NOTES ON OPERATOR CATEGORIES

SHIGERU YAMAGAMI

Department of Mathematical Sciences Ibaraki University Mito, 310-8512, JAPAN http://suuri.sci.ibaraki.ac.jp/˜ yamagami/

Abstract. The bicategory of normal functors between W*-categories is monoidally equivalent to the bicategory of W*-bimodules.

Introduction

Related to subfactor theory two kinds of tensor categories are used in describing quantum symmetry ([6, 9]): Given a von Neumann algebra M, we have the tensor category _{M}Bimod_{M} of M-M bimodules on the one hand and the tensor category End(M) of endomorphisms on the other hand. In spite of different appearances, they have a close similarity: For an endomorphism ρ of M, assign the bimodule L^{2}(M)ρ, where L^{2}(M)ρ is the standard Hilbert space of M with the right action modified by ρ. Then we have canonical isomorphisms L^{2}(M)ρ ⊗_{M} L^{2}(M)σ → L^{2}(M)(ρ ∘ σ) for endomorphisms ρ and σ, which turn out to satisfy the condition of multiplicativity for monoidal functors and defines a fully faithful imbedding of the opposite of End(M) into the tensor category _{M}Bimod_{M}.

If M is of an infinite type, this monoidal imbedding gives an equivalence of categories, i.e., they contain the same information as categorical data. For a finite von Neumann algebra, however, the imbedding is not surjective because of the independant sizes of left and right modules in that case.

Furthermore, the catgory Bimod of bimodules is more flexible than End(M) in the sense that we can choose different algebras for left and right actions. The bimodules, together with the associative tensor products, then constitute a so-called bicategory (see §2 for explanations of the notion).

With these observations in mind, the author has suspected the possibility to extend the tensor category End(M) in such a way that the above imbedding can be extended to an equivalence of bicategories. The major purpose of this article is to give an affirmative answer to the above question in the framework of W*-categories: the bicategory of normal functors between W*-categories is monoidally equivalent to the bicategory Bimod of W*-bimodules.

On the way of describing this result, we shall also review some of basic facts on operator categories ([11], [3], [7]), where we have fully used tensor products of W*-modules (the relative tensor product) to obtain concise expressions.

Technically we have to use the modular theory in operator algebras on occasions, which is, however, algebraic (and formal) in nature rather than analytic if properly formulated (see [16]).

As backgrounds of the subject, we refer, for example, to [15] on operator algebras and to [8, 10] for generalities on category theory.

1. W*-categories

By a linear category, we shall mean an essentially small category for which homsets are vector spaces over the complex number field ℂ and the operation of taking compositions is complex-linear in variables involved.

The restriction of essential smallness reflects our standing position that we shall not study operator algebras with the help of categorical languages but work with categories themselves of operator algebraic natures.

A functor between linear categories is said to be linear if the operation on morphisms is linear. Recall that a functor is said to be faithful if it is injective on hom-sets (no commitment to objects).

A linear category is called a *-category if it is furnished with conjugate-linear involutions on hom-sets satisfying (i) f^{∗} : Y → X for f : X → Y and (ii) (g ∘ f)^{∗} = f^{∗} ∘ g^{∗} for f : X → Y and g : Y → Z.

A *-category is called a C*-category if hom-sets are Banach spaces such that (i) ∥g ∘ f∥ ≤ ∥g∥ ∥f∥ for morphisms of the form f : X → Y and g : Y → Z, (ii) ∥f^{∗}f∥ = ∥f∥^{2} and (iii) f^{∗}f ≥ 0 for any morphism f.

Note here that, by the conditions (i) and (ii), each End(X) is a C*-algebra and the meaning of positivity in (iii) is that for C*-algebras.

A C*-category C is called a W*-category if each Banach space Hom(X, Y) is the dual of a Banach space. ‘Preduals’ are uniquely determined by the C*-category C as soon be checked (an analogue of Sakai’s characterization, [13]).

A typical example of W*-categories is the category Rep(A) of *-representations of a C*-algebra A in Hilbert spaces of a specified class with hom-sets given by intertwiners of representations.

When A is separable (i.e., having a countable norm-dense subset), we can define the much smaller W*-category SRep(A) of *-representations of A in separable Hilbert spaces of a specified class with hom-sets given by intertwiners of representations. If A = ℂ, Rep(A) (resp. SRep(A)) is the category of Hilbert spaces (resp. separable Hilbert spaces) of a specified class Hilb (resp. SHilb) whose morphisms are bounded linear operators.

Another typical example of W*-category is the category P(M) of projections in a W*-algebra M: objects of P(M) are projections in M with hom-sets given by Hom(e, f) = fMe for projections e and f in M.

