Frobenius structures over Hilbert C*-modules

We study the monoidal dagger category of Hilbert C*-modules over a commutative C*-algebra from the perspective of categorical quantum mechanics. The dual objects are the finitely presented projective Hilbert C*-modules. Special dagger Frobenius structures correspond to bundles of uniformly finite-dimensional C*-algebras. A monoid is dagger Frobenius over the base if and only if it is dagger Frobenius over its centre and the centre is dagger Frobenius over the base. We characterise the commutative dagger Frobenius structures as finite coverings, and give nontrivial examples of both commutative and central dagger Frobenius structures. Subobjects of the tensor unit correspond to clopen subsets of the Gelfand spectrum of the C*-algebra, and we discuss dagger kernels.


Introduction
Categorical quantum mechanics [38] provides a powerful graphical calculus for quantum theory. It achieves this by stripping the traditional Hilbert space model of much detail. Nevertheless, the main examples remain based on Hilbert spaces, and relations between sets. The latter can be extended to take scalars in arbitrary quantales [2]. This article extends scalars in the former from complex numbers to arbitrary commutative C*-algebras. In other words, we study the monoidal category of Hilbert modules over a commutative C*-algebra. This provides a genuinely new model, that is interesting for various reasons.
• Just like commutative C*-algebras are dual to locally compact Hausdorff spaces, we prove that Hilbert modules are equivalent to bundles of Hilbert spaces over locally compact Hausdorff spaces (in Section 4). This gives a very naive model of algebraic quantum field theory [30,8]: instead of a single Hilbert space of states, we may have Hilbert spaces over every point of a base space that vary continuously. • We prove that the abstract scalars hide more structure than previously thought: subobjects of the tensor unit correspond to clopen subsets of the base space (see Section 3). This exposes a rich approach to causality [16,35], and opens the possibility of handling relativistic quantum information theory categorically. See also [22], which additionally characterises open subsets of the base space in purely categorical terms. This also invites questions about contextuality [1,3], that might now be addressed within categorical quantum mechanics using regular logic [37].
• E is complete in the norm x E = x | x A .
A function f : E → F between Hilbert A-modules is called bounded by f ∈ R when f (x) F ≤ f x E for all x ∈ E. It is called adjointable when there exists a function f † : F → E satisfying f (x) | y F = x | f † (y) E for all x ∈ E and y ∈ F .
Write Hilb bd C for the category of Hilbert C-modules and bounded C-linear functions. A dagger category is a category C with a functor † : C op → C satisfying X † = X on objects and f † † = f on morphisms. Write Hilb C for the dagger category of Hilbert C-modules and adjointable functions.
For so-called self-dual Hilbert A-modules E, F , these two types of morphisms coincide: Hilb bd A (E, F ) = Hilb A (E, F ) [43, 3.3-3.4]. If E and F are Hilbert C-modules over a commutative C*-algebra C, another Hilbert C-module E ⊗F is given by completing the algebraic tensor product E ⊗ C F with the following inner product and (right) C-module structure: For more details, see Appendix A. A dagger monoidal category is a monoidal category that is also a dagger category in which (f ⊗ g) † = f † ⊗ g † and the coherence isomorphisms are unitary.
Proposition 2.2. The category Hilb bd C is symmetric monoidal, and Hilb C is a symmetric monoidal dagger category.
Proof. If f : E 1 → E 2 and g : F 1 → F 2 are bounded maps between Hilbert Cmodules, we may define f ⊗g : E 1 ⊗F 1 → E 2 ⊗F 2 as the continuous linear extension of x ⊗ y → f (x) ⊗ g(y). If f, g were adjointable, then f ⊗ g is adjointable with adjoint f † ⊗ g † : Clearly id ⊗ id = id and (f • g) ⊗ (h • k) = (f ⊗ h) • (g ⊗ k), making the tensor product into a functor Hilb bd C × Hilb bd C → Hilb bd C . There are functions λ E : C ⊗E → E, ρ E : E⊗C → E, and α E,F,G : E⊗(F ⊗G) → (E ⊗ F ) ⊗ G, that continuously extend their algebraic counterparts. Thus they satisfy the pentagon and triangle equalities. It is clear that α E,F,G is unitary, but this is not immediate for λ E and ρ E . Recall the precise description of the tensor product in Appendix A: it involves the * -homomorphism C → L(E) that sends f to x → xf . This * -homomorphism is nondegenerate [42, page 5]: if f n is an approximate unit for C, and x ∈ E, then lim n x − xf n | x − xf n = lim , so that λ E is an isometric surjection C ⊗ E → E, and hence unitary [42,Theorem 3.5]. Similarly, there are unitaries σ E,F : E ⊗F → F ⊗E satisfying the hexagon equality. Thus Hilb bd C and Hilb C are symmetric monoidal with unit C. A zero object is an object that is initial and terminal at the same time. If a category has a zero object, there is a unique map 0 : E → F that factors through the zero object between any two objects. A category has finite biproducts when it has a zero object and any two objects E 1 , E 2 have a product and coproduct E 1 ⊕ E 2 with projections p n : E 1 ⊕ E 2 → E n and injections i n : E n → E 1 ⊕ E 2 satisfying p n • i n = id and p m • i n = 0 for m = n. A dagger category has finite dagger biproducts when it has finite biproducts and i n = p † n . Lemma 2.3. The category Hilb bd C has finite biproducts; Hilb C has finite dagger biproducts.
Proof. Clearly the zero-dimensional Hilbert C-module {0} is simultaneously an initial and terminal object. Binary direct sums [42, p5] are well-defined Hilbert C-modules. Since the category Vect of vector space has finite biproducts, the universal property is satisfied via the forgetful functor Hilb bd C → Vect, and it suffices to show that direct sums are well-defined on morphisms. Clearly, if f and g are bounded, then so is f ⊕ g. Similarly, f and g are adjointable maps between Hilbert C-modules, so is f ⊕ g: (f ⊕ g)(x 1 , y 1 ) | (x 2 , y 2 ) = f (x 1 ) | x 2 + g(y 1 ) | y 2 = x 1 | f † (x 2 ) + y 1 | g † (y 2 ) = (x 1 , y 1 ) | (f † ⊕ g † )(x 2 , y 2 ) .
Finally, the injections E → E ⊕ F given by x → (x, 0) are clearly adjoint to the projections E ⊕ F → E given by (x, y) → x.
Can we turn a Hilbert C-module into a Hilbert D-module? It turns out that such a change of base needs not just a map D → C to alter scalar multiplication, but also a map C → D to alter inner products. Recall that the multiplier algebra of a C*algebra A is the unital C*-algebra M  See also Appendix B.
Proposition 2.5 (Localization). Let f : C ։ D be a conditional expectation of a unital commutative C*-algebra C onto a unital commutative subalgebra D ⊆ C. There is a functor Loc f : Hilb bd C → Hilb bd D ; if f is strict then it is (strong) monoidal and restricts to a dagger functor Loc f : Hilb C → Hilb D .
Proof. The functor acts on objects E by localization [42, p57]: This clearly respects identity morphisms and composition, making Loc f a welldefined functor. It also preserves daggers when they are available: x C )f ( y | y C ) = 0 for any y ∈ F , and so x ⊗ y ∈ N E⊗F f . The adjoint of this map is given by These maps are clearly each others inverse.
For the unitary map D → Loc f (C), recall that Loc f (C) is the completion of by the Schwartz inequality for completely positive maps [44,Exercise 3.4] and [55,Theorem 1]. The required coherence diagrams are easily seen to commute. Thus Loc f is a (strong) monoidal functor.
Remark 2.6. Not every conditional expectation is strict. For example, take C = C 2 , and regard D = C as a subalgebra of C via z → (z, z). Then f (u, v) = u + v defines a conditional expectation f : C ։ D. But taking a = (1, 0), and b = (0, 1) We will be using Urysohn's lemma for locally compact spaces often [48, 2.12].
Lemma 2.7 (Urysohn). If X is a locally compact Hausdorff space, and K ⊆ V ⊆ X with K compact and V open, then there exists a continuous function ϕ : X → [0, 1] that is 1 on K and is 0 outside a compact subset of V .
