(R)\), and if simple objects of \(\mathcal {C}\) remain simple in \({{\mathrm{Bim}}}(R)\) (this agrees with the usual notion of fully faithfulness from category theory). In the next theorem, we will see that if we restrict the von Neumann algebra *R* to be a hyperfinite factor which is not of type \(\mathrm I\), then every unitary fusion category admits a fully faithful representation in \({{\mathrm{Bim}}}(R)\). We begin with the following well-known lemma:

### Lemma 3.4

Let *R* be a hyperfinite factor which is not of type \(\mathrm{I}\), and let \(R_{\mathrm{II}_1}\) be a hyperfinite \(\mathrm{II}_1\)-factor. Then, \(R\,{\bar{\otimes }}\, R_{\mathrm{II}_1}\cong R\).

### Proof

If *R* is either of type \(\mathrm{II}_1\) or \(\mathrm{II}_\infty \), then the result follows from the uniqueness of the hyperfinite \(\mathrm{II}_1\) and \(\mathrm{II}_\infty \) factors [23, Thm. XIV]. We may therefore assume that *R* is of type \(\mathrm{III}\).

Let \(\sigma :\mathbb {R}\rightarrow \mathrm {Aut}(R)\) be the modular flow of *R*. The *flow of weights* [5] is the dual action of \(\mathbb {R}\) on the von Neumann algebra \(S(R):=Z(R\rtimes _\sigma \mathbb {R})\).^{4} By the work of Connes [3], Haagerup [13], and Krieger [19] (see also [32, Chapt. XVIII] ), the map \(R\mapsto S(R)\) establishes a bijective correspondence between isomorphism classes of hyperfinite type \(\mathrm{III}\) factors, and isomorphism types of ergodic actions of \(\mathbb {R}\) on abelian von Neumann algebras, provided one excludes the standard action of \(\mathbb {R}\) on \(L^\infty (\mathbb {R})\). (The latter is the flow of weights of the hyperfinite \(\mathrm{II}_1\) and \(\mathrm{II}_\infty \) factors.)

^{5}It follows that

### Theorem 3.5

Let *R* be a hyperfinite factor which is not of type \(\mathrm{I}\). Then, every unitary fusion category \(\mathcal {C}\) admits a fully faithful representation \(\mathcal {C}\rightarrow {{\mathrm{Bim}}}(R)\).

### Proof

*R*be an arbitrary hyperfinite factor which is not of type \(\mathrm{I}\). By Lemma 3.4, we have \(R\,{\bar{\otimes }}\, R_{\mathrm{II}_1}\cong R\). We may therefore compose the above embedding with the map\(\square \)

The above result raises the question of uniqueness. We believe that the following conjecture should follow straightforwardly from Popa’s uniqueness theorems for hyperfinite finite depth subfactors of types \(\mathrm{II}_1\) [25, 26] and \(\mathrm{III}_1\) [28]. However, we do not attempt to prove it here as it would take us too far afield.

### Conjecture 3.6

Let \(\mathcal {C}\) be a unitary fusion category, and let *R* be a hyperfinite factor which is either of type \(\mathrm{II}_1\) or \(\mathrm{III}_1\). Then, any two fully faithful representations \(\mathcal {C}\rightarrow {{\mathrm{Bim}}}(R)\) are equivalent in the sense of Definition 3.3.

## 4 The commutant of a fusion category

Throughout this section, we fix a factor *R* (not necessarily hyperfinite), a unitary fusion category \(\mathcal {C}\), and a representation \(\mathcal {C}\rightarrow {{\mathrm{Bim}}}(R)\). To simplify the notation, we will assume that the representation is fully faithful and identify \(\mathcal {C}\) with its image in \({{\mathrm{Bim}}}(R)\), but the fully faithfulness condition is actually not required for the results of this section. It will however be needed later on, in Sect. 5.

### 4.1 Constructing objects in \(\mathcal {C}'\)

### Proposition 4.1

\(e_{\Delta }=(e_{\Delta ,a}:\Delta \boxtimes a\rightarrow a\boxtimes \Delta )_{a\in \mathcal {C}}\) is a unitary half-braiding.

### Proof

*a*by construction. To see that \(e_{\Delta ,a}\) is unitary, we use the Bigon and Fusion relations: The verification that \(e_{\Delta ,a}\circ e_{\Delta ,a}^*={{\mathrm{id}}}_{a\boxtimes \Delta }\) is similar.