A typical example of C*-category is the category A-Mod of Hilbert A-modules (again in a specified class) with A a C*-algebra.

Given *-categories C and D, a functor F : C → D is called a *-functor if F is linear in morphisms and preserves the *-operation. If both of C and D are C*- categories, then a *-functor F : C → D is norm-decreasing:

∥F (f)∥^{2} = ∥F (f^{∗}f)∥ ≤ ∥f^{∗}f∥ = ∥f∥^{2}

for a morphism f : X → Y in C. The kernel of F (which is a C*-subcategory of C) is then an analogue of closed *-ideals in C*-algebras and we have the exact sequence

0 → Ker(X, Y) → Hom(X, Y) → F(Hom(X, Y)) → 0

of C*-categories.

Given *-categories C and D, we define the category Hom(C, D) with objects given by *-functors and morphisms consisting of natural transformations. Recall here that we have stuck to essentially small categories, which enables us to keep Hom(C, D) essentially small. When D is a C*-category, morphisms are restricted to be bounded so that the category Hom(C, D) of functors is again a C*-category.

Given an object X in a C*-category C and a positive linear functional φ on the C*-algebra End(X), we introduce the *-functor F_{φ} : C → Hilb by the GNSconstruction: For an object Y in C, the Hilbert space F_{φ}(Y) is the completion of the vector space Hom(X, Y) with respect to the positive-semidefinite inner product

(y|y′) = φ(y^{∗}y′) for y, y′ : X → Y.

We shall often use the notation yφ^{1/2} to distinguish the morphism y in C with the associated element in the Hilbert space F_{φ}(Y).

For a morphism f : Y → Z in C, F_{φ}(f) : F_{φ}(Y) → F_{φ}(Z) is the bounded linear map defined by

F_{φ}(f)(yφ^{1/2}) = (fy)φ^{1/2}.

These are well-defined by the positivity and the C*-norm assumption in the definition of C*-categories.

Note that, if X ≅ Y and positive linear functionals φ : End(X) → ℂ, ψ : End(Y) → ℂ are related by a *-isomorphism between End(X) and End(Y), then the functors F_{φ} and F_{ψ} are unitarily equivalent.

Since (infinite) direct sums are permitted in the C*-category Hilb, we can define the *-functor F : C → Hilb as the direct sum of {F_{φ}}, where φ ∈ End(X)_{+}^{∗} with the family {X} representing the isomorphism classes of objects in C. Then F is faithful on each C*-algebra End(X) and we have

∥F (f)∥^{2} = ∥F (f^{∗}f)∥ = ∥f^{∗}f∥ = ∥f∥

for a morphism f : X → Y, which implies that F is isometric on hom-sets. Thus, given a finite family {X_{i}}_{1≤i≤n} of objects in the C*-category C, the algebra of matrix form

F (Hom(X_{1}, X_{1})) … F (Hom(X_{n}, X_{1}))
⋮  ...  ⋮
F (Hom(X_{1}, X_{n})) … F (Hom(X_{n}, X_{n}))

is a C*-algebra on the Hilbert space

⨁_{i=1}^{n} F (X_{i}),

which is *-isomorphic to the *-algebra

(\begin{matrix}Hom(X_{1}, X_{1}) & … & Hom(X_{n}, X_{1}) \\ ⋮ & ... & ⋮ \\ Hom(X_{1}, X_{n}) & … & Hom(X_{n}, X_{n}) \end{matrix}) .

Proposition 1.1. Given a C*-category C and a finite family {X_{i}}_{1≤i≤n} of objects in C, the vector space

Hom(X_{1}, X_{1}) … Hom(X_{n}, X_{1})
⋮  ...  ⋮
Hom(X_{1}, X_{n}) … Hom(X_{n}X_{n})

is a C*-algebra with each matrix component isometrically identified with the Banach spaces Hom(X_{i}, X_{j}).

The above proposition is used to enlarge a C*-category C so that it allows finite direct sums; consider the category Ĉ for which objects are finite families of objects in C and hom-sets are given by

Hom(\{X_{i}\}, \{Y_{j}\}) = (\begin{matrix}Hom(X_{1}, Y_{1}) & … & Hom(X_{m}, Y_{1}) \\ ⋮ & ... & ⋮ \\ Hom(X_{1}, Y_{n}) & … & Hom(X_{m}, Y_{n}) \end{matrix})

with the norm induced from a bigger C*-algebra of matrix form. Corollary 1.2. A C*-category is a W*-category if and only if

(\begin{matrix}End(X) & Hom(Y, X) \\ Hom(X, Y) & End(Y) \end{matrix})

is a W*-algebra for any pair (X, Y) of objects in C.