Example 2.8. Any point t in a locally compact Hausdorff space X gives rise to a strict conditional expectation as follows. The completely positive map f : C 0 (X) → C evaluates at t. The * -homomorphism g : C → M (C 0 (X)) is determined by g(z)(ϕ) = zϕ. This clearly satisfies M (f ) • g(z) = z, and is strict because f is multiplicative. This localization at t ∈ X is the setting Proposition 2.5 will be applied in below. Remark 2.9. We will also use the previous lemma in the form of Tietze's extension theorem: if X is a locally compact Hausdorff space, and K ⊆ X compact, then any function in C(K) extends to a function in C 0 (X).

Scalars
Can we get more information about X from Hilb C0(X) by purely categorical means? We first investigate scalars: endomorphisms I → I of the tensor unit in a monoidal category. They form a commutative monoid. In the presence of biproducts, they form a semiring, and in the presence of a dagger, they pick up an involution [38].
Lemma 3.1. If X is a locally compact Hausdorff space, there is a * -isomorphism between scalars of Hilb C0(X) and C b (X), the bounded continuous complex-valued functions on X. The same holds for Hilb bd C0(X) . Proof. Recall that a closed ideal I ⊆ A of a C*-algebra is essential when aI = {0} implies a = 0 for all a ∈ A. We claim that C 0 (X) is an essential ideal of the C*-algebra L(C 0 (X)) of scalars of Hilb C0(X) . Seeing that C 0 (X) is an ideal in L(C 0 (X)) comes down to showing that for each f ∈ C 0 (X) and scalar s ∈ L(C 0 (X)), there exists g ∈ C 0 (X) such that for all h ∈ C 0 (X) we have hg = s(h)f ; choose g = s(f ). Seeing that the ideal is essential comes down to showing that for each scalar s ∈ L(C 0 (X)), if s(f )g = 0 for all f, g ∈ C 0 (X), then s = 0; given f ∈ C 0 (X), choosing g = s(f ) * shows that s(f ) * s(f ) = 0 implies s(f ) 2 = 0 and hence s(f ) = 0. It follows that the scalars of Hilb C0(X) are precisely the multiplier algebra of C 0 (X), which is C b (X), see [42, page 14-15].
It follows that for compact X, the scalars in Hilb C(X) simply form C(X) itself: any f ∈ C(X) gives a scalar by multiplication, and all scalars arise that way.

Remark 3.2. If
A is a noncommutative C*-algebra, then Hilb A is a perfectly well-defined dagger category. However, it cannot be monoidal with A as monoidal unit. That is, Proposition 2.2 does not generalise to noncommutative A. After all, there is an injective monoid homomorphism A ֒→ Hilb A (A, A) that sends a to b → ba, which contradicts commutativity of the latter monoid [41, Proposition 6.1].
Next we investigate subobjects. A (dagger) subobject of E is a monomorphism u : U → E (satisfying u † • u = id) considered up to isomorphism of U .
There is an isomorphism of partially ordered sets between clopen subsets of a locally compact Hausdorff space X and (dagger) subobjects of the tensor unit C 0 (X) in Hilb C0(X) .
Proof. We will first establish a bijection between clopen subsets of X and subobjects Given a clopen subset U ⊆ X, take E = {f ∈ C 0 (X) | f (U ) = 0}. This is a well-defined Hilbert C 0 (X)-module under the inherited inner product f | g = f * g.
Conversely, the image of a complemented subobject E C 0 (X) is a closed ideal of C 0 (X), and hence is of the form E = {f ∈ C 0 (X) | f (U ) = 0} for a closed subset U ⊆ X. Because the same holds for E ⊥ and C 0 (X) = E ⊕ E ⊥ , the closed subset U must in fact be clopen. Taking into account that subobjects are defined up to isomorphism, these two constructions are each other's inverse.
Finally, we prove that any subobject of C 0 (X) in Hilb C0(X) is complemented, so that every subobject is a dagger subobject by Lemma 2.3. See also [26, is a well-defined object in Hilb C0(X) , but the inclusion i : E ֒→ C 0 (X) is not necessarily a well-defined morphism. Suppose i were adjointable, so that f (t) * g(t) = f (t) * i † (g)(t) for all t ∈ X and f, g ∈ C 0 (X) with f (U ) = 0. If t ∈ U , Urysohn's lemma provides a continuous function f : X → [0, 1] such that f (U ) = 0 and f (t) = 1. Hence i † (g)(t) = g(t) for t ∈ X \ U . But to make i † well-defined, i † (g)(t) = 0 for t ∈ U , and i † (g) must be continuous. Letting g range over an approximate unit for C 0 (X) shows that U must be clopen.
It follows that there is a bijection between the clopen subsets of a locally compact Hausdorff space X and self-adjoint idempotent scalars in Hilb C0(X) .
Lemma 3.4. The monoidal categories Hilb C0(X) and Hilb bd C0(X) are monoidally well-pointed: if f, g : Proof. Any element x ∈ E gives rise to a morphism C 0 (X) → E given by ϕ → xϕ with adjoint x | − E .

Hilbert Bundles
Hilbert modules are principally algebraic structures. This section discusses a geometric description, in terms of bundles of Hilbert spaces. While most of this material is well-known [19], we state it in a way that is useful for our purposes. We will use the following definition of vector bundle in a Hilbert setting. (a) all fibres E t for t ∈ X are Hilbert spaces; (b) any t 0 ∈ X has an open neighbourhood U ⊆ X, a natural number n, and sections s 1 , . . . , s n : U → E such that: The dimension of the Hilbert bundle is the function that assigns to each t ∈ X the cardinal number dim(E t ). The Hilbert bundle is finite when its dimension function is bounded: Notice that a Hilbert bundle is a vector bundle. Notice also that any Hilbert bundle over a compact space X is necessarily finite: because X is covered by the open neighbourhoods of each t 0 ∈ X given by (b), there is a finite subcover, and the supremum of dim(E t ) is a maximum ranging over that finite index set and is therefore always finite. Definition 4.1 is a simplification of a few variations in the literature, that we now compare. The ε-tube around a local section s of a bundle p : E → X whose fibres are normed vector spaces is defined as A bounded section s is a section whose norm s = sup t∈X s(t) is bounded. (1) all fibres E t for t ∈ X are Banach (Hilbert) spaces; (2) addition is a continuous function {(x, y) ∈ E 2 | p(x) = p(y)} → E; (3) scalar multiplication is a continuous function C × E → E; (4) the norm is a continuous function E → C; (5) each x 0 ∈ E has a local section s with s(p(x 0 )) = x 0 , and x 0 has a neighbourhood basis T ε (s) ∩ E U for some neighbourhood U ⊆ X of p(x 0 ). We say p has locally finite rank when: (6) any t 0 ∈ X has a neighbourhood U ⊆ X and n ∈ N such that dim(E t ) = n for all t ∈ U . Finally, a field of Hilbert spaces is finite when the dimension of its fibres is bounded. -{s(t) | s ∈ ∆} ⊆ E t is dense for all t ∈ X; -for every s, s ′ ∈ ∆ the map x → s(x) | s ′ (x) Et is in C(X); -∆ is locally uniformly closed: if s ∈ t∈X E t and for each ε > 0 and each t ∈ X, there is an s ′ ∈ ∆ such that s(t ′ ) − s ′ (t ′ ) < ε on a neighbourhood of t, then s ∈ ∆; this is equivalent because we can recover E as t∈X E t with the topology generated by the basic open sets T ε (s) ∩ E U for ε > 0, and U ⊆ X open, and s ∈ ∆; this topology makes ∆ into the set of bounded sections; • [21, Definition 2.1] explicitly takes p to be open, which follows from (5), because it also considers a weaker version of (5); • [51, Definition 3.4] takes s in (5) to be a global section, because it also considers spaces X that are not functionally separated; for locally compact Hausdorff spaces X this is equivalent; • finite fields of Hilbert spaces are usually called uniformly finite-dimensional, and automatically have locally finite rank.
None of these variations matter for the material below. Proof. First assume that p : E → X is a field of Hilbert spaces of locally finite rank. Condition (a) of Definition 4.1 is precisely condition (1) of Definition 4.3.
X for the category of fields of Hilbert spaces and fibrewise linear bundle maps, HilbBundle bd X for the full subcategory of Hilbert bundles, and FHilbBundle bd X for the full subcategory of finite Hilbert bundles.