### Proposition 4.2

The assignment \(\Lambda \mapsto (\Delta , e_\Delta )\) defines a functor \({{\mathrm{Bim}}}(R)\rightarrow \mathcal {C}'\).

### Proof

### Remark 4.3

The construction of \(\underline{\Delta }(\Lambda )=(\Delta (\Lambda ),e_{\Delta (\Lambda )})\) works under the greater generality of a rigid C*-tensor category (in particular semisimple) represented in \({{\mathrm{Bim}}}(R)\), not necessarily fully faithfully. The half-braiding (10) is unitary by Proposition 4.1, and thus bounded.

### 4.2 The endomorphism algebra

In this section, we fix a bimodule \(\Lambda \in {{\mathrm{Bim}}}(R)\). Our goal is to compute the endomorphism algebra of \(\underline{\Delta }(\Lambda )\). As in the previous section, we will write \(\Delta \) for the underlying object of \(\underline{\Delta }(\Lambda )\).

### Theorem 4.4

### Remark 4.5

The map \(f_{\overline{a}}:\Lambda \boxtimes \overline{a} \rightarrow \overline{a}\boxtimes \Lambda \), which appears in the right-hand side of (11) requires the choice of an isomorphism between \(\overline{a}\) and the unique element of \({{\mathrm{Irr}}}(\mathcal {C})\) to which it is isomorphic. It is important to note that, because \(\overline{a}\) appears in both the domain and the codomain, the map \(f_{\overline{a}}\) does not depend on that choice.

### Remark 4.6

If we take \(\Lambda =\bigoplus _{x\in {{\mathrm{Irr}}}(C)} x\), then the two Eqs. (11) and (12) are exactly the ones describing Ocneanu’s tube algebra [7, 16].

### Proof of Theorem 4.4

*T*commutes with (a scalar multiple of) the half-braiding, and finally Lemma 2.17.

### Remark 4.7

The map \(f\mapsto T_f:\bigoplus _{a\in {{\mathrm{Irr}}}(\mathcal {C})} {{\mathrm{Hom}}}_{{{\mathrm{Bim}}}(R)}(\Lambda \boxtimes a, a\boxtimes \Lambda )\rightarrow {{\mathrm{End}}}_{\mathcal {C}'}(\underline{\Delta }(\Lambda ))\) makes sense in the greater generality of a rigid C*-tensor category represented in \({{\mathrm{Bim}}}(R)\). In particular, the operator \(T_f\) is always bounded (this follows from Open image in new window being unitary, and hence bounded).

## 5 Absorbing objects

*no zero-divisors*if for every nonzero object

*X*and every objects \(Y_1, Y_2\), the maps

### Example 5.1

The tensor category \({{\mathrm{Bim}}}(R)\) has no zero-divisors. Indeed, since *R* is a factor, every nonzero module is faithful, and the claim follows from Lemma 2.13.

### Example 5.2

*X*and a morphism \(f:Y_1\rightarrow Y_2\) such that \({{\mathrm{id}}}_X\otimes f=0\). We need to show that \(X\not \cong 0\) implies \(f=0\). Since

*X*is nonzero, \({{\mathrm{ev}}}_X\) is an epimorphism (indeed a projection onto a direct summand). The morphism \({{\mathrm{ev}}}_X\otimes {{\mathrm{id}}}_{Y_1}\) is then also an epimorphism, and we may reason as follows:

### Definition 5.3

*X*is called

*right absorbing*if for every nonzero object \(Y\in \mathcal {C}\), we have \(X\otimes Y\cong X\),*left absorbing*if for every nonzero object \(Y\in \mathcal {C}\), we have \(Y\otimes X\cong X\), and*absorbing*if*X*is both right and left absorbing.

Clearly, if \(\mathcal {C}\) admits an absorbing object, then such an object is unique up to (non-canonical) isomorphism. Note also that if a category has both right absorbing and left absorbing objects, then any such object is in fact absorbing.

If \(\mathcal {C}\) is equipped with a conjugation, then *X* is right absorbing if and only if \(\overline{X}\) is left absorbing. In this case, any right absorbing object is automatically absorbing, and isomorphic to its conjugate. By taking \(Y=1\oplus 1\), we can also readily see that any absorbing object satisfies \(X\oplus X \cong X\).