Let C be a W*-category. Since Hom(X, Y) is realized as a corner of a W*-algebra, we can define their L^{p}-extensions (see [4] for L^{p}-theory on von Neumann algebras, cf. also [16, 17]): the Banach space L^{p}(X, Y) is defined to be the upper-right corner of the L^{p}-space associated to the W*-algebra in the above corollary. We then have the bounded bilinear map L^{q}(X, Y) × L^{p}(Y, Z) → L^{r}(X, Z) if 1/r = 1/p + 1/q and L^{∞}(X, Y) is identified with Hom(Y, X). We also have the duality L^{p}(X, Y)^{∗} = L^{q}(Y, X) with 1/p + 1/q = 1 and the predual of Hom(X, Y) is identified with the Banach space L^{1}(X, Y). Proposition 1.3. Given a finite family {X_{i}} of objects in a W*-category, the matrix algebra

Hom(X_{1}, X_{1}) … Hom(X_{n}, X_{1})
⋮  ...  ⋮
Hom(X_{1}, X_{n}) … Hom(X_{n}, X_{n})

is a W*-algebra. Moreover, any W*-category is (in a unique way) extended so that it admits finite direct sums.

Let C be a *-category and D be a W*-category. Then Hom(C, D) is again a W*-category (C being assumed to be essentially small and natural transformations being restricted to be bounded). For the identity functor idC, End(idC) is then a commutative W*-algebra Z(C) given by

{t = {t_{X}}; t_{X} ∈ End(X), ft_{X} = t_{Y}f for any f ∈ Hom(X, Y) in C}.

We call Z(C) the center of C. If C is a full subcategory of M-Mod including a faithful representation, then Z(C) is naturally isomorphic to the center of M.

For a morphism f : X → Y in C, the central support of f is defined to be the projection c(f) in Z(C) which is minimal among the projections c in Z(C) satisfying c_{Y} f = f = fc_{X}. For a family {f_{i}} of morphisms, we define its central support as the smallest projection in Z(C) majorizing all c(f_{i})’s: c = ⋁_{i} c(f_{i}). For a family {X_{i}} of objects, its central support is defined to be the central support of the family {1_{X_{i}}}.

The central support c of f is calculated by

c_{Z} = ⋁_{g:Z→X} s(fg),

where s(fg) denotes the support projection of the element g^{∗}f^{∗}fg in End(Z). In particular, if Hom(X, Y) = {0}, i.e., X and Y are disjoint, then c(1_{X})c(1_{Y}) = 0.

A family {U_{i}} of objects in a *-category C is said to be generating if the associated family of hom-functors Hom(U_{i}, ·) : C → Vec is faithful, i.e., the algebraic direct sum

⨁_{i} Hom(U_{i}, ·) : C → Vec

is faithful. A single object U in C is called a generator if the one-object family {U} is generating, i.e., the hom-functor Hom(U, ·) : C → Vec is faithful. Example 1.4. Any representative family of isomorphism classes of objects is generating. Lemma 1.5. Let {U_{i}} be a family of objects in a W*-category C. Then the following conditions are equivalent.

(i) The family {U_{i}} is generating.

(ii) For any object X and any projection 0 ≠ p ∈ End(X), we can find a U_{i}

such that Hom(U_{i}, pX) = pHom(U_{i}, X) ≠ {0}. (iii) The central support of {U_{i}} is equall to the family {1_{X}} of identity mor-

phisms. (iv) For any object X in C, we can find a family of partial isometries {u_{i,j} :

U_{i} → X} such that

∑_{i,j} u_{i,j}u_{i,j}^{∗} = 1_{X}.

Proof. (i) ⇒ (ii). If there exist an object X and a projection 0 ≠ p ∈ End(X) such that Hom(U_{i}, pX) = {0} for any i, then p can not be distinguished with 0 under the hom-functor Hom(U_{i}, ·), which contradicts with the condition (i).

(ii) ⇒ (iii). If the central support {c_{X}} of {U_{i}} is different from {1_{X}}, we can find an object X such that c_{X} ≠ 1_{X}, which satisfies (1_{X} − c_{X})f = 0 for any i and any f : U_{i} → X.

(iii) ⇒ (iv). The formula for the central support together with polar decompositions shows that for any object X, we can find i and a non-zero partial isometry u : U → X. Now the maximality argument gives the result.

(iv) ⇒ (i) is obvious.