A bundle map f : p → p ′ between fields of Hilbert spaces is adjointable when it is adjointable on each fibre, and the map E ′ t ∋ y → f † (y) ∈ E t is continuous. Write FieldHilb X , HilbBundle X , and FHilbBundle X for the wide dagger subcategories of adjointable maps.
There is a version of the Serre-Swan theorem for Hilbert bundles, that we now embark on proving. If p : E ։ X is a field of Hilbert spaces, we say a function s : X → E vanishes at infinity when for each ε > 0 there is a compact U ⊆ X such that s(t) Et < ε for t ∈ X \ U .
Proposition 4.7. Let X be a locally compact Hausdorff space. There is a functor It restricts to a functor Γ 0 : FieldHilb X → Hilb C0(X) that preserves daggers.
Proof. Pointwise multiplication makes Γ 0 (p) into a right C 0 (X)-module. For s, s ′ ∈ Γ 0 (p) and t ∈ X, the nondegenerate inner product s | s ′ (t) = s(t) | s ′ (t) Et takes values in C 0 (X) by the Cauchy-Schwarz inequality. Finally, Γ 0 (p) is complete: if s n is a Cauchy sequence in Γ 0 (p), then s n (t) is a Cauchy sequence in E t for each t ∈ X, and hence converges to some s(t); since the convergence is uniform this defines a continuous function s : X → E, that satisfies p • s = 1 X and vanishes at infinity by construction. Thus Γ 0 (p) is a well-defined Hilbert C 0 (X)-module.
Let f : p → p ′ be a morphism of fields of Hilbert spaces.
. This is clearly C 0 (X)-linear, bounded, and functorial. It is also well-defined: if s ∈ Γ 0 (p), then f • s ≤ f s vanishes at infinity too.
A morphism f : p → p ′ in FieldHilb X is adjointable precisely when there is a bounded bundle map f † : p ′ → p that provides fibrewise adjoints: for all t ∈ X, s ∈ Γ 0 (p), and s ′ ∈ Γ 0 (p ′ ). That is, f is adjointable if and only if Γ(f ) is. Thus the functor Γ 0 preserves daggers. Proof. We first show that the functor Γ 0 is faithful. Suppose f = g, say f (x) = g(x) and p(x) = t. There exists a local continuous section s U : U → E of p over some open set U ⊆ X because p is a field of Hilbert spaces. Local compactness of X ensures there is a compact neighbourhood of t within U , which in turn contains an open neighbourhood V ⊆ X of t. Urysohn's lemma provides a continuous function r : X → [0, 1] that vanishes on X \ V and satisfies r(t) = 1. Now define s Next we show that the functor Γ 0 is also full. Suppose f : It is also fibrewise linear because if p(x) = p(y) then f (s x + s y )(p(x)) = f (s x+y )(p(y)). Moreover g is continuous by the definition of the topology on the field of Hilbert spaces E. Hence g is a well-defined morphism of fields of Hilbert spaces. Finally, if s ∈ Γ 0 (p) and t ∈ X, Finally, we show that Γ 0 is essentially surjective. Let H be a C 0 (X)-Hilbert module. Set E = t∈X Loc t (H), and let p be the canonical projection E ։ X. Because X is locally compact Hausdorff, it is compactly generated: a subset U ⊆ X is open if and only if U ∩ K is open in K for all compact subsets K ⊆ X. Hence the topology on X is determined by the topology of its compact subspaces. It follows from [17, II.1.15] and [51, Lemma 3.01(iv), Lemma 3.09, and Proposition 3.10] that there is a unique weakest topology on E making p into a field of Hilbert spaces.
As in Lemma 3.4, we may regard elements of H as adjointable maps C 0 (X) → H.
To complete the proof that Γ 0 is essentially surjective, it now suffices to show that {s x | x ∈ X} ⊆ Γ 0 (p) is dense. Let s ∈ Γ 0 (p) and ε. Then there exists a compact subset K ⊆ X such that s(t) < ε for t ∈ X \ K. Urysohn's lemma provides a function X → [0, 1] that vanishes at infinity such that f (t) = 1 for t ∈ K. By multiplying with this function it suffices to find x ∈ H so that the continuous local section s x : K → X satisfies s x (t) − s(t) < ε for t ∈ K. This can be done by the method of the proof of [51, Theorem 3.12]. Therefore s x (t) − s(t) < ε for all t ∈ X. Thus Γ 0 (p) ≃ H, and Γ 0 is essentially surjective.
Corollary 4.9. The category FieldHilb bd X is a symmetric monoidal category for any topological space X, where the tensor product of E → X and F → X is Proof. The tensor product E⊗E ′ becomes a well-defined object by letting ∆ E⊗F be the closure of the pre-Hilbert C 0 (X)-module of all finite sums of bounded sections vanishing at infinity [20,Section 18] or [13,Definition 15.3]. Via Lemma 4.5, this restricts to the monoidal product on FHilbBundle X as in the statement. Defining tensor products of morphisms is straightforward, as are associators and unitors, and checking the pentagon and triangle equations. The dagger is also clearly well-defined in FHilbBundle X , making it a symmetric monoidal dagger category. By construction of Proposition 2.2, the functors Γ 0 are (strong) monoidal.

Dual objects
In this section we investigate dual objects in the category Hilb C0(X) . From now on we will restrict ourselves to locally compact Hausdorff spaces X that are paracompact.
In other words, finitely presented projective Hilbert C-modules are orthogonal direct summands of C n . Any (algebraically) finitely generated projective Hilbert C-module is an example. When X is compact, a Hilbert C(X)-module is finitely presented projective if and only if it is finitely generated as a C(X)-module and a projective object in the category of C(X)-modules [58, Theorem 5.4.2].
Lemma 5.2. Any bounded C-linear map between finitely presented projective Hilbert C-modules is adjointable.
and hence an m-by-n matrix of bounded C-linear maps C → C. But the latter are adjointable by Lemma 3.1. So, by Lemma 2.3, also g is adjointable. But then It follows that the full subcategories of Hilb C and Hilb bd C of finitely presented projective Hilbert C-modules coincide. We write FHilb C for this category.
It is clearly sesquilinear and positive semidefinite by Lemma 3.1. It is also nondegenerate: We call E * the dual Hilbert C-module of E. There is also a categorical notion of dual object.
Definition 5.4. Objects E, E * in a monoidal category are called dual objects when there are morphisms η : I → E * ⊗ E and ε : E ⊗ E * → I making the following diagrams commute: In a symmetric monoidal dagger category, dual objects are dagger dual objects when If an object has a (dagger) dual, then that dual is unique up to unique (unitary) isomorphism. We now show that dual Hilbert C-modules are dual objects in the finitely presented projective case over a paracompact space X.
Theorem 5.5. Let X be a paracompact locally compact Hausdorff space X. For a Hilbert C 0 (X)-module E, the following are equivalent: (a) E has a dagger dual object in Hilb C0(X) ; (b) E ≃ Γ 0 (p) for a finite Hilbert bundle p; (c) E is finitely presented projective.
Proof. (a)⇒(b): Assume that E has a dagger dual object E * . Then also all its localisations E t = Loc t (E) are dagger dualisable, and so [3,Corollary 19] each E t is a finite-dimensional Hilbert space. Now regard E as a field of Hilbert spaces over X as in Theorem 4.8. Then η : C 0 (X) → E * ⊗ E and ε : E ⊗ E * → C 0 (X) are bundle maps and hence bounded. But then dim (b)⇒(c): Let p : E ։ X be a finite Hilbert bundle. Then every t ∈ X has a neighbourhood U t and a homeomorphism g t : C nt × U t → p −1 (U t ) that is fibrewise unitary. This forms a cover {U t } of X. Because X is paracompact, we may pick a locally finite refinement U j , and a partition of unity f j : X → [0, 1] subordinate to it: f j vanishes outside U j and j f j (t) = 1 for all t ∈ X. Because p is finite, the numbers n t are bounded by some n ∈ N, and the functions g t extend to continuous ; these are still continuous and fibrewise coisometric.
First, notice that C 0 (X) is its own dagger dual object, and therefore so is C 0 (X) n . Explicitly, because it holds at each t ∈ X and therefore globally by Theorem 4.8. It follows that the idempotent (id ⊗ ε) Theorem 5.6. There is a monoidal equivalence of compact (dagger) categories for any paracompact locally compact Hausdorff space X.