Let \(\mathsf {Hilb}\) be the category of separable Hilbert spaces.

### Example 5.4

The Hilbert space \(\ell ^2(\mathbb N)\) is absorbing in \(\mathsf {Hilb}\).

### Example 5.5

*y*and

*z*of \(\mathcal {C}\), there exists an

*x*such that

*z*occurs as a summand of \(x\otimes y\). The object \(y\otimes (\bigoplus _{x\in {{\mathrm{Irr}}}(\mathcal {C})} x)\) therefore contains each simple object at least once. It follows that \(y\otimes (\bigoplus _{x\in {{\mathrm{Irr}}}(\mathcal {C})} x\otimes \ell ^2(\mathbb N))\) contains each simple object infinitely many times. The same remains true when

*y*gets replaced by an arbitrary nonzero object of \(\mathcal {C}\otimes _{\mathsf {Vec}}\mathsf {Hilb}\).

### Example 5.6

*G*be an infinite countable group, and let \(\mathsf{Rep}(G)\) denote the category of unitary representation of

*G*whose underlying Hilbert spaces is separable. Then,

*V*is a unitary representation with orthonormal basis \(\{v_i\}_{i\in I}\), then \(e_g\otimes e_i \mapsto (g\cdot v_i)\otimes e_g\) defines a unitary isomorphism \(\ell ^2(G)\otimes \ell ^2(I)\rightarrow V\otimes \ell ^2(G)\). It follows that \(V\otimes \ell ^2(G)\otimes \ell ^2(\mathbb N)\cong \ell ^2(G)\otimes \ell ^2(I\times \mathbb {N})\cong \ell ^2(G)\otimes \ell ^2(\mathbb N)\).

Let *R* be a separable factor, and let \({{\mathrm{Bim}}}(R)\) be the category of *R*–*R*-bimodules whose underlying Hilbert space is separable. Let also \(\mathrm {Mod}(R)\) be the category of left *R*-modules whose underlying Hilbert space is separable. We say that \(H\in \mathrm {Mod}(R)\) is *infinite* if it is nonzero and satisfies \(H\oplus H\cong H\). It is well known that an infinite module exists and is unique up to isomorphism.

### Example 5.7

### Remark 5.8

If we had taken \({{\mathrm{Bim}}}(R)\) to be the category of *all* bimodules, with no restriction on cardinality, then it would not admit an absorbing object (and similarly for the previous examples).

Absorbing objects are useful because *they control half-braidings*:

### Proposition 5.9

Let \(\Omega \) be an absorbing object of \(\mathcal {C}\), and let \((X,e_X)\) be an object of \(\mathcal {C}'\). Then, \(e_X\) is completely determined by its value on \(\Omega \).

### Proof

*Y*be a nonzero object of \(\mathcal {C}\). Since \(e_X\) is a half-braiding, we have a commutative diagramFix an isomorphism \(\phi :Y\boxtimes \Omega \rightarrow \Omega \). The following square is commutativeand so we get an equation

### 5.1 The absorbing object of \(\mathcal {C}'\)

*R*equipped with a fully faithful representation \(\mathcal {C}\rightarrow {{\mathrm{Bim}}}(R)\) of some unitary fusion category \(\mathcal {C}\). Our next goal is to show that \(\mathcal {C}'\) admits absorbing objects. Recall the construction

### Theorem 5.10

The functor \(\underline{\Delta }\) sends absorbing objects to absorbing objects. In particular, the category \(\mathcal {C}'\) admits absorbing objects.

The proof of this theorem will depend on Theorem 5.12, proved in next section, according to which the endomorphism algebra of \(\underline{\Delta }(\Lambda )\) is a factor whenever \(\Lambda \) is absorbing in \({{\mathrm{Bim}}}(R)\). We begin with the following technical lemma:

### Lemma 5.11

Suppose that \(\underline{\Omega }=(\Omega , e_\Omega )\in \mathcal {C}'\) is such that \(\Omega \) is absorbing in \({{\mathrm{Bim}}}(R)\), and such that \(\underline{\Omega }\oplus \underline{\Omega }\;\!\cong \;\! \underline{\Omega }\) in \(\mathcal {C}'\). Then, \(\underline{\Omega }\) is (non-canonically) isomorphic to \(\underline{\Delta }(\Omega )\).