□

Lemma 1.6 ([12, Lemma 2.1]). Let X, Y and Z be objects in C. Let T : L^{2}(Y, X) → L^{2}(Z, X) be a bounded linear map satisfying T (ξx) = T (ξ)x for ξ ∈ L^{2}(Y, X) and x ∈ End(X). Then we can find an element y ∈ Hom(Y, Z) such that T (ξ) = yξ for ξ ∈ L^{2}(Y, X).

Proof. Let M be the W*-algebra associated to the family {X, Y, Z} in Proposition 1.3 with e, f and g projections of M to the component X, Y and Z respectively. Consider a bounded operator T on L^{2}(M) satisfying T (ξ) = gT (fξe)e for ξ ∈ L^{2}(M) and T (ξexe) = T (ξ)exe for x ∈ M.

Since the commutant of a reduction is the induction of the commutant, the second condition on T implies we can find y ∈ M satisfying T (ξ) = yξ. Now the restrictions on T for the domain and range reveals that we can replace y by the element gyf and we are done. □

Proposition 1.7. Given a generating family {U_{i}} in a W*-category C, let M be the opposite of the von Neumann algebra ⨁_{i,j} Hom(U_{i}, U_{j}). Then the functor F : C → Hilb defined by

F (X) = ⨁_{i} L^{2}(X, U_{i})

with the obvious right action of M^{∘} on F (X) gives a fully faithful imbedding of C into the W*-category M-Mod.

Proof. The surjectivity of the functor maps on morphisms is a consequence of the previous lemma because local solutions admit a convergent net by the compactness argument.

On the other hand, if f : X → Y vanishes on the Hilbert space F (X), then fHom(U_{i}, X)φ_{i}^{1/2} = {0} for any i and any φ_{i} ∈ End(U_{i})_{∗}^{+}, whence fHom(U_{i}, X) = {0} for i. Since {U_{i}} is generating, this implies f = 0. □

Lemma 1.8. For a von Neumann algebra M, the following are equivalent. (i) M has the separable predual. (ii) M has a faithful normal representation on a separable Hilbert space. (iii) The standard space L^{2}(M) is separable.

Proof. Non-trivial is (i) ⇒ (iii).

Let {f_{i}}_{i≥1} be a countable dense set of M_{∗}. Then we can find a family {x_{i} ∈ M}_{i≥1} such that ∥x_{i}∥ ≤ 1 and |f_{i}(x_{i})| ≥ ∥f_{i}∥/2, which turns out to be total in M with respect to the weak*-topology. In fact, if not, we can find an element x ∈ M and a functional f ∈ M_{∗} satisfying f(x) = 1 and f(x_{i}) = 0 for i ≥ 1. For each integer n ≥ 1, choose i_{n} ∈ ℕ so that ∥f_{i_{n}} − f∥ ≤ 1/n. Then the inequality

\frac{1}{2}∥f_{i_{n}}∥ ≤ |f_{i_{n}}(x_{i_{n}})| = |f_{i_{n}}(x_{i_{n}}) − f(x_{i_{n}})| ≤ ∥f_{i_{n}} − f∥ ≤ \frac{1}{n}

implies ∥f_{i_{n}}∥ → 0 and hence f = lim_{n} f_{i_{n}} = 0, which contradicts the choice f(x) = 1.

The separability of the predual M_{∗} also ensures that we can find a faithful positive functional φ in M_{∗}. Now the set {x_{i}φ^{1/2}}_{i≥1} is total in L^{2}(M) by the Kaplansky’s density theorem and we are done. □ Remark .

(i) A similar argument shows that, if a Banach space X has the separable dual Banach space, then X itself is separable. Thus, the separabilty of L^{p}(M) for some 1 < p < +∞ implies the separabilty of L^{q}(M) with 1/p+1/q = 1 and then the predual is separable as a continuous image of L^{p}(M)×L^{q}(M). (ii) The argument in the above proof reveals that a von Neumann algebra is countably generated if it has the separable predual. The converse implication is, however, not true; the dual Banach space of the separable C*-algebra C[0, 1] is identified with the space M[0, 1] of complex measures in the interval [0, 1], which is not separable because ∥δ_{s} − δ_{t}∥ = 2 if s ≠ t in [0, 1]. Then the double dual von Neumann algebra M[0, 1]^{∗} is countably generated with the non-separable predual M[0, 1].

Definition 1.9. A W*-category is said to be locally separable if each Hom(X, Y) has the separable predual. A locally separable W*-category is said to be separable if it admits a generating family indexed by a countable set.

Given a W*-algebra M, we denote by Rep(M) the W*-category of normal *- representations of M on Hilbert spaces in a specified class. When M has the separable predual, we denote by SRep(M) the W*-category of normal *-representations of M on separable Hilbert spaces in a specified class. Proposition 1.10.