Proof. By Theorem 5.5, the monoidal subcategory FHilb C0(X) of Hilb C0(X) is compact. Because (strong) monoidal functors preserve dual objects, the image under Γ 0 in FieldHilb (bd) X is also compact by Corollary 4.9. The dual of E → X is given by (E * ) t = (E t ) * (with topology given by [17,II.1.15]). By Lemma 4.5 the functor Γ 0 therefore restricts as in the statement, and is an equivalence by Theorem 4.8.
It follows that FHilb C0(X) is a symmetric monoidal dagger category. It is also easy to see that FHilb C0(X) has finite dagger biproducts.

Frobenius structures
We now start the study of dagger Frobenius structures in the category Hilb C0(X) . Many of the results below hold for arbitrary (non-dagger) Frobenius structures, but we focus on dagger Frobenius structures, and leave open the generalisation to purely algebraic proofs. Definition 6.1. A dagger Frobenius structure in Hilb C is a Hilbert C-module E with morphisms η : C → E and µ : E ⊗ E → E satisfying: The dagger Frobenius structure (E, µ, η) is called: Dagger Frobenius structures are their own dagger dual, with unit µ † • η : I → E ⊗ E. Hence dagger Frobenius structures in Hilb C0(X) live in FHilb C0(X) for paracompact X.
Remark 6.2. For C = C, special dagger Frobenius structures correspond to finitedimensional C*-algebras [56]. Any dagger Frobenius structure E in Hilb C has an (anti-linear) involution i : [56, 4.4]. We will occassionally use the graphical calculus, where µ is drawn as , and η as , dagger becomes horizontal reflection, tensor product becomes drawing side by side, and composition becomes vertical stacking. For more details we refer to [49]. The involution is thus drawn as follows. (2) One of our first aims is to generalise this to arbitrary C. Definition 6.3. A * -homomorphism between Frobenius structures in Hilb C is a morphism f that preserves the involution (2) and the multiplication: Frob C for the category of specialisable dagger Frobenius structures in Hilb C with * -homomorphisms, and SFrob C for the full subcategory of special dagger Frobenius structures.
Proposition 6.4. The categories Frob C and SFrob C are monoidally equivalent (via the inclusion of the latter into the former).
Proof. It follows from Proposition 2.5 that there is a (strong) monoidal dagger functor Hilb C0(X) → Hilb for each t ∈ X. Such functors preserve dagger Frobenius structures, as well as speciality and specialisability. Example 6.6. Any finite-dimensional C*-algebra A is a special dagger Frobenius structure in FHilb, and gives rise to a special dagger Frobenius structure C 0 (X, A) in Hilb C0(X) over a locally compact Hausdorff space X. Frobenius structures isomorphic to one of this form are called trivial.
In particular, M n (C 0 (X)) ≃ C 0 (X, M n ) is a special dagger Frobenius structures in Hilb C0(X) . It follows from Lemma 2.3 that direct sums of such matrix algebras are special dagger Frobenius structures in Hilb C0(X) , too, and up to isomorphism this accounts for all trivial Frobenius structures.
Example 6.7. If X is a paracompact locally compact Hausdorff space, and E is a finitely presented projective Hilbert C 0 (X)-module, then L(E) = E * ⊗ E is a specialisable dagger Frobenius structure.
Proof. This follows from Theorem 5.5 and [15, Proposition 2.11]; take multiplication Notice that trivial Frobenius structures in Hilb C0(X) in general need not be direct summands of C 0 (X) n . There are endomorphism algebras that are not direct sums of matrix algebras in Hilb C(X) . For example, take X = 2. Then M n (C) is a corner algebra of M n (C 2 ), but it is not isomorphic to a direct summand of the latter. It is nevertheless the endomorphism algebra of the Hilbert C(X)-module C n , but still trivial as a Frobenius structure.
The rest of this section develops nontrivial examples of commutative and central dagger Frobenius structures in Hilb C0(X) . We need some topological preliminaries. Definition 6.8. A bundle is a continuous surjection p : Y ։ X between topological spaces. Write Y U = p −1 (U ) for U ⊆ X, and Y t = p −1 (t) for the fibre over t ∈ X. The bundle is finite when there is a natural number n such that all fibres have cardinality at most n. A (local) section over U is a continuous function s : U → Y satisfying p • s = id U ; a global section is a section over X. A bundle is a covering when every t ∈ X has an open neighbourhood U ⊆ X such that Y U is a union of disjoint open sets that are each mapped homeomorphically onto U by p. Example 6.9. Write S 1 = {z ∈ C | |z| = 1} for the unit circle. For any natural number n, the map p : S 1 → S 1 given by p(z) = z n is a finite covering.
The map z → z n is also a finite covering on the unit disc {z ∈ C | |z| ≤ 1}.
Example 6.11. If p : Y ։ X is a covering between locally compact Hausdorff spaces, then C 0 (Y ) is a right C 0 (X)-module with scalar multiplication C 0 (Y ) × C 0 (X) → C 0 (Y ) given by g · f : y → g(y) f (p(y)).
If p is finite, then C 0 (Y ) is a Hilbert C 0 (X)-module under Proof. The module axioms are clearly satisfied. The inner product f | g is welldefined when p has finite fibres; it is continuous because p is a covering, and vanishes at infinity because f and g do so and p is finite. It is clearly sesquilinear and positive definite. We need to prove that C 0 (Y ) is complete in this inner product. Let {g n } be a Cauchy sequence in C 0 (Y ). Say that the fibres of p have cardinality at most N . For ε > 0 and large m, n: for all y ∈ Y , so {g n (y)} is a Cauchy sequence in C. Because this convergence is uniform, we obtain a continuous function g ∈ C 0 (Y ) satisfying g(y) = lim g n (y) pointwise, and hence also lim n g n = g in C 0 (Y ). Lemma 6.12. If p : Y ։ X is a finite covering between locally compact Hausdorff spaces, then the Hilbert C(X)-module C(Y ) of Example 6.11 is a nondegenerate special dagger Frobenius structure in Hilb C(X) .
These are clearly equal to each other.
We will see in Section 8 below that in fact every commutative special dagger Frobenius structure in Hilb C0(X) is of the form of the previous lemma.
Example 6.13. Applying Lemma 6.12 to the double cover of Example 6.9 with It clearly consists of two homeomorphic connected components, one of which is the diagonal {(a, a) ∈ S 1 × S 1 | a ∈ S 1 }, as in Lemma 6.10, and the other one is {(a, −a) ∈ S 1 × S 1 | a ∈ S 1 }. This enables the definition of the comultiplication µ † as a map of C(X)-modules. However, as the double cover p is not trivial, it has no global sections e i . Therefore there cannot be a description of the comultiplication µ † in terms of e i → e i ⊗ e i as in the case X = 1; this is only the case over local neighbourhoods of points t ∈ X.
Remark 6.14. The previous example shows that not every special dagger Frobenius structure in Hilb C0(X) is of the form End(E i ) for projective Hilbert C 0 (X)modules E i . If that were the case, since the rank of the previous example can uniquely be written as a sum of squares as 2 = 1 + 1, then it would have to be a direct sum of two Hilbert C 0 (X)-modules of rank 1. But then it would have nontrivial idempotent central global sections, which it does not.
We end this section with nontrivial examples of central (noncommutative) special dagger Frobenius structures in Hilb C . Definition 6.15. A dagger Frobenius structure E in Hilb C is central when Z(E) = {x ∈ E | ∀y ∈ E : xy = yx} = 1 E · C and it is faithful as a right C-module: f ∈ C vanishes when 1 E f = 0 (or equivalently, when xf = 0 for all x ∈ E). Example 6.16. Write D = {z ∈ C | |z| ≤ 1} for the unit disc, S 1 = {z ∈ C | |z| = 1} for the unit circle, and X = S 2 = {t ∈ R 3 | t = 1} for the 2-sphere. Let n ≥ 2 be a natural number, and consider Then E is a C(X)-module via the homeomorphism X ≃ D/S 1 ; more precisely, if q : D → X is the quotient map, then multiplication E × C(X) → E is given by (x · f )(z) = x(t) · f (q(t)). Moreover, E is a Hilbert C(X)-module under x | y (t) = tr(x(t) * y(t)). Finally, pointwise multiplication makes E a nontrivial central special dagger Frobenius structure in Hilb C(X) .