### Proof

The coevaluation map \(\mathrm {coev}_x:L^2R\rightarrow x\boxtimes \overline{x}\) is, up to a constant, the inclusion of a direct summand. So \(\varphi \) is manifestly injective. By polar decomposition in \(\mathcal {C}'\), the map \(\varphi \) therefore induces a unitary isomorphism between \(\underline{\Omega }\) and a certain subobject of \(\underline{\Delta }(\Omega )\).

Now, the subobjects of \(\underline{\Delta }(\Omega )\) are in one-to-one correspondence with the projections in \(M:={{\mathrm{End}}}_{\mathcal {C}'}(\underline{\Delta }(\Omega ))\), which is a factor by Theorem 5.12. Let \(p\in M\) be the projection corresponding to \(\underline{\Omega }\). Since \(\underline{\Omega }\oplus \underline{\Omega }\cong \underline{\Omega }\) and \(\underline{\Omega }\ne 0\), that projection is infinite (its range is an infinite module). So there is a partial isometry \(u\in M\) with \(p=uu^*\) and \(u^*u=1\). The latter provides an isomorphism \(u:\underline{\Delta }(\Omega )\rightarrow \underline{\Omega }\) in \(\mathcal {C}'\). \(\square \)

### Proof of Theorem 5.10

*X*be an arbitrary nonzero object of \(\mathcal {C}'\). We wish to show that \(\underline{\Omega }:=\underline{\Delta }(\Lambda )\boxtimes X\) is isomorphic to \(\underline{\Delta }(\Lambda )\). Let \(\Omega \) denote the underlying object of \(\underline{\Omega }\). If we could show that \(\underline{\Omega }\) satisfies the hypotheses of Lemma 5.11, then we could reason as follows:

So let us show that \(\underline{\Omega }\) satisfies the hypotheses of Lemma 5.11. Since \(\Lambda \) is absorbing in \({{\mathrm{Bim}}}(R)\), the object \(\Omega =\bigoplus _x x\boxtimes \Lambda \boxtimes \overline{x}\boxtimes X\) is clearly absorbing in \({{\mathrm{Bim}}}(R)\). And since \(\Lambda \oplus \Lambda \cong \Lambda \) in \({{\mathrm{Bim}}}(R)\) and \(\Lambda \mapsto \underline{\Delta }(\Lambda )\boxtimes X\) is a linear functor, the same holds true for \(\underline{\Omega }\), namely \(\underline{\Omega }\oplus \underline{\Omega }\;\!\cong \;\! \underline{\Omega }\).\(\square \)

### 5.2 The endomorphism algebra is a factor

The goal of this section is to prove that when \(\Lambda \) is absorbing, the endomorphism algebra of \(\underline{\Delta }(\Lambda )\) is a factor (a von Neumann algebra with trivial center). We emphasize the fact that, for the above result to hold, it is essential that the representation \(\mathcal {C}\rightarrow {{\mathrm{Bim}}}(R)\) be fully faithful (this is used in the last paragraph of the proof of Theorem 5.13).

### Theorem 5.12

If \(\Lambda \) is absorbing in \({{\mathrm{Bim}}}(R)\), then \({{\mathrm{End}}}_{\mathcal {C}'}(\underline{\Delta }(\Lambda ))\) is a factor.

It will be easier to prove the following stronger result:

### Theorem 5.13

### Proof

*H*for \(\ell ^2(\mathbb {N})\), we have

*R*–

*R*-bimodule) therefore extends to an action of the spatial tensor product \(R\,{\bar{\otimes }}\, R^{\text {op}}\). We may therefore treat \(\Lambda _0\) as a left \((R\,{\bar{\otimes }}\, R^{\text {op}})\)-module. Writing 1 for \(L^2(R)\), we then have canonical isomorphisms

But \(1\otimes x\) and \(x\otimes 1\) are irreducible \((R\,{\bar{\otimes }}\, R^{\text {op}})\)–\((R\,{\bar{\otimes }}\, R^{\text {op}})\)-bimodules, and \(1\otimes x\not \cong x\otimes 1\) unless \(x=1\). The maps \(f_x\) can therefore only be nonzero when \(x=1\), in which case it must be a scalar. \(\square \)

*R*-modules with a right

*R*-module:

### Proposition 5.14

The algebras \({{\mathrm{End}}}({}_RH)\) and \({{\mathrm{End}}}(K_R)\) are each other’s relative commutants in \({{\mathrm{End}}}_{\mathcal {C}'}(\underline{\Delta }(\Lambda ))\).