(i) A W*-category C is equivalent to Rep(M) for some W*-algebra M if and only if C admits subobjects (projections are associated to subobjects) and direct sums indexed by sets in a specified class.

(ii) A W*-category C is equivalent to SRep(M) for some W*-algebra M of separable predual if and only if C is separable and admits countably many direct sums as well as subobjects.

Proof. Clearly the condition is necessary. On the other hand, we have the fully faithful imbedding F : C → M-Mod by Proposition 1.7 and any M-module is isomorphic to F (X) for some object X in C by the condition (cf. the structure theorem of normal representations [1, 15]). □

For functors between W*-categories, it is reasonable to restrict to normal ones: a *-functor F : C → D of W*-categories is said to be normal if F is weak*-continuous on each Hom(X, Y).

Lemma 1.11 ([11, Prop. 4.7]). Let {U_{i}} be a generating family of objects in C. A *-functor F : C → D between W*-categories is normal if and only if it is normal on each W*-algebra End(U_{i}).

Proof. Assume that F is normal on End(U_{i})’s.

We will first show that F is normal on each End(X): for any φ ∈ End(F(X))_{∗}^{+}, the functional End(X) ∋ x → φ(F(x)) is weak*-continuous. To see this, it suffices to check the σ-strong continuity on the unit ball of End(X) by a well-known result ([1, Theorem I.3.1]). Let x_{α} → x in the σ-strong topology with x_{α} and x in the unit ball. Here choose a family of partial isometries {u_{i,j} : U_{i} → X} satisfying ∑ u_{i,j}u_{i,j}^{∗} = 1_{X}.

Then u_{i,j}(x_{α} − x)^{∗}(x_{α} − x)u_{i,j}^{∗} ∈ End(U_{i}) converges to 0 in the weak*-topology and hence by the normality of F on End(U_{i}), we have

F (u_{i,j})^{∗}F (x_{α} − x)^{∗}F (x_{α} − x)F (u_{i,j}) = F (u_{i,j}^{∗}(x_{α} − x)(x_{α} − x)u_{i,j}) → 0

in the weak*-topology, i.e., F (x_{α}−x)F (u_{i,j}) converges to 0 in the σ-strong topology for each index (i, j). Thus

(φ^{1/2}|F (x_{α} − x)F (u_{i,j}u_{i,j}^{∗})φ^{1/2}) → 0 as α → ∞

for each (i, j).

Since ∥F (x_{α} − x)∥ ≤ 2 and

∑(φ^{1/2}|(F (u_{i,j}u_{i,j}^{∗})φ^{1/2}) = (φ^{1/2}|φ^{1/2})

is finite, the usual argument shows that φ(F (x_{α} − x)) = (φ^{1/2}|F (x_{α} − x)φ^{1/2}) converges to 0, proving the normality of F on End(X).

Now let f_{α} be a net in Hom(X, Y) which converges to 0 in σ*-strong topology. Then f_{α}^{∗}f_{α} → 0 in weak*-topology and hence F (f_{α})^{∗}F (f_{α}) = F (f_{α}^{∗}f_{α}) → 0, i.e., F (f_{α}) → 0 in σ-strong topology. Thus the *-homomorphism

(\begin{matrix}End(X) & Hom(Y, X) \\ Hom(X, Y) & End(Y) \end{matrix}) → (\begin{matrix}End(F(X)) & Hom(F(Y), F(X)) \\ Hom(F(X), F(Y)) & End(F(Y)) \end{matrix})

between W*-algebras induced from F is σ-strongly continuous. Then it is weak*- continuous by the well-known result on topologies of W*-algebras ([1, Theorem I.3.1]) and we are done. □

Corollary 1.12. An equivalence of W*-categories is automatically normal.

Proof. This follows from the automatical normality of *-isomorphisms between W*- algebras (the nomality being the condition on the order structure of hermitian elements). □

Let M and N be W*-algebras and H be an N-M bimodule. Then we can define a normal *-functor Rep(M) → Rep(N) by

F (_{M}X) = _{N}H ⊗_{M} X

(see [14] further information on relative tensor products, cf. also [16, 17]).

Conversely we have the following reformulation of Rieffel’s theorem on Morita equivalences. Proposition 1.13. Any normal *-functor F : Rep(M) → Rep(N) (resp. F : SRep(M) → SRep(N) with M and N having separable preduals) is unitarily equivalent to the one associated to an N-M bimodule H (resp. a separable N-M bimodule).