Proof. See [6, Theorem 5.8] for the fact that E is the Hilbert module of sections of a nontrivial finite C*-bundle. Use Theorem 7.7 below to see that it is a nontrivial special dagger Frobenius structure.
To see that E is central, notice that because if y ∈ Z(E) does not take values in Z(M n ) at some z ∈ D, there are three cases: if |z| < 1, then x does not commute with some y ∈ E at z; if z = 1, then x does not commute with some y ∈ E at z; and if |z| = 1, then it also does not take values in Z(M n ) at z = 1.

C*-bundles
Next we apply the bundle perspective to dagger Frobenius structures. They form C*-algebras themselves, as the following lemma shows. Proof. First of all, E is clearly a Banach space, as an object in Hilb C0(X) . It is also an algebra with multiplication µ : E ⊗ E → E. In fact, it becomes a Banach algebra because µ † µ is a projection by speciality [3, Lemma 9]: Finally, this satisfies the C*-identity because it does so locally at each t ∈ X by Corollary 6.5: The outer equalities use Theorem 5.6.
The C*-algebras induced by dagger Frobenius structures have more internal structure: they are in fact a bundle of C*-algebras, as made precise in the following definition.
(1) all fibres E t for t ∈ X are finite-dimensional (commutative) C*-algebras; (2) any t 0 ∈ X has an open neighbourhood U ⊆ X, a finite-dimensional C*algebra A, and a homeomorphism ϕ : U × A → E U , such that the map ϕ(t, −) : A → E t is a * -isomorphism for each t ∈ U ; (3) the dimension of the fibres is bounded.
If X is compact, then condition (3) is superfluous. Proof. Let p : E → X be a finite C*-bundle. The fibre over t 0 ∈ X is a finitedimensional C*-algebra, and hence canonically of the form M n1 ⊕ · · · ⊕ M n k up to isomorphism. It is a finite-dimensional Hilbert space under the inner product (a 1 , . . . , a k ) | (b 1 , . . . , b k ) = tr(a * 1 b 1 ) + · · · + tr(a * k b k ). Condition (2) also gives an open neighbourhood U of t 0 , a finite-dimensional C*algebra A = M n1 ⊕ · · · ⊕ M n k , and a homeomorphism ϕ : U × A → E U . Take n = dim(A), and let the standard matrix units constitute an orthonormal basis e 1 , . . . , e n of A. Define continuous sections s i : U → E by s i (t) = ϕ(t, e i ). Now {s i (t)} forms an orthonormal basis of E t for all t ∈ U by (2).
Just as Definition 4.1 was a simplification of Definition 4.3, the previous definition is a simplification of the notion of field of C*-algebras in the literature [23,24,21,53,54,9]: a field p : E ։ X of Banach spaces where each fibre is a C*algebra, where multiplication gives a continuous function {(x, y) ∈ E 2 | p(x) = p(y)} → E, and where involution gives a continuous function E → E. A field of C*-algebras is uniformly finite-dimensional when each fibre is finite-dimensional, and the supremum of the dimensions of the fibres is finite. Lemma 7.4. A finite C*-bundle is the same thing as a uniformly finite-dimensional field of C*-algebras.
Proof. By Lemma 7.3, any finite C*-bundle is a finite Hilbert bundle, and hence a finite field of Banach spaces of locally finite rank by Lemma 4.5. Similarly, multiplication and involution are continuous functions by the same argument as in the proof of Lemma 4.5.
The converse is similar to Lemma 4.5 for the most part. Let p : E ։ X be a uniformly finite-dimensional field of C*-algebras. Let t 0 ∈ X. Take A = E t0 , say of the form M n1 ⊕ · · · ⊕ M n k , and let x 1 , . . . , x n be the orthonormal basis of A constituted by standard matrix units. Condition (5) gives sections s i : U → X with s i (t 0 ) = x i . Take U = U 1 ∩ · · · ∩ U n ∩ {t ∈ X | {s i (t)} linearly independent}; this is an open subset of X. Define ϕ : U × A → E U by linearly extending (t, s i ) → s i (t). This is a homeomorphism, and ϕ(t, −) is a * -isomorphism by construction.
Example 7.5. If X is a paracompact locally compact Hausdorff space, and E a finitely presented projective Hilbert C 0 (X)-module, then L(E) = E * ⊗ E ≃ Hilb C0(X) (E, E) is a finite C*-bundle.
Proof. Notice that Hilb C0(X) is a C*-category [28,Example 1.4], and a monoidal category by Proposition 2.2. Thus it is a tensor C*-category, and hence a 2-C*category (with a single object). The result follows from [60, Proposition 2.7]. Definition 7.6. A morphism of finite C*-bundles is a bundle map that is fibrewise a * -homomorphism. Write FCstarBundle X for the category of finite C*-bundles with their morphisms.
Theorem 7.7. There is an equivalence of monoidal dagger categories for any paracompact locally compact Hausdorff space X.
Proof. By Proposition 6.4, we may use SFrob C0(X) instead of Frob C0(X) . Write ∆ for the adjoint of Γ 0 of Theorem 5.6. Let (E, µ, η) be a special dagger Frobenius structure in Hilb C0(X) . Equivalently, the embedding R : E → L(E) E E E * and the involution i : E → E * of equation (2) satisfy i•R = R * •i [34, Corollary 9.7]. By Example 7.5, ∆(E * ⊗ E) is a finite C*-bundle over X. Now, because both i and R are defined purely in terms of tensor products, composition, and dagger, the above equations also hold fibrewise by Theorem 5.6. Hence ∆(E) is a finite Hilbert bundle, which embeds into ∆(E * ⊗ E) with ∆(R), and is closed under the involution ∆(i). We conclude that ∆(E) is in fact a finite C*-bundle. The same reasoning establishes the converse: if p is a finite C*-bundle, then Γ 0 (p) is a special(isable) dagger Frobenius structure in Hilb C(X) . Compare [20,Definition 21.7]. See also [59].
The rest of this section derives from the previous theorem some corollaries of interest to categorical quantum mechanics.
Recall that the phase group of a dagger Frobenius structure E consists of all morphisms φ : A group bundle is a bundle E ։ X whose every fibre is a group, and such that each point t 0 ∈ X has a group G and a neighbourhood on which fibres are isomorphic to G.
Corollary 7.8. The phase group of a dagger Frobenius structure E in FHilb C0(X) is a group bundle U (E) ։ X whose fibres are the unitary groups of fibres of E.
Proof. The general case follows easily from the case X = 1, which is a simple computation [38].
For example, for the trivial Frobenius structure C 0 (X) in FHilb C0(X) , the phase group is the trivial bundle U (1) × X ։ X. Definition 7.9. A completely positive map between finite C*-bundles over X is a bundle map that is completely positive on each fibre. Write FCstarBundle cp X for the category of finite C*-bundles and completely positive maps.
In general, there is a construction that takes a monoidal dagger category C to a new one CP[C], see [15]. Objects in CP[C] are special dagger Frobenius structures in C.
for some object G and some morphism g : Proof. The correspondence on objects is already clear from Theorem 7.7. By definition, morphisms in CP(Hilb C0(X) ) are morphisms in Hilb C0(X) that satisfy (3). Because the equivalence is monoidal, these correspond to morphisms between finite C*-bundles that satisfy the same condition. By Theorem 7.7 the condition also holds in each fibre. Hence [15] these morphisms are completely positive maps in each fibre.

Commutativity
By Theorem 7.7, commutative special dagger Frobenius structures in Hilb C0(X) correspond to commutative finite C*-bundles over X. In this section we phrase that in terms of Gelfand duality, generalizing [45]. We first reduce to nondegenerate Frobenius structures. Lemma 8.1. Let X be a locally compact Hausdorff space. Any (specializable) dagger Frobenius structure in Hilb C0(X) is determined by a nondegenerate (specializable) one in Hilb C0(U) for a clopen subset U ⊆ X Proof. Let E ∈ Frob C0(X) . By Theorem 7.7 it corresponds to a finite C*-bundle. (Note that this does not need paracompactness.) So t → dim(E t ) is a continuous function X → N, and U = {t ∈ X | dim(E t ) > 0} is clopen. We need to show that the restricted finite C*-bundle over U is nondegenerate. Note that dim(E t ) is the value of the scalar η † • µ • µ † • η ∈ C b (X) at t. In particular, it takes values in N, and if t ∈ U , then it is invertible.