### Proof

We will only prove that \(Z_{{{\mathrm{End}}}_{\mathcal {C}'}(\underline{\Delta }(\Lambda ))}({{\mathrm{End}}}({}_RH))={{\mathrm{End}}}(K_R)\). The other claim is symmetric and can be proved in a completely analogous way.

*H*, and let

*f*be an element of \({{\mathrm{End}}}_{\mathcal {C}'}(\underline{\Delta }(\Lambda ))\). Let \(f_a:\Lambda \boxtimes a\rightarrow a \boxtimes \Lambda \) be the maps which correspond to \(f\in {{\mathrm{End}}}_{\mathcal {C}'}(\underline{\Delta }(\Lambda ))\) under the bijection established in Theorem 4.4. The statement that

*b*and

*f*commute is then equivalent to the statement that for every \(a\in {{\mathrm{Irr}}}(\mathcal {C})\), the following equality holds in \({{\mathrm{Hom}}}(H\otimes _{\mathbb C}K\boxtimes _R a,a\boxtimes _R H\otimes _{\mathbb C}K)\):Treating

*K*as a left \(R^{{{\mathrm{op}}}}\)-module and letting \(R'\) be the commutant of

*R*on

*H*(so that

*H*is an

*R*–\(R'{}^{{{\mathrm{op}}}}\)-bimodule), we may “fold” the above diagram (as we did to get (15)):where Open image in new window. It follows that \(f_a\) is not just in

*H*is an invertible

*R*–\(R'{}^{{{\mathrm{op}}}}\)-bimodule, and so

### 5.3 Algebras acting on cyclic fusions

*x*, the isomorphism (16) is independent of those choices.)

### Lemma 5.16

Let \(\Lambda _1\) and \(\Lambda _2\) be coarse bimodules. Then, \(N_1={{\mathrm{End}}}_{{{\mathrm{Bim}}}(R)}(\Lambda _1)\) and \(N_2={{\mathrm{End}}}_{{{\mathrm{Bim}}}(R)}(\Lambda _2)\) are each other’s commutants on \(\big [ \Lambda _1 \boxtimes _R \Lambda _2 \boxtimes _R - \big ]_{\mathrm{cyclic}}\).

### Proof

### Proposition 5.17

Let \(\Lambda _1\) and \(\Lambda _2\) be coarse bimodules. Then, \(M_1={{\mathrm{End}}}_{\mathcal {C}'}(\underline{\Delta }(\Lambda _1))\) and \(M_2={{\mathrm{End}}}_{\mathcal {C}'}(\underline{\Delta }(\Lambda _2))\) are each other’s commutants on \(H=\bigoplus _{x\in {{\mathrm{Irr}}}(\mathcal {C})} \big [ x \boxtimes \Lambda _1 \boxtimes \overline{x} \boxtimes \Lambda _2 \boxtimes - \big ]_{\mathrm{cyclic}} \).

### Proof

*f*be in \(M_2'\). Since

*f*commutes with \({{\mathrm{End}}}_{{{\mathrm{Bim}}}(R)}(\Lambda _2)\subset M_2\), it follows from Lemma 5.16 that \(f\in {{\mathrm{End}}}_{{{\mathrm{Bim}}}(R)}(\Delta (\Lambda _1))\). We therefore have the following situation:It remains to show that

*f*commutes with the half-braiding. Write \(\Lambda _2\) as \({}_R(H_2)\otimes _{\mathbb {C}} (H_1)_R\), for some right/left

*R*-modules \(H_1\) and \(H_2\). We then have a canonical isomorphism

*g*of the formfor

*R*-module maps \(v:H_2\rightarrow a\boxtimes H_2\) and \(u:H_1\boxtimes a\rightarrow H_1\), Eq. (17) becomes:This being true for any

*u*and

*v*, it follows thatFinally, fusing with \(H_1\) and \(H_2\) are faithful operations by Lemma 2.13, and so the above equation implies \(e_{\Delta (\Lambda _1),a}\circ (f\boxtimes {{\mathrm{id}}}_a)=({{\mathrm{id}}}_a\boxtimes f)\circ e_{\Delta (\Lambda _1),a}\), as desired.\(\square \)

## 6 Proof of the main theorem

### Lemma 6.1

The restriction functor \((\mathcal {C}\otimes _\mathsf{Vec}\mathsf{Hilb})'\rightarrow \mathcal {C}'\) is an equivalence.

### Proof

### Theorem

(Theorem A) Let \(\mathcal {C}\) be a unitary fusion category, and let \(\alpha :\mathcal {C}\rightarrow {{\mathrm{Bim}}}(R)\) be a fully faithful representation. Then, \(\alpha ^{\mathsf{Hilb}}\) exhibits \(\mathcal {C}\otimes _\mathsf{Vec}\mathsf{Hilb}\) as a bicommutant category.

### Proof

We will show that \(\mathcal {C}''\) is equivalent to \(\mathcal {C}\otimes _\mathsf{Vec}\mathsf{Hilb}\). The result will then follow since \(\mathcal {C}''=(\mathcal {C}\otimes _\mathsf{Vec}\mathsf{Hilb})''\) by Lemma 6.1. We first note that the “inclusion” functor \(\iota :\mathcal {C}\rightarrow \mathcal {C}''\) (described in Sect. 3)

The functor \(\iota ^{\mathsf{Hilb}}\) is fully faithful:

The functor \(\iota ^{\mathsf{Hilb}}\) is essentially surjective:

- 1.
If \((X,e_X)\) is an object of \(\mathcal {C}''\), then its underlying bimodule

*X*lies in \(\mathcal {C}\otimes _\mathsf{Vec}\mathsf{Hilb}\) (the essential image of \(\iota ^{\mathsf{Hilb}}\)). - 2.
Let \((X,e_X^{\scriptscriptstyle (1)})\) and \((X,e_X^{\scriptscriptstyle (2)})\in \mathcal {C}''\) be two objects with same underlying bimodule

*X*. Then, \(e_{X,\underline{\Omega }}^{\scriptscriptstyle (1)}=e_{X,\underline{\Omega }}^{\scriptscriptstyle (2)}\). - 3.
Given an object \((X,e_X)\in \mathcal {C}''\), then \(e_X=\big (e_{X,\underline{Y}}:X\boxtimes Y\rightarrow Y\boxtimes X\big )_{\underline{Y}=(Y,e_Y)\in \mathcal {C}'}\) is uniquely determined by \(e_{X,\underline{\Omega }}\).

### Proposition 6.2

The underlying bimodule of an object of \(\mathcal {C}''\) lies in \(\mathcal {C}\otimes _\mathsf{Vec}\mathsf{Hilb}\).

### Proof

*R*, two left actions, and two right actions: In order to keep track of all these copies of

*R*, we denote them \(R_1\), \(R_2\), \(R_3\), \(R_4\), respectively.

*e*is a morphism in \({{\mathrm{Bim}}}(R)\), meaning that it is an \(R_1\)–\(R_4\)-bimodule map. This map also has the property of being natural with respect to endomorphisms of \(\underline{\Delta }(\Lambda _0)\). Restricting attention to

*e*being an \(R_3\)–\(R_2\)-bimodule map (or rather an \(R_2^{{{\mathrm{op}}}}\)–\(R_3^{{{\mathrm{op}}}}\)-bimodule map). All in all, we learn that there is an isomorphism of

*quadri-modules*:

*X*as an element of \(\mathcal {C}\otimes _\mathsf{Vec}\mathsf{Hilb}\).

Let \(\mathcal {C}'_{abs} \subset \mathcal {C}'\) be the full subcategory of absorbing objects of \(\mathcal {C}'\). This is a non-unital tensor category, and it makes sense to talk about half-braidings with \(\mathcal {C}'_{abs}\) (the axioms of a half-braiding never mention unit objects).

### Lemma 6.3

Let \(\underline{\Omega }=(\Omega ,e_\Omega )\in \mathcal {C}'\) be an absorbing object, let *X* be a right *R*-module, and let \( u:X\boxtimes \Omega \rightarrow X\boxtimes \Omega \) be a right module map that commutes with \({{\mathrm{id}}}_X\otimes {{\mathrm{End}}}_{\mathcal {C}'}(\underline{\Omega })\). Then, \(u=v\boxtimes {{\mathrm{id}}}_{\Omega }\) for some right module map \(v:X\rightarrow X\).

### Proof

*u*commutes with \({{\mathrm{End}}}_{\mathcal {C}'}(\underline{\Delta }(\Lambda ))\), it lies in \({{\mathrm{End}}}_{\mathcal {C}'}(\underline{\Delta }(\Lambda _2))\). Now, we also know that

*u*commutes with \(R^{{{\mathrm{op}}}}={{\mathrm{End}}}({}_RL^2R)\). By Proposition 5.14, it therefore comes from some element of \({{\mathrm{End}}}(X_R)\), which we may call

*v*. In other words, \(u=v\boxtimes {{\mathrm{id}}}_{\Omega }\).\(\square \)

### Proposition 6.4

An object \(X\in {{\mathrm{Bim}}}(R)\) admits at most one half-braiding with \(\mathcal {C}'_{abs}\).

### Proof

*u*commutes with \({{\mathrm{id}}}_X\otimes {{\mathrm{End}}}_{\mathcal {C}'}(\underline{\Omega })\). By Lemma 6.3, we may therefore write it as \(u=v\boxtimes {{\mathrm{id}}}_{\Omega }\) for some \(v\in {{\mathrm{End}}}_{{{\mathrm{Bim}}}(R)}(X)\). All in all, we get a commutative diagramFix an isomorphism \(\phi : \underline{\Omega }\boxtimes \underline{\Omega } \rightarrow \underline{\Omega }\) in \(\mathcal {C}'\), and let us denote by the same letter the corresponding isomorphism \(\Omega \boxtimes \Omega \rightarrow \Omega \). By combining the “hexagon” axiom with the statement that the half-braiding is natural with respect to \(\phi \), we get the following commutative diagrams (as in the proof of Proposition 5.9):andHorizontally precomposing (19) withyields the following diagramThe latter is almost identical to (18), but for the top right arrow. All maps in sight being isomorphisms, it follows that \({{\mathrm{id}}}_\Omega \boxtimes e_1={{\mathrm{id}}}_\Omega \boxtimes e_2\). At last, by Lemma 2.13, we conclude that \(e_1=e_2\).\(\square \)

## Footnotes

- 1.
- 2.
The formula for the inner product makes most sense if one rewrites formally \([D\phi :D\psi ]_t\) as \(\phi ^{it}\psi ^{-it}\) and \(\phi (a)\) as \(\mathrm {Tr}(\phi a)\). It then simplifies to \(\mathrm {Tr}(\phi ^{1+it}\psi ^{-it})|_{t=i/2}=\mathrm {Tr}(\phi ^{1/2}\psi ^{1/2})\). Similarly, for next formula, one may replace formally \(\sigma _t^\psi (b)\) by \(\psi ^{it}b\psi ^{-it}\). Note that these formal symbols are genuinely meaningful and can be implemented as (unbounded) operators on some Hilbert space, see, e.g., [34].

- 3.
Later on, we will restrict attention to separable von Neumann algebras (i.e., ones which admit faithful actions on separable Hilbert spaces), in which case we will take \({{\mathrm{Bim}}}(R)\) to be the category of

*R*–*R*-bimodules whose underlying Hilbert space is separable. The reason for that restriction will become evident in Sect. 5. - 4.
Unlike the modular flow, which depends on a choice of state, the crossed product \(R\rtimes _\sigma \mathbb {R}\) does not depend on any choices, up to canonical isomorphism.

- 5.
The result in [5] is only stated for type \(\mathrm{III}\) factors, but the proof never uses the type \(\mathrm{III}\) assumption.

## Notes

### Acknowledgments

This project began at the 2015 Mathematisches Forschungsinstitut Oberwolfach workshop on Subfactors and conformal field theory. The authors would like to thank the organizers and MFO for their hospitality. André Henriques was supported by the Leverhulme trust and the EPSRC grant “Quantum Mathematics and Computation” during his stay in Oxford. David Penneys was partially supported by an AMS-Simons travel Grant and NSF DMS Grant 1500387.

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