The bimodule H is uniquely determined up to unitary isomorphisms by the functor F: H = _{N}F (_{M}L^{2}(M))_{M} with the right action of M = End(_{M}L^{2}(M))^{∘} on H given by the normal *-homomorphism End(_{M}L^{2}(M)) → End(_{N}F(_{M}L^{2}(M))) induced by the functor.

Proof. Let_{M}X be an M-module. By the structure theorem of normal *-homomorphism ([1, Theorem I.4.3]), we can find an index set I and a projection in Mat_{I}(M) so that _{M}X ≅ _{M}L^{2}(M)^{⊕I}p. We shall then construct a unitary intertwiner

NF (_{M}L^{2}(M)) ⊗_{M} X → _{N}F (_{M}X)

by the commutativity of

H ⊗_{M} X  ⟶  F (X)
↓  ↓
H ⊗_{M} L^{2}(M)^{⊕I}p ⟶ F (_{M}L^{2}(M)^{⊕I}p).

Here the bottom line is a unitary map given by the composition of

NF (_{M}L^{2}(M)) ⊗_{M} L^{2}(M)^{⊕I}p = _{N}F (_{M}L^{2}(M))^{⊕I}p
= _{N}F (_{M}L^{2}(M)^{⊕I})F (p) = _{N}F (_{M}L^{2}(M)^{⊕I}p).

By the multiplicativity of F on morphisms, the unitary map H ⊗_{M} X → F (X) is independent of the choice of an isomorphism _{M}X → _{M}L^{2}(M)^{⊕I}p and behaves naturally for intertwiners. □

A W*-category is said to be of type I if the matrix W*-algebra

(\begin{matrix}End(X) & Hom(Y, X) \\ Hom(X, Y) & End(Y) \end{matrix})

is of type I (in the sense of Murray-von Neumann) for any pair (X, Y) of objects.

The following is an easy reformulation of the structure theorem on W*-algebras of type I. Proposition 1.14 (cf. [11, Theorem 8.10]). Assume that a W*-category C is saturated for taking subobjects. Then C is of type I if and only if we can find a generating family {U_{i}} satisfying (i) End(U_{i}) is commutative and (ii) Hom(U_{i}, U_{j}) = {0} for i ≠ j.

2. W*-Tensor Categories

Recall that a tensor category is a linear category T together with a functor Φ : T × T → T and a natural isomorphism (the associativity constraint) a : Φ(Φ × idT) → Φ(idT×Φ) satisfying the so-called pentagonal condition. The tensor product notation is often used to denote the functor Φ: Φ(X, Y) = X⊗Y and Φ(f, g) = f ⊗g for objects X, Y and morphisms f, g in T.

When T is a *-category and Φ preserves the *-operation, i.e., (f ⊗ g)^{∗} = f^{∗} ⊗ g^{∗} for morphisms f, g in T, T is called a *-tensor category. A *-tensor category is called a C*-tensor category if it is based on a C*-category and the associativity constraint is unitary.

A W*-tensor category is, by definition, a C*-tensor category T with the tensor product functor Φ : T×T → T is binormal in the sense that it is separately normal on each variables.

Similar adjective definitions work for bicategories; we can talk about *-bicategories, C*-bicategories and W*-bicategories.

Recall that a bicategory consists of labels, A, B, C, …, categories Hom(A, B) indexed by pairs of labels, functors

Φ_{A,B,C} : Hom(B, A) × Hom(C, B) → Hom(C, A)

indexed by triplets of labels together with natural isomorphisms

a_{A,B,C,D} : Φ_{A,C,D}(Φ_{A,B,C} × id_{Hom(D,C)}) → Φ_{A,B,D}(id_{Hom(B,A)} × Φ_{B,C,D})

indexed by quadruplets of labels and satisfying the pentagonal relation.

The functor Φ_{A,B,C} is often denoted by the notation of composition, which reflects the view-point that a bicategory is a ‘categorization’ of hom-sets as homcategories.

Here is another view-point from which we regard the hom-category Hom(A, B) as a resemblance of the category of B-A bimodules (if labels represent algebras) with the notation Hom(A, B) = _{B}M_{A}. Then the functor Φ_{A,B,C} is consequently denoted by the tensor product notation: For objects X in _{A}M_{B} and Y in _{B}M_{C}, Φ_{A,B,C}(X, Y) is denoted by X ⊗_{B} Y. Similarly for morphisms.

A typical example of W*-bicategory is provided by bimodules with normal actions of W*-algebras.

A bicategory is said to be strict if the natural isomorphisms a_{A,B,C,D} are identities, i.e.,

Φ_{A,C,D}(Φ_{A,B,C} × id_{Hom(D,C)}) = Φ_{A,B,D}(id_{Hom(B,A)} × Φ_{B,C,D})

and a_{A,B,C,D} is the identity for each quadruplet.