Next, we show that any nondegenerate specialisable dagger Frobenius structure in Hilb C0(X) is induced by a finite bundle p : Y ։ X.
Proposition 8.2. Let X be a paracompact locally compact Hausdorff space. Any commutative nondegenerate specialisable dagger Frobenius structure in Hilb C0(X) is isomorphic as a * -algebra to C 0 (Y ) for some locally compact Hausdorff space Y through a finite bundle p : Y ։ X.
Proof. By Proposition 6.4 we may assume that the given dagger Frobenius structure E is special. It then follows from Lemma 7.1 that E is of the form C 0 (Y ) for some locally compact Hausdorff space Y . Applying Lemmas 3.4 and 3.1 to the unit law µ • (η ⊗ η) = η • λ shows that the map η : C 0 (X) → C 0 (Y ) is multiplicative. Being a morphism in Hilb C0(X) it is also additive. It preserves the involution by definition of dual objects. Hence η is a * -homomorphism, which is nondegenerate as in Proposition 2.2. Therefore η is of the form −•p : C 0 (X) → C 0 (Y ) for a continuous map p : Y → X. Because η : C 0 (X) → C 0 (Y ) is injective by nondegeneracy, p is surjective.
The complex vector space C 0 (p −1 (t)) contains at least as many linearly independent elements as distinct elements y i of p −1 (t), namely the continuous extension of y j → δ ij by Tietze's extension theorem. But C 0 (Y ) is finitely presented projective as a C 0 (X)-module by Theorem 5.5, so there is a natural number n and some E ∈ FHilb C0(X) such that for each t ∈ X we have C 0 (p −1 (t)) ⊕ E t ≃ C n by localising as in Proposition 2.5. Thus dim(C 0 (p −1 (t))) ≤ n, and hence p −1 (t) has cardinality at most n, for each t ∈ X.
To show that p must in fact be a finite covering, we first prove p is an open map. Proof. By Theorem 7.7 a specialisable dagger Frobenius structure E in Hilb C0(X) corresponds to a finite C*-bundle, whose fibres have uniformly bounded dimension. We need to show that p is open; suppose for a contradiction that it is not. Let V ⊆ Y be an open set such that p(V ) ⊆ X is not open. Fix a limit point t 0 ∈ p(V ) of X \ p(V ), and pick s 0 ∈ V with p(s 0 ) = t 0 . Urysohn's lemma now provides a continuous function y : Y → [0, 1] with y(s 0 ) = 1 that vanishes at infinity and outside V . Now η † (y)(t) = 0 if and only if p(s)=t y(s) = 0 for all t ∈ X, so η † (y) vanishes on X \ p(V ). But η † (y)(t 0 ) > 0 by Lemma  Next, we show that p : Y ։ X must also be a closed map. When Y is compact and X is Hausdorff this is automatic because continuous images of compact spaces are compact and compact subsets of Hausdorff spaces are closed; we show that it also holds when Y is only locally compact. Proof. Suppose V ⊆ Y is closed. We want to show that U = p(V ) ⊆ X is closed. Let t α be a net in U that converges to t ∈ X. Pick s α in p −1 (t α ) ∩ V . Say p −1 (t) ∩ V = {s 1 , . . . , s n }. Pick compact neighbourhoods V i ⊆ V of s i (possible because Y is locally compact). Then s α is eventually in i V i (because this finite union is compact). So a subnet of s α converges to one of the s i ∈ V . But then, by continuity of p, a subnet of t α converges to p(s i ) ∈ U . But then t = p(s i ) is in U (because X is Hausdorff).
Finally, we can show that the p must be a finite covering.
Proposition 8.5. Let X be a paracompact locally compact Hausdorff space. Any nondegenerate commutative specialisable dagger Frobenius structure in Hilb C0(X) is of the form C 0 (Y ) for a finite covering p : Y ։ X.
Proof. We simplify [45,Theorem 4.4]. By Theorem 5.5, C 0 (Y ) ⊕ E ≃ C 0 (X) n for some n ∈ N and E ∈ FHilb C0(X) . Hence k t = | dim(C 0 (Y ) t )| = |p −1 (t)| ≤ n for all t ∈ X. Because t → k t is a continuous function X → N by Remark 4.2, the subsets X k = {t ∈ X | k t = k} ⊆ X are closed and open for k = 1, . . . , n. That is, X = X 1 ⊔ · · · ⊔ X n is a finite disjoint union of clopen subsets, on each of which the fibres of p have the same cardinality. Now for t ∈ X, by Lemma 8.4 and [45, Lemma 2.2], we can choose a neighbourhood U ⊆ X over which p −1 (U ) is a disjoint union of open subsets V 1 , . . . , V k ⊆ Y that each contain a preimage of t. By replacing U by p(V i ), and intersecting V i with p −1 (p(V j )), we may assume that each p : V i → U is surjective. But then, because all fibres have the same size, it cannot happen that one of the V i has two points of a fibre, as then another V j must have none (because there are only finitely many points in the fibre), whence p : V j → U would not be surjective. So each p : V i → U is a closed and open bijection, and hence a homeomorphism.
This completely characterises commutative specialisable dagger Frobenius structures in Hilb C0(X) for paracompact connected X. The category Covering X is symmetric monoidal under Cartesian product. Write cFrob C0(X) for the full subcategory of nondegenerate commutative objects in Frob C0(X) , and write Covering X for the category of finite coverings and bundle maps. Theorem 8.6. For any paracompact locally compact Hausdorff space X there is an equivalence cFrob C0(X) ≃ Covering X of symmetric monoidal dagger categories.
Proof. Combine Lemma 8.3 and Lemma 6.12 to establish the equivalence. Monoidality follows because the tensor product is the coproduct of commutative C*-algebras, and so C 0 (X) ⊗ C 0 (Y ) ≃ C 0 (X) + C 0 (Y ) ≃ C 0 (X × Y ) by duality.
Alternatively, we could include degenerate objects in cFrob C0(X) and objects p in Covering X to be non-surjective.

Transitivity
In this section we reduce the study of special dagger Frobenius structures to the study of central ones and commutative ones, by proving a transitivity theorem that adapts [18,Theorem II.3.8] to the setting of dagger Frobenius structures. We start with combining Frobenius structures E over Z and Z over C into a Frobenius structure E over C.
Lemma 9.1. Let C and Z be commutative C*-algebras with paracompact spectrum. If E is a nondegenerate (specialisable) dagger Frobenius structure in Hilb Z , and Z is a nondegenerate (specialisable) dagger Frobenius structure in Hilb C , then E is a nondegenerate (specialisable) dagger Frobenius structure in Hilb C .
Proof. By Theorem 7.7, there is a finite C*-bundle p : E ։ Spec(Z), and a commutative finite C*-bundle Z ։ X = Spec(C). By Theorem 8.6, the latter corresponds to a branched covering q : Spec(Z) → X. We will show that r = q • p is a finite C*bundle E ։ X. First of all, the fibre of r over t ∈ X is r −1 (t) = u∈q −1 (t) p −1 (u), a finite direct sum of finite-dimensional C*-algebras, and hence a finite-dimensional C*-algebra. Now let t 0 ∈ X. Say q −1 (t 0 ) = {u 1 , . . . , u n } ∈ Spec(Z). Pick open neighbourhoods U i ⊆ Spec(Z) of u i , finite-dimensional C*-algebras A i , and homeomorphisms a 1 ), . . . , ϕ n (u n , a n ) where a = (a 1 , . . . , a n ), and t = q(u i ) for u i ∈ U i . Then, for each t ∈ V , say t = q(u i ) with u i ∈ U i , the function is a * -isomorphism. It is clear that r is nondegenerate when p and q are, and that r is specialisable when p and q are.
Next, we consider the converse: if E is a Frobenius structure over C, does it decompose into Frobenius structures E over Z and Z over C? Our proof of the former below will use the following algebraic lemma. Proof. Specialisable dagger Frobenius structures are symmetric [15,Proposition 2.7], and adapting [4] to monoidal categories then shows that (E, µ, η) is strongly separable [18]. By [31,Theorem 1], there is a direct sum E ≃ Z(E) ⊕ [E, E] of C-modules. It now suffices to prove that this direct sum is orthogonal, as it then follows that both summands are Hilbert modules [58,Section 15.3]. But if z ∈ Z(E) and x, y ∈ E, then where the second equation uses that dagger Frobenius structures are H*-algebras; see [2,Lemma 5], which does not depend on commutativity.
It follows that the projection p 1 : E → Z(E) is cyclic: p 1 (xy) = p 1 (yx). It also follows that if E is a specialisable dagger Frobenius structures, its centre Z(E) is a well-defined Hilbert module. We leave open the question whether special(isable) dagger Frobenius structures in arbitrary monoidal dagger categories correspond to monoid-comonoid pairs E with E ≃ Z(E) ⊕ F a dagger biproduct, where Z(E) is defined by an equaliser. Example 9.3. Consider the special dagger Frobenius structure E = M n in Hilb. Then Z(E) = C, and [E, E] = {y ∈ M n | tr(y) = 0} (see [5]) and indeed n tr(x)) = 1 n tr(x) + x − 1 n tr(x) = x. Any special dagger Frobenius structure E in Hilb C0(X) is a C*-algebra according to Lemma 7.1. Therefore so is Z(E), and it makes sense to talk about the monoidal category Hilb Z(E) . Lemma 9.4. If E is a special dagger Frobenius structure in Hilb C0(X) , then it is also an object in Hilb Z(E) .
Proof. First of all, E is certainly a Z(E)-module; let us verify that it is a Hilbert Z(E)-module. As the inner product, take x | y = p 1 (x * y), using the projection p 1 : E → Z(E) induced by Lemma 9.2, and the involution (2). By Lemma 9.2, p 1 has norm one, and hence is a conditional expectation [55]. Thus the inclusion p † 1 : Z(E) → E is a * -homomorphism, and p 1 is completely positive. Because (completely) positive maps preserve the involution [50, p2], we have y | x * = p 1 (y * x) * = p 1 (x * y) = x | y for x, y ∈ E. Because p 1 is Z(E)-linear, also x | y + y ′ = x | y + x | y ′ and x | yz = x | y z for x, y, y ′ ∈ E and z ∈ Z(E). Hence the inner product is Z(E)-sesquilinear.
Again because p 1 is (completely) positive, x | x ≥ 0 for any x ∈ E. To see that the inner product is in fact positive definite, first consider the case where X = 1 and E = M n . Then p 1 : M n → C n takes the diagonal of a matrix. So if x ∈ M n , and p 1 (x * x) = 0, then x = 0, so certainly p 1 (x) = 0. This generalises to finitedimensional C*-algebras E. Next we use Corollary 6.5 to go back to the case of general E: if x ∈ E satisfies p 1 (x * x) = 0, then for all t ∈ X we have Loc t (p 1 (x)) = 0. So, by Theorem 4.8, in fact p 1 (x) = 0. Thus − | − is a well-defined Z(E)-valued inner product on E.
The inner product is complete because by Lemma 7.1. Hence E is a well-defined Hilbert Z(E)-module.
Lemma 9.5. If E is a special dagger Frobenius structure in Hilb C0(X) , then it is also a special dagger Frobenius structure in Hilb Z(E) .
Proof. By definition, the tensor product of E with itself in Hilb C0(X) , denoted E ⊗ C0(X) E, is the completion of the algebraic tensor product E ⊙ C0(X) E in the C 0 (X)-valued inner product x 1 ⊗ y 1 | x 2 ⊗ y 2 = x 1 | y 1 x 2 | y 2 . Similarly, converges in the former inner product, then it does so in the latter inner product too: Here, the equality uses that (2) is a C*-involution locally as in Corollary 6.5, and the inequality uses that p 1 has norm one. Because the multiplication µ is in fact Z(E)-bilinear, it factors through q. This gives a map µ Z that makes the following diagram of modules commute.
where we draw solid lines for E and dashed lines for Z(E); the first and third equalities use the Frobenius law, and the second and fourth equalities use associativity, naturality of the swap map, and the fact that Z(E) is commutative. Thus because cyclicity of p 1 allows us to change x * y * w into y * x * w under this map. On the other hand, We can now verify the laws for special dagger Frobenius structures for µ Z . Unitality of µ Z follows directly from unitality of µ because η factors through Z(E). Speciality is also easy: It follows from associativity of µ that and hence on all of E ⊗ Z(E) E ⊗ Z(E) E, making µ Z associative. The Frobenius law follows similarly: the two morphisms The last step is to prove that if E is Frobenius over C, then so is its centre Z(E). Lemma 9.6. Let C be a commutative C*-algebra with a paracompact spectrum. If E is a special dagger Frobenius structure in Hilb C , then Z(E) is a specialisable dagger Frobenius structure in Hilb C .
Proof. By Theorem 7.7, E corresponds to a finite C*-bundle p : E ։ X. Define q : Z(E) → X by restriction; we will prove that it is a commutative finite C*-bundle. Clearly, q is still continuous and surjective, because it maps 1 ∈ Z(E t ) to t ∈ X. Also, Z(E) t = Z(E t ) is a commutative finite-dimensional C*-algebra. Now let t 0 ∈ X. Pick an open neighbourhood U of t 0 in X, a finite-dimensional C*-algebra A, and a map ϕ : Finally, we can state the transitivity theorem.
Theorem 9.7. Let X be a paracompact locally compact Hausdorff space, and E a monoid in Hilb C0(X) . The following are equivalent: (i) E is a special dagger Frobenius structure in Hilb C0(X) ; (ii) E is a special dagger Frobenius structure in Hilb Z(E) , and Z(E) is a specialisable dagger Frobenius structure in Hilb C0(X) .
Proof. Combine Lemmas 9.1, 9.5, and 9.6. The only thing left to prove is that E is special over Z(E) precisely when it special over C 0 (X). But this is already included in the proof of Lemma 9.5.
The latter algebra in (ii) is commutative, the former is central. We leave open the question to which monoidal dagger categories the previous theorem can be generalised [39]. We also leave open the question whether it can be made functorial, that is, how the categories and Frobenius structures in (ii) of the previous theorem depend on E and X.

Kernels
A dagger category with a zero object has dagger kernels when every morphism f : E → F has a kernel k : K → E satisfying k † • k = id E [33]. Similarly, it has dagger equalisers when every pair of morphisms f, g : E → F has an equaliser e satisfying e † • e = id. In this section we show that FHilb C0(X) has dagger kernels, and discuss when Hilb C0(X) has dagger kernels.
Proposition 10.1. If X is a locally compact Hausdorff space, Hilb bd C0(X) has kernels; the kernel of f : E → F is given by (the inclusion of ) ker(f ) = {x ∈ E | f (x) = 0}.
Proof. We prove that ker(f ) is always a well-defined object in Hilb bd C0(X) . The inherited inner product x | y K = x | y E is still sesquilinear and positive definite. If (x n ) is a Cauchy sequence in ker(f ), it is also a Cauchy sequence in E, and hence has a limit x ∈ E. Because f is adjointable, it is bounded and hence continuous, so that f (x) = lim n f (x n ) = 0 and x ∈ ker(f ). Thus ker(f ) is complete.
The inclusion ker(f ) ֒→ E is bounded because it is fibrewise contractive, and hence a well-defined morphism. It inherits the universal property from the category of vector spaces.
Proposition 10.2. If X is a paracompact locally compact Hausdorff space, then FHilb C0(X) has dagger kernels; the kernel of f : E → F is given by (the inclusion of ) Proof. First, notice that K = ker(f ) is indeed a well-defined object of FHilb C0(X) by Theorem 5.6: for a subbundle ker(f ) of a finite Hilbert bundle E is a finite Hilbert bundle. By Theorem 5.5, this means there exists L ∈ FHilb C0(X) such that K ⊕ L ≃ C 0 (X) m for some natural number m. Next, because the map t → dim(E t ) is continuous, we can write X as a disjoint union of clopen subsets on which the fibres of E and F have constant dimension. Thus we may assume that E = C 0 (X) n for some natural number n. Now the inclusion k : K → E is adjointable if and only if the map [k, 0] : K ⊕ L ≃ C 0 (X) m → C 0 (X) n is. But this follows from Lemma 5.2 because k is bounded.