A typical example of strict bicategory is provided by categories of functors: Let F, F′ : C → D and G, G′ : D → E be functors. Then, given natural transformations s : F → F′ and t : G → G′, we can associate the natural transformation G ∘ F → G′ ∘ F′ by the commutative diagram

G(F (X)) ⟶^{G(s_{X})} G(F′(X))
t_{F (X)}↓  ↓^{t_{F′(X)}}
G′(F (X)) ⟶_{G′(s_{X})} G′(F′(X)) .

Thus F = DFC, G = EGD and EG ⊗D FC = E(G ∘ F)C with the unit objects given by identity functors. Note that the commutativity of the above diagram expresses the identity (t ⊗D 1_{F′})(1_{G} ⊗D s) = (1_{G′} ⊗D s)(t ⊗D 1_{F}). The (strict) associativity for tensor products of morphisms is also immediate.

In particular, the category End(C) of functors from C into itself is a strict monoidal category. Example 2.1. If C is a one-object category, objects of End(C) are endomorphisms of the algebra A = End(pt) with hom-sets given by

Hom(ρ, σ) = {a ∈ A; aρ(x) = σ(x)a, ∀x ∈ A}.

The monoidal structure takes the form ρ ⊗ σ = ρ ∘ σ for ρ, σ ∈ End(A) and a ⊗ b = aρ(b) = ρ′(b)a for a ∈ Hom(ρ, ρ′), b ∈ Hom(σ, σ′).

When C is a W*-category, A is a W*-algebra and objects in End(C) are normal *-endomorphisms of A.

The tensor category End(C) is also denoted by End(A) (see [6] for more information on End(A)). Lemma 2.2 ([3, Theorem 7.13], [2, Lemma 2.1]). Let C, D be W*-categories and {U_{i}} be a generating family in C. Let U be the full subcategory of C consisting of objects in {U_{i}}.

Then the restriction (of functors and natural transformations)

Hom(C, D) ∋ F ↦ F |U ∈ Hom(U, D)

gives a fully faithful imbedding of W*-categories. Here Hom(C, D) and Hom(U, D) are W*-categories of normal *-functors and natural transformations.

More concretely, given a natural transformation {t_{i} : F (U_{i}) → G(U_{i})} between normal *-functors F |U and G|U, the natural transformation t_{X} : F (X) → G(X) is recovered by the formula

t_{X} = ∑_{i,j} F (u_{i,j})t_{i}G(u_{i,j})^{∗},

where {u_{i,j} : U_{i} → X} is a family of partial isometries satisfying ∑_{i,j} u_{i,j}u_{i,j}^{∗} = 1_{X}.

Proof. Let X be an object in C. Since {U_{i}} is a generating family, we can find a family {u_{i,j} : U_{i} → X} of partial isometries such that ∑_{i,j} u_{i,j}u_{i,j}^{∗} = 1_{X}.

Given a natural transformation t : F → G, we have t_{X}F (u_{i,j}) = G(u_{i,j})t_{i} for any (i, j) and then

t_{X} = ∑_{i,j} t_{X}F (u_{ij}u_{i,j}^{∗}) = ∑_{i,j} G(u_{i,j})t_{i}F (u_{i,j})^{∗}

by the normality of F. Thus the restriction is injective on natural transformations.

Conversely, given a natural transformation {t_{i} : F (U_{i}) → G(U_{i})} between F |U and G|U and an object X in C, write

t_{X} = ∑_{i,j} G(u_{i,j})t_{i}F (u_{i,j})^{∗},

which is a morphism in Hom(F(X), G(X)).

If Y is another object in C with a family {v_{k,l} : U_{k} → Y } of partial isometries satisfying ∑_{k,l} v_{k,l}v_{k,l}^{∗} = 1_{Y}, we associate another morphism t_{Y} : F (Y) → G(Y). Then, for any morphism f : X → Y in C, we have

G(f)t_{X} = ∑_{i,j} G(fu_{i,j})t_{i}F (u_{i,j})^{∗}

= ∑_{i,j} ∑_{k,l} G(v_{k,l}v_{k,l}^{∗}fu_{i,j})t_{i}F (u_{i,j})^{∗}

= ∑_{i,j} ∑_{k,l} G(v_{k,l})t_{k}F (v_{k,l}^{∗}fu_{i,j})F (u_{i,j})^{∗}

= t_{Y} F (f)

by the naturality of {t_{i}} and the normality of F, G.

If we take f = 1_{X} (X = Y particularly), then the above formula means that the morphism t_{X} is well-defined. Thus, the natural rtansformation {t_{X}} is recovered from {t_{i}}. □

If we restrict ourselves to W*-categories, normal *-functors and bounded natural transformations, then we obtain the strict W*-bicategory Funct. Recall that the tensor category End(M) in Example 2.1 is a part of Funct. In accordance with our separability notation, we denote by SFunct the W*-bicategory of normal functors between separable W*-categories. Moreover we have the W*-bicategory Funct_{I} of normal functors between W*-categories of type I.

We shall now give a fully faithful imbedding of Funct into the bicategory Bimod of W*-bimodules.

To this end, we first enlarge relevant W*-categories so that they admit generators; we shall take a generator UA for each W*-category A and define the W*-algebra A by A = End(UA) (B = End(UB) and son on). Then, by the imbedding theorem of W*-categories, we have fully faithful imbedding ΦA of A into the category of right A-modules:

ΦA : X ↦ L^{2}(X, UA)_{A}.

To each functor F : A → B, we associate the right B-module by

Φ(F) = L^{2}(F (UA), UB).

Note here that the functor F has the unique extension to the enlargements of A and B (cf. Lemma 2.2). The Hilbert space Φ(F) admits the left action of A by

aξ = F (a)ξ, a ∈ A, ξ ∈ L^{2}(F (UA), UB),

which clearly commutes with the right action of B. Thus Φ(F) is an A-B module.

Moreover, given a natural transformation t : F → F′ in Hom(A, B), the left multiplication of t_{UA} ∈ A defines an A-B intertwiner between Φ(F) and Φ(F′). In this way, we obtain a *-functor Φ : Funct → Bimod. By the previous lemma, Φ is fully faithful.

Given a functor F : A → B and an object X in A, we define a linear map m_{X,F} : L^{2}(X, UA) ⊗_{A} Φ(F) → L^{2}(F (X), UB) by

L^{2}(X, UA)⊗_{A}L^{2}(F (UA), UB) ∋ xφ^{1/2}⊗_{φ^{−1/2}}fψ^{1/2} ↦ F (x)fψ^{1/2} ∈ L^{2}(F (UA), UB),

which is clearly B-linear and isometric because of

(xφ^{1/2} ⊗_{φ^{−1/2}} fψ^{1/2}|xφ^{1/2} ⊗_{φ^{−1/2}} fψ^{1/2}) = (fψ^{1/2}|F (x^{∗}x)fψ^{1/2}) = ∥F (x)fψ^{1/2}∥^{2},

where φ ∈ A_{∗}^{+} and ψ ∈ B_{∗}^{+}.

To see the surjectivity of m_{X,F}, choose a family {x_{i} : UA → X} of partial isometries satisfying ∑_{i} x_{i}x_{i}^{∗} = 1_{X}. Then, for y : UB → F (X), the normality of F shows that

y = F (∑_{i} x_{i}x_{i}^{∗}) y = ∑_{i} F (x_{i})F (x_{i}^{∗})y = ∑_{i} F (x_{i})f_{i},

where f_{i} = F (x_{i}^{∗})y is in Hom(UB, F(UA)). Thus the image of m_{X,F} is dense in L^{2}(F (X), UB) and m_{X,F} gives a unitary map.

Let G : B → C be another normal *-functor. We can then define a unitary map m_{F,G} : Φ(F) ⊗_{B} Φ(G) → Φ(G ∘ F) by

m_{F (UA),G} : L^{2}(F (UA), UB) ⊗_{B} L^{2}(G(UB), UC) → L^{2}(GF (UA), UC),

which is A-C linear by the definition of m_{X,F} and the left action of A. The explicit form of these multiplication maps also shows the identity

m_{F (X),G}(m_{X,F} ⊗ 1_{Φ(G)}) = m_{X,GF} (1_{Φ(X)} ⊗ m_{F,G}).

In other words, the following diagram commutes.

Φ(X) ⊗_{A} Φ(F) ⊗_{B} Φ(G) ⟶ Φ(X) ⊗_{A} Φ(GF)
↓  ↓
Φ(F (X)) ⊗_{B} Φ(G) ⟶ Φ(GF (X)).

Summarizing the argument so far, we obtain the following. Theorem 2.3. The opposite of the bicategory Funct (SFunct or Funct_{I} respectively) is monoidally equivalent to the bicategory Bimod of W*-bimodules (SBimod of separable W*-bimodules or CBimod of W*-bimodules with actions of commutative W*-algebras respectively).

Proof. We have a fully faithful monoidal imbedding Funct → Bimod by the above argument, whose image covers every bimodule up to unitary isomorphisms by Proposition 1.13. □

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