When we consider Hilbert modules that are not necessarily finitely presented projective, dagger kernels do not always exist. If they do, the base space X must be totally disconnected, that is, its connected components must be singletons. If X is compact this is equivalent to C(X) being a C*-algebra of real rank zero. Proposition 10.3. Let X be a locally compact Hausdorff space. If Hilb C0(X) has dagger kernels, then X is totally disconnected.
Proof. Let U ⊆ X be a closed set containing distinct points x, y ∈ X. Since X is Hausdorff, x and y have disjoint open neighbourhoods V x and V y . Now {y} is compact and V y is open, so Urysohn's lemma constructs f ∈ C 0 (X) with f (y) = 1 and f (X \ V y ) = 0 so f (x) = 0. Regard f as a morphism C 0 (X) → C 0 (X) by h → f h; it has adjoint h → f * h. As in Lemma 3.3, f has a dagger kernel of the form K = {h ∈ C 0 (X) | h(W ) = 0} for a clopen W ⊆ X. Now U x = U ∩ (X \ W ) and U y = U ∩ W are both open in U , satisfy U = U x ∪ U y and U x ∩ U y = ∅, and are not empty because x ∈ U x and y ∈ U y . Therefore U is not connected. That is, X is totally disconnected.
Remark 10.4. If X is totally disconnected, does Hilb C0(X) have dagger kernels? The question is whether the inclusion ker(f ) ֒→ E is adjointable. The luxury of finitely presented projectivity as used in the proof of Proposition 10.2 is not available. In general it would suffice for ker(f ) to be self-dual [43, 3.3-3.4], but it is unclear whether ker(f ) is self-dual when E and F are self-dual and X is totally disconnected; for related functional-analytic problems see [26,25]. We leave this question open.
Remark 10.5. Which categories C embed into FHilb C0(X) or Hilb C0(X) for some X? We might generalise the strategy of [32, 7.2] that worked for Hilb while removing an inelegant cardinality restriction on the scalars: it suffices that C is symmetric dagger monoidal; has finite dagger biproducts; has dagger equalisers of cotuples [f, g], [g, f ] : E ⊕ E → F for f, g : E → F ; makes every dagger monomorphism a dagger kernel; is well-pointed, and is locally small. The scalars C(I, I) then form a unital commutative * -ring, and we would need an additional condition guaranteeing that it embeds into a commutative complex *-algebra C b (X) for some X. To embed into FHilb C0(X) , we additionally require every object in the category C to have a dagger dual object. As a sanity check that these properties do indeed characterise categories C embedding into FHilb C0(X) for some X, note that the category FHilb C0(X) itself satisfies all of these properties [42, 3.6].

Appendix A. Bimodules and bicategories
We start by briefly recalling Hilbert bimodules and their tensor products; for more information we refer to [42]. Recall that the adjointable maps E → E on a Hilbert A-module E form a C*-algebra L(E).
Definition A.1. Let A and B be C*-algebras. A Hilbert (A, B)-bimodule is a (right) Hilbert B-module E together with a * -homomorphism ϕ : A → L(E) that is nondegenerate, in the sense that ϕ(A)(E) is dense in E. A morphism of Hilbert (A, B)-bimodules is an adjointable map f : E → F of (right) Hilbert B-modules that intertwines, i.e. f (a(x)) = a(f (x)) for a ∈ A and x ∈ E.
A Hilbert C-module is simply a Hilbert space, and a morphism of C-modules is simply an adjointable map between Hilbert spaces. A Hilbert A-module is the same as a Hilbert (C, A)-bimodule, and a morphism of Hilbert (C, A)-bimodules is the same as an adjointable map of Hilbert A-modules. Hence a Hilbert (A, C)-bimodule is precisely a * -representation of A, and a morphism of Hilbert (A, C)-bimodules is precisely an intertwiner.
Definition A.2. The tensor product E ⊗ B F of a Hilbert (A, B)-bimodule E and a Hilbert (B, C)-bimodule F is the algebraic tensor product of C-modules E ⊗ C F made into a Hilbert A-C-bimodule under the inner product x ⊗ y | x ′ ⊗ y ′ E⊗ C F = y | x | x ′ E (y ′ ) F by quotienting out {x ∈ E ⊗ C F | x | x E⊗ C F = 0} and completing, with the map A → L(E ⊗ B F ) sending a to x ⊗ y → a(x) ⊗ y.
Notice that this quotient automatically enforces xb ⊗ y = x ⊗ by in E ⊗ B F for x ∈ E, y ∈ F , and b ∈ B. So E ⊗ B F may alternatively be constructed as the algebraic tensor product E ⊙ B F over B of A-B-bimodules and B-C-bimodules by quotienting out the same subspace and completing in the same inner product.
The tensor product E ⊗ F of Hilbert A-modules E and F over a commutative A is got by regarding them as Hilbert (C, A)-bimodules. If A is commutative, F is also a Hilbert (A, A)-bimodule, via the map A → L(F ) that sends a to y → ya. The tensor product E ⊗ A F of Hilbert bimodules then is a Hilbert (C, A)-bimodule and hence a Hilbert A-module E ⊗ F . Explicitly, it is the completion of the algebraic tensor product E ⊗ C F with the following inner product and (right) A-module structure: x 1 ⊗ y 1 | x 2 ⊗ y 2 = x 1 | x 2 y 1 | y 2 , (x ⊗ y)a = x ⊗ (ya).
Note that this inner product is indeed already nondegenerate [42,Proposition 4.5]. the inner product (x i,j ) | (y i,j ) E (j) = i x i,j | y i,j Hi,j ; it becomes a Hilbert C m -C n -bimodule by the * -representation C m → L(E) sending (z i ) to (x i,j ) → (z i x i,j ).
It is also well-defined on 2-cells: the map f : x i,j → (f i,j (x i,j )) is adjointable because i f i,j (x i,j ) | y i,j Ki,j = i x i,j | f † i,j (y i,j ) Hi,j ; and it is intertwining because f i,j (z i x i,j ) = z i f i,j (x i,j ). This is clearly functorial on homcategories.
The pseudofunctorial data consists of 2-cells C n → n i,j=1 δ i,j C for identities, and ( a,b H a,b ) ⊗ C n ( c,d K c,d ) → i,j,k H i,k ⊗ K k,j for composition. By construction ( a,b H a,b ) ⊗ C n ( c,d K c,d ) is a,b,c,d H a,b ⊗ K c,d , where we identify ((x a,b ) ⊗ (y c,d )) with 0 when x a,b y b,d = 0 for all a and d. Hence there are natural candidates for both, that are adjointable intertwiners, and furthermore are in fact unitary. The coherence diagrams clearly commute.
Finally, this pseudofunctor is clearly injective on 0-cells, and moreover, it is an equivalence on homcategories; see also [11,Proposition 8.1.11].
Thus 2FHilb is a full subbicategory of Hilb * . In other words, Hilb * is a conservative infinite continuous extension of the finite discrete 2FHilb that is more suitable for local quantum physics.

Appendix B. Complete positivity
Write cCstar cp for the category of commutative C*-algebras and (completely) positive linear maps. By Gelfand duality, its objects are isomorphic to C 0 (X) for locally compact Hausdorff spaces X. We now consider morphisms. The set Radon(X) becomes a locally compact Hausdorff space [52,Chapter 13] under the following, so-called vague, topology: a net µ n converges to µ if and only if X f dµ n converges to X f dµ for all measurable f : X → C.
Definition B.2. Write Radon for the following category.
• Objects are locally compact Hausdorff spaces X.
• Morphisms X → Y are continuous functions X → Radon(Y ).
• Composition of f : X → Radon(Y ) and g : Y → Radon(Z) is given by where g U : Y → C for measurable U ⊆ Z is defined by y → g(y)(U ). • The identity on X sends x to the Dirac measure δ x .  Proof. The proof of [27,Theorem 5.1] shows that F (X) = C 0 (X) and F (f )(h)(x) = f (x)(h) define an equivalence F : R → Cstar op cp , for the following category R: