Toward a non-commutative Gelfand duality: Boolean locally separated toposes and Monoidal monotone complete C^{*}-categories

# Toward a non-commutative Gelfand duality: Boolean locally separated toposes and Monoidal monotone complete C∗-categories

Simon Henry
###### Abstract

** Draft Version ** To any boolean topos one can associate its category of internal Hilbert spaces, and if the topos is locally separated one can consider a full subcategory of square integrable Hilbert spaces. In both case it is a symmetric monoidal monotone complete -category. We will prove that any boolean locally separated topos can be reconstructed as the classifying topos of “non-degenerate” monoidal normal -representations of both its category of internal Hilbert spaces and its category of square integrable Hilbert spaces. This suggest a possible extension of the usual Gelfand duality between a class of toposes (or more generally localic stacks or localic groupoids) and a class of symmetric monoidal -categories yet to be discovered.

[.]

footnotetext: Keywords. Boolean locally separated toposes, monotone complete C*-categories, reconstruction theorem. footnotetext: 2010 Mathematics Subject Classification. 18B25, 03G30, 46L05, 46L10 .footnotetext: email: simon.henry@imj-prg.fr

## 1 Introduction

This is a draft version. It will be replaced by a more definitive version within a few months. In the meantime any comments is welcome.

This paper is part of a program (starting with the author’s phd thesis) devoted to the study of the relation between topos theory and non-commutative geometry as two generalizations of topology. The central theme of this research project is the construction explained in section 2 which naturally associate to any topos the -category of Hilbert bundles over .

In the previous paper [7] we studied “measure theory” of toposes and compare it to the theory of -algebras (von Neumann algebras) through this construction of . At the end of the introduction of this previous paper one can find a table informally summing up some sort of partially dictionary between topos theory and operator algebra theory. The goal of the present work is somehow to provide a framework for making this dictionary a concrete mathematical result, by showing that a boolean locally separated topos can actually be completely reconstructed from any of the two symmetric monoidal111The term monoidal is taken with a slightly extended meaning: has in general no unit object -categories and of Hilbert bundles and square-integrable Hilbert bundles.

More precisely, we will show that if is a boolean locally separated topos, then is the classifying topos for non-degenerate222see definitions 5.4 and 5.7. normal symmetric monoidal representations of either and . These are not geometric theory, not even first order theory in fact, but we will prove an equivalence of categories between points of and these representations over an arbitrary base topos, see theorems 5.5 and 5.8 for the precise statement.

Sections 2,3 and 4 contain some preliminaries which are mostly, but not entirely, recall of previous work.

Section 5 contains the statement of the two main theorems of the present paper: theorem 5.5, which is the reconstruction theorem from the “unreduced” category and theorem 5.8 which is the reconstruction theorem from the “reduced” category of square integrable Hilbert spaces.

Sections 6 and 7 contain respectively the proof of the “reduced” theorem 5.8 and the proof that the reduced theorem imply the unreduced theorem 5.5.

The key arguments, which actually use the specificity of the hypothesis “boolean locally separated” seems to all be in subsection 6.1, and the rest of the proof seems to us, at least in comparison, more elementary (it is mostly about some computations of multilinear algebra) and also more general (the specific hypothesis are essentially333Lemma 6.3.2 seems to be the unique exception not used anymore). In fact, we have unsuccessfully tried to prove a result in this spirit for several years, and results of subsection 6.1 have always been the main stumbling block.

Finally we conclude this paper with section 8, where we explain how the results of this paper might maybe be extended into a duality between certain monoidal symmetric -categories and certain geometric objects (presumably, localic groupoids or localic stacks) which would be a common extension of the usual Gelfand duality, the Gelfand duality for -algebras, the Doplicher-Roberts reconstruction theorem for compact groups, and of course the results of the present paper. This would somehow constitute a sort of non-commutative Gelfand duality. Of course the existence of such a duality, and even its precise statement are at the present time highly conjectural, but we will try to highlight what are the main difficulties on the road toward such a result.

This conjectured duality is formally extremely similar to the reconstruction theorems obtained for algebraic stacks, as for example [11], [3].

## 2 General preliminaries

We will make an intensive use of the internal logic of toposes (i.e. the Kripke-Joyal semantics for intuitionist logic in toposes) in this paper. A reader unfamiliar with this technique can read for example sections , and of [13] which give a relatively short and clear account of the subject. Other possible references are [2, chapter 6], [12, chapter VI], or [10, D1 and D4].

Because this paper is mostly about boolean toposes and monotone complete -algebras we will assume that the base topos (that is the category of set) satisfies the law of excluded middle, but we won’t need to assume the axiom of choice.

It is reasonable to think that these results can also be formulated and proved over a non-boolean basis, but the gain in doing so would be very small: over a non-boolean basis, any boolean topos and any monotone complete -algebra is automatically defined over a boolean sub-locale of the terminal object, and hence can be dealt with in a boolean framework.

This being said, a large part of the proofs will take place internally in a non boolean topos and hence will have to avoid the law of excluded middle anyway.

We will also often have to juggle between internal and external logic or between the internal logic of two different toposes. We will generally precise what statement has to be interpreted “internally in ” or “externally”, but we also would like to emphasis the fact that most of the time the context makes this completely clear: if an argument start by “let ” and that is not a set but an object of a topos it obviously means that we are working internally in . This convention, is in fact completely similar with the usual use of context444“Context” is taken here in its formal “type theoretic” meaning, i.e. the “set” of all variable that has been declared at a given point of a proof. in mathematics: when a mathematicians says something like “let ” (for a set) then what follows is actually mathematics internal to the topos of sets over , indeed everything being said after implicitly depends on a parameter , and if the conclusion actually does not depends on then it will be valid independently of the context only if was non-empty.

Our convention is hence that when we say something like “let ” where is an object of a topos then what follows is internal to the topos , or equivalently, internal to with a declared variable , and this being true until the moment where this variable is “removed” from the context (in classical mathematics, this is usually left implicit because, as soon as is non-empty, it is irrelevant, but in our case we will generally make it precise by saying that we are now working externally).

We will of course say explicitly in which topos we are working as soon as we think that it actually improve the readability, but we also think that this perspective makes (once we are used to it) the change from working internally into one topos from another topos as simple as introducing and forgetting abstract variables in usual mathematics and makes the text easier to read.

All the toposes considered are Grothendieck toposes, in particular they all have a natural numbers object (see [10, A2.5 and D5.1]) “” or “” which is just the locally constant sheaf equal to .

A set, or an object of a topos, is said to be inhabited if (internally) it satisfies . For an object of a topos it corresponds to the fact that the map from to the terminal object is an epimorphism.

An object of a topos is said to be a bound of if subobjects of form a generating family of (i.e. is any object of admit a covering by subobject of ). Every Grothendieck topos admit a bound (for exemaple take the direct sum of all the representable sheaves for a given site of definition), in fact the existence of a bound together with the existence of small co-products characterize Grothendieck toposes among elementary toposes.

Let be an arbitrary topos. is the object of “continuous555also called Dedekind complex numbers complex numbers” that is where is the object of continuous/Dedekind real numbers as defined for example in [10, D4.7]. In any topos (with a natural number object), is a locale ring object.

When is a decidable proposition, i.e. if one does have or not-, one denotes the real number defined by if and =0 otherwise. An object is said to be decidable if for all and in the proposition is decidable, in which case we denote for . i.e. is one if and zero otherwise.

By a “Hilbert space of ”, or a -Hilbert space we mean an object of , endowed with a -module structure and a scalar product (linear in the second variable and anti-linear in the first), which satisfies internally all the usual axioms for being a Hilbert space, completeness being interpreted in term of Cauchy filters, or equivalently Cauchy approximations but not Cauchy sequences.

denotes the -category666see for exemple [6] for the definition of -category. whose objects are Hilbert spaces of and whose morphisms are “globally bounded operators”, that is linear maps which admit an adjoint and such that it exists an external number satisfying (internally) for all , . The norm is then the smallest such constant (if we were not assuming the law of excluded middle in the base topos it would be an upper semi-continuous real number), the addition and composition of operators is defined internally , is the adjoint of internally, and this form a -category.

Because the tensor product of two Hilbert spaces can easily be defined, even in intuitionist mathematics, the category is endowed with a symmetric monoidal structure.

Precise definition of a symmetric monoidal category, a symmetric monoidal functor, and a symmetric monoidal natural transformation can be found in S.MacLane’s category theory books777Only in the second edition (1998). [15], chapter XI, section and .

Briefly, a symmetric monoidal category is a category endowed with a bi-functor , a specific “unit” object and isomorphisms , and which are natural and satisfies certain coherence conditions. A symmetric monoidal functor (“braided strong monoidal functor” in MacLane’s terminology) is a functor between symmetric monoidal categories with a natural isomorphism which has to satisfy a certain number of coherence and compatibility relations. Finally a symmetric monoidal transformation (the adjective symmetric is actually irrelevant for natural transformation) is a natural transformation between symmetric monoidal functor which satisfy coherence conditions, stating that, up to the previously defined natural isomorphisms, is the same as and is the identity.

Moreover, when we are talking about (symmetric) monoidal -categories or symmetric monoidal -functor between such categories we are always assuming that all the structural isomorphisms are in fact isometric isomorphisms, i.e. their inverse is their adjoint. For example, it is clearly the case for .

Finally, if is a geometric morphism between two toposes, and is a Hilbert space of then is a “pre-Hilbert” space of but fails in general to be complete and separated, we denote by its separated completion. is a symmetric monoidal -functor from to .

## 3 Monotone complete C∗-categories and boolean toposes

We recall that a -algebra is said to be monotone complete if every bounded directed net of positive operators has a supremum. A positive linear map between two monotone complete -algebras is said to be normal if it preserves supremum of bounded directed set of positive operators.

The theory of monotone complete -algebras is extremely close to the theory of -algebras, in fact it is well know that a monotone complete -algebra having enough normal positive linear form is a -algebra (see [16, Theorem 3.16]).

When is a boolean topos, is a monotone complete -category in the sense that it has bi-products and the -algebra of endomorphisms of any object is monotone complete. Indeed, because is boolean, the supremum of a bounded net of operators can be computed internally, and as the supremum is unique it “patches up” into an externally defined map, see [7, section 2].

Monotone complete -categories are extremely close to the -categories studied in [6], in fact most of the result of [6] which does not involve the existence of normal states (or the modular time evolution) also hold for monotone complete -categories. We will review some of these results:

If is a monotone complete -category then we define the center of as being the commutative monotone complete -algebra of endomorphisms of the identity functor of . In the more general situation might fail to be a set and be a proper class, but we will not be concern by this issue because we proved in [7, 3.6] that has a generator and hence, by results of [6], the algebra can be identified with the center of the algebra of endomorphisms of this generator.

If is an object of a monotone complete -category then we define its central support by:

 c(A)B:=supf:An→B∥f∥<1ff∗∈Hom(B,B).

If the monotone complete category we are working with admit bi-product then denotes the bi-product of -copies of and if not, then one can still make sense of a map from to as the data of maps from to , is a,d is defined as . One can check that the supremum involved in the definition of is directed by showing that the set of such “” is in order preserving bijection with the set of where is an arbitrary maps from to without condition on the norm, the bijection being obtained by multiplying by a convenient function of .

Equivalently, can be defined as the smallest projection in such that , but we will need the fact that it is a directed supremum.

One says that an object is quasi-contained in an object if . An object is said to be a generator of a monotone complete -category if and only if i.e. if every other object is quasi-contained in .

One can for example check that if two normal functors agree on a generator and its endomorphisms then they are isomorphic: it is an easy consequence of results of [6] for -categories and the proof can extended to monotone complete -categories easily.

We conclude this section by briefly mentioning what quasi-containement mean in the case of :

#### 3.1

Proposition : Let be a boolean topos and two Hilbert spaces of , then is weakly contained in if and only if there exists a set of bounded operators from to such that internally in the functions in spam a dense subspace of .

If this is true, we will say that is covered by the maps in .

Proof :

If is weakly contained in then hence . Rewriting this using the definition of one gets:

 IdH=supf:(H′)n→H∥f∥<1ff∗∈Hom(H,H)

But as we mentioned earlier, supremums of directed nets in are computed internally, this means that this supremum converge internally for the strong operator topology. In particular, for any one has which is arbitrarily close to when run through the (external) set:

 F0={f|f:(H′)n→H,∥f∥<1}

Hence taking to be the set of “component” of maps in , the sum of the images of maps in spam all of .

Conversely, assume that is spammed by a family of external maps . For any , and in particular, for any one has . The projector is hence (internally in ) equal to the identity on the image of all the maps and hence on all of , i.e. which proves that .

## 4 Locally separated toposes and square integrable Hilbert spaces

We recall (see [14, chapter II]) that a topos is said to be separated if its diagonal map (which is localic by [10, B3.3.8]) is proper, i.e. if, when seen as a -locale though the diagonal maps, is compact.

In [7, theorem 5.2] we proved that a boolean topos is separated if and only if it is generated by internally finite objects. One will use a slightly modified form of this result :

#### 4.1

Theorem : Let be a boolean separated topos, then admit a generating family of objects such that for each there exists an interger such that internally in , the cardinal of is smaller than .

One will say that such objects are of bounded cardinal.

Proof :

This theorem is an immediate consequence of [7, theorem 5.2]: is generated by a familly of internally finite object , but for each object of this generating familly and for each natural number one can define whose cardinal is internally bounded by , and as is internally finite the form a covering family of and hence the form a generating family fulfilling the property announced in the theorem.

An object of boolean topos is said to be separating if the slice topos is separated. A boolean topos is said to be locally separated if it admit an inhabited separating object, or equivalently if any object can be covered by separating objects, see [7] section 5 for more details.

If is an object of a boolean topos then one can define the -Hilbert space of square sumable sequences indexed by . It can also be done in a non boolean topos but it require to be a decidable888In order to define the scalar product of two generators or to define the sum of a sequence we need that for any or . object. Internally in , the space has generators for such that .

#### 4.2

Proposition : Let be a bound of a boolean topos and be a separating object of then is quasi-contained in .

Proof :

Let be a bound and any separating object. As is a bound, can be covered by maps with . Using the fact that is separated and boolean, we know that it is generated by objects of bounded cardinal and the image of a map whose domain is finite is also finite, and of smaller cardinal, hence admit a covering by sub-objects with bounded cardinal in , hence we can freely assume that is itself of bounded cardinal in .

One can then define (for each such map ) a map by if and if . The fact that the object is finite with bounded cardinal over mean that there exists an (external) integer such that each fiber of has cardinal smaller than , this is exactly what we need to know to construct the adjoint of and to prove that is bounded (and hence extend into an operator).

Now as such maps cover , one has internally in : “ such that is in the image of ”, where denote the external set of such map which are finite and of bounded cardinal in . In particular the joint image of all the for contains all the generators of and hence spam a dense subspace of , which concludes the proof by proposition 3.1.

We denote by the full subcategory of of objects which are weakly contained in for some separating object . Objects of are said to be square-integrable999because when is the topos of -sets for some discrete group , then is the category of unitary representations of while is precisely the category of square integrable representations of .

Results of [7] (especially section 7) suggest that the square integrable Hilbert spaces of are the one that are clearly related to the geometry of .

Because of proposition 4.2, for any separating bound of the Hilbert space is a generator of , hence extending proposition 7.6 of [6] to monotone complete -category (or restricting ourselves to topos which are integrable in the sense of [7, section 3]) gives us that when is a boolean and locally separated topos, is equivalent to the category of reflexive Hilbert modules over . For this reason we can call this algebra “the” (or “a”) reduced101010For example, if is the topos of -sets for a discrete group one obtains the usual (reduced) von Neumann algebra of the group this way. algebra of (it is unique up to Morita equivalence).

Finally, The fact that it is possible to define internally the tensor product of two Hilbert spaces yields a symmetric monoidal structure on . Moreover as one can see that is stable by tensor product111111in 7.2.2 we will actually prove the stronger result that the tensor product of an arbitrary Hilbert space with a square integrable Hilbert space is square integrable. Hence, is also endowed with a symmetric monoidal structure, but without a unit object (unless is separated).

## 5 Statement of the main theorems

#### 5.1

Definition : Let be a monotone complete -category, any topos (not necessary boolean) a representation of in is a -functor from to . It is said to be normal if for any supremum of a bounded directed net of positive operator in , converge internally in the weak operator topology to .

A representation of in is also the same as a representation of in the category of Hilbert space internally in (where is the geometric morphism from to the point). Also if is boolean, then is monotone complete and a representation is normal in the sense of this definition if and only if it is normal as a -functor between -category.

#### 5.2

When has additional structure (for example is monoidal) we will by default assume that the representation preserve these structures, for exemple:

Definition : If is a boolean topos and an arbitrary topos, a representation of in is a normal representation of the monotone complete -category in such that the underlying -functor is symmetric and monoidal.

#### 5.3

Definition : We will say that a Hilbert space is inhabited if one has such that internally in , or equivalently, if holds internally in .

#### 5.4

Definition : If is a boolean topos and an arbitrary topos, a representation of is said to be non-degenerate if for any inhabited object of the Hilbert space is inhabited in .

It is important to notice that detecting whether a representation of is non-degenerate or not can be done completely from the (monoidal) category without knowing the topos . Indeed, an object of is inhabited in if and only if the functor is faithful.

As an example of a “degenerate” representation, one can consider the topos of -sets for some infinite discrete groupe , then is the category of unitary representation of . Let be the representation of into defined by:

 ρ(H)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯{h∈H|h belongs to a finite dimensional sub-% representation }

One has so it cannot be non-degenrate. One easily checks that it is a symmetric normal -functor, and it is monoidal because of the following observation:

Lemma : Let be two representations of . Assume that contains a (non trivial) finite dimensional sub-representation, then both and contains a non trivial finite dimensional sub-representation.

Proof :

Let a non-trivial finite dimensional sub-representation. let the dual of the representation . Because is finite dimensional and non-trivial contains a non-zero invariant vector. In particular contains a non-zero invariant vector which corresponds to a non zero -linear Hilbert-Schmidt from to , is hence a non-zero compact self-adjoint -linear automorphism of which is hence going to have some non-trivial finite dimensional -stable eigenspaces. This concludes the proof of the lemma.

There is in particular a representation of in itself called the tautological representation and given by the identity functor. This representation is non-degenerate. Moreover if is a (non-degenerate) representation of in a topos and is any geometric morphism then , given by composing the functor by the functor is a (non-degenerate) representation of in . In particular, any geometric morphism from induce a non-degenerate representation of in .

#### 5.5

Theorem : Let be a boolean locally separated topos, then is the classifying topos for non-degenerate representations of and the tautological representation is the universal non-degenerate representation. More precisely, for any topos there is an equivalence of category from the category of geometric morphism to the category121212See 5.6 below. of non-degenerate representations of in which associate to any geometric morphism the representation .

In particular, the topos is uniquely determined (up to unique isomorphism) from the symmetric monoidal -category of Hilbert bundles over .

#### 5.6

This theorem, as it is stated, does not completely make sense yet because we did not say what are the morphisms of representations of in . It appears that the good notion of morphisms is the following:

Definition : If and are two representations of in a morphism from to is a collection of isometric inclusions for each object of which is natural in and such that, up to the structural isomorphisms, is the identify of and is .

Because we do not assume that the have adjoints they are not morphisms in and hence it would not make sense strictly speaking to says they form a symmetric monoidal natural transformation, but this is essentially what this definition means.

It can be proved directly, using the fact that as a generator (as proved in [7, 3.6]), that morphisms between two representations and actually form a set and not a proper class, but we do not need to know that and one can instead obtain this result as a corollary of the theorem.

#### 5.7

There is also a form of this theorem for the category instead of , which we will use as an intermediate step in the proof of theorem 5.5. The problem to state it directly is that is not exactly a monoidal category because it does not have a unit object. In particular, one cannot ask monoidal functors to preserve the unit object, but it appears that the condition “non-degenerate” actually completely replace the preservation of the unit object, and moreover, in this case, the definition of non-degenerate can be weakened:

Definition : A non-degenerate representation of in is a normal representation of the monotone complete -category in such that:

• satisfies all the axioms of the definition of a symmetric monoidal functor not involving the unit object.

• There exists an object of such that is inhabited.

The morphisms of such representations are defined exactly as in 5.6.

Similarly to 5.5, there is a tautological representation of in and it is possible to pullback any representation of along a geometric morphism

#### 5.8

Theorem : The tautological representation of in is the universal non-degenerate representation of . I.e., for any topos the association induce an equivalence of categories between geometric morphisms from to and non-degenerate representations of in .

In the rest of the article, all the representations considered will always be implicitly assumed to be non-degenerate.

These two theorems together, suggest that it should be possible to reconstruct directly from . This indeed seems to be the case, here is a sketches of proof:

We will see in 7.2.2, that any in induce by tensorisation an endofunctor:

 MH:=(H⊗_):Hred(T)→Hred(T)

which satisfies a “multiplier” condition of the form (this isomorphism being functorial and satisfying some coherence conditions). Conversely, if is an endofunctor of satisfying the same condition than , then for any separating object of , is going to be a -module in the sense of definition 7.1.3, hence, by proposition 7.1.4, is of the form for a Hilbert space in , but using the coherence condition on the isomorphisms one should be able to prove that all the are canonically isomorphic and hence (if is inhabited) that if of the form . One can then conclude the is exactly the tensorization by by assuming that is a separating bound and hence that is a generator of . One also has to check that endomorphisms of acts as they should on but this will follow by the same “matrix elements” argument as in the proof of 6.4.2.

Hence, should be, in some sense, the “category of multiplier” of the (non-unital) monoidal category .

We decided not to include a precise form of this last result in the present version of this paper, because its proof, and in fact even its proper formulation, require some work on the precise coherence conditions required on such a “multiplier” that seems to be out of the scope of the present paper, and we are not sure that such a statement has, despite its elegance, any interesting applications.

## 6 From representations of Hred to geometric morphisms

In this section we consider an arbitrary representation of in a topos , and we will prove that it induces a geometric morphism from to . In all this section we will work internally in . Hence objects of will be called sets and objects of will simply be called Hilbert spaces, the price of this being to avoid the use of the law of excluded middle and of the axiom of choice.

### 6.1 Construction of the geometric morphism on separating objects

#### 6.1.1

In this subsection, we fix a separating object , and we will show that is of the form for a well determined decidable set . We will denote by the space . This space is endowed with an operator which comes from the operator in :

 Δ:l2(X)→l2(X)⊗l2(X)ex↦ex⊗ex

is co-commutative and co-associative and its adjoint defines a multiplication on denoted , which is commutative and associative. Moreover hence .

In , the operator is defined by is and if , or, tu put it another way, it is just the point-wise multiplication of -indexed sequences.

In , for , we denote by the linear map on defined by . we denote by , and the usual norm of will be denoted by . One has because of the previous bound on the norm of a product.

Finally, let be the closure in the space of bounded linear map from to of the of algebra formed by the for . It is a commutative algebra. the term “closure” is taken here in the sense that:

 C:={f:H→H|∀ϵ>0,∃h∈H,∥f−mh∥<ϵ}

In fact, it should be called “fiberwise closure” or “weak closure” because the resulting set is not closed in the sense the complementary of an open, see [9] for more details about these notions.

#### 6.1.2

Before going further we need to make a few external constructions that will also be usefull in the next subsetions.

Because there exists an object of such that is inhabited, and that the for separating are generators of there exists a separating object such that is also inhabited. We fix such an object .

A sub-object is said to be of degree smaller than over if internally in for all there is at most distinct elements such that , i.e. if is of cardinal smaller than as an object of . It is said to be of finite degree if it is of degree smaller than for some external natural number , i.e. if, as an object of its cardinal is bounded.

Theorem 4.1 applied to imply that sub-objects of finite degree over cover . We denote by the (external) ordered set of sub-objects of of finite degree. As, if and are of finite degree then is also of finite degree, is a directed pre-ordered set.

#### 6.1.3

For let (in ):

 VU:l2(X0)→l2(X0)⊗l2(X)ex↦∑x′ s.t. (x,x′)∈Uex⊗ex′
 PU:l2(X0)⊗l2(X)→l2(X0)⊗l2(X)ex⊗ex′↦(I(x,x′)∈U)ex⊗ex′

We also denote by and their images by , and we denote the space .

One easily check that:

• has an adjoint given by

• is well defined and bounded (if has degree , then has norm smaller than ).

• is a -indexed net of projections whose supremum is the identity of , and as is normal it is also the case in that the supremum of the is the identify of .

• . Indeed this is easily checked internally in on elements of the form , extended by linearity and continuity and transported to by .

#### 6.1.4

The role of these operators and is essentially to provide “locally” an approximate unit for the product . Indeed, let us fix an element such that , which is always possible by assumption on , then one can define , and, because of the formula relating and one has after a short computation that for any :

 h∗1U=(h∗0⊗IdH)(PU(h0⊗h)),

which (because converge strongly to the identity) converge in norm to . In particular:

Lemma : In , the association from to is injective.

Indeed, using (internally) our approximate unit one gets that:

 h=limU∈Fh∗1U=limU∈Fmh(1U).

#### 6.1.5

We can now prove:

Lemma : The algebra generated by the is a -algebra.

The term -algebra is taken here in the same sense as, for example, in [1]. I.e. it is complete in the sense of Cauchy approximation or Cauchy filters, and the “norm” is describe either by the data of the “rational ball” corresponding morally to the set of such that of equivalently as a function with value into the object of upper semi-continuous real number.

Proof :

All we have to do is to prove that every element of have an adjoint which belongs to . As the adjunction preserve the norm and that the space of bounded linear map is complete (in the norm topology) it is enough to prove it for a dense family of elements of .

Let be an arbitrary element of .

We define:

 hU=h∗1U=(h∗0⊗IdH)(PU(h0⊗h))∈H

As mentioned earlier, converges in norm to .

We also define:

 h′U=(h∗0⊗h∗⊗IdH)(IdH0⊗Δ)(VU)(h0)

We will prove that for all one has:

 ⟨h′U∗a,b⟩=⟨a,hU∗b⟩

this will show that has an adjoint which belong to . As for any , is approximated by the (indeed, ) the for and are dense in , hence this will conclude the proof.

In order to do so, consider the following two operators and from to defined by:

 W1:=(IdH0⊗Δ∗)∘(IdH0⊗Δ∗⊗IdH)∘(VU⊗IdH⊗IdH)
 W2:=(V∗U⊗IdH)∘(IdH0⊗Δ∗⊗idH)∘(IdH0⊗IdH⊗Δ)

Two short computations show that:

 ⟨a,hU∗b⟩=⟨h0⊗a,W1(h0⊗h⊗b)⟩
 ⟨h′U∗a,b⟩=⟨h0⊗a,W2(h0⊗h⊗b)⟩

Moreover, both and are image by of operators also denoted and defined by the same formulas in (replacing and by and ) and one can easily check in that indeed, if a short computation shows that:

 W1(ex⊗ey⊗ez)=I(y=z)I((x,y)∈U)ex⊗ey=W2(ex⊗ey⊗ez)

which proves that in and hence also in and hence that which concludes the proof as mentioned above.

#### 6.1.6

One of the key observation is that this algebra is “geometrically” attached to the representation , in order to make this clear let us denote instead of in what follows.

Lemma : Let be any topos above . One has a representation of in , and a canonical isomorphism:

 Cp♯ρ≃p♯Cρ.

Proof :

It is clear that , obtained as the composition of with , is a representation in the sense of definition 5.7. Moreover, the -algebra is naturally acting on simply because is acting on . Moreover if , the operator on is equal to the operator (this is clear from the definition of ). Hence the image of in is generated by operators of the form for but as is dense in this proves that the image of in the algebra of endomorphism of is indeed the algebra generated by the for , i.e. the algebra , giving a canonical surjection from to . But this surjection is also isometric and hence is an isomorphism because of the relation:

 ∥mh∥=limn→∞∥h∗h∗⋯∗h∥1n

which is proved in the proof of lemma 6.1.7 below and clearly preserved by .

In particular, if one compute in the tautological representation of in it is exactly , hence if our representation corresponds to a pullback along some geometric morphism the previous lemma imply that will be and hence using the non-unital Gelfand duality proved in [8] one has that can be reconstructed out of as the spectrum of . In order to show that an arbitrary fo comes from a geometric morphism one needs to prove that the spectrum of is a discrete decidable locale, and we will do that by showing that is locally positive (i.e. that is an open map) and that the diagonal map is both open (hence by [10, C3.1.15] that is discrete) and closed (hence that is decidable).

#### 6.1.7

Lemma : The norm of any element of is a continuous real number and is locally positive (or “open”, or “overt”).

Proof :

Let such that , as this proves that in the sense that there exists a rational number such that . Let denotes the -th power of for the product . As is an element of a commutative -algebra one has hence .

Moreover:

 ∥h(n)∥2=∥mhh(n−1)∥2⩽∥mh∥∥h(n−1)∥2⩽∥h∥∞∥h(n−1)∥2

Hence by induction on :

 ∥h(n+1)∥2⩽∥h∥n∞∥h∥2

As one has:

 q∥h∥n∞⩽∥h∥n+1∞⩽∥h(n+1)∥2⩽∥h∥n∞∥h∥2

For some rational number such that . Taking the -th root one obtains that:

 q1n∥h∥∞⩽∥h(n+1)∥1n2⩽∥h∥∞∥h∥1n2⩽∥h∥∞(q′)1n

And equivalently that:

 1q′1/n∥h(n+1)∥1n2⩽∥h∥∞⩽1q1/n∥h(n+1)∥1n2

As is bounded, the difference between the upper bound and the lower bound can be made arbitrarily small, hence can be approximated arbitrarily close by continuous real number (and hence also by rational number), which proves that is a continuous real number. One also gets the identity:

 limn→∞∥h(n+1)∥1n2=∥h∥∞

which was required in the proof of the previous lemma.

As and the such that are dense in (for any and any either is of positive norm or it is of norm smaller than and hence is approximated by ) and as the for are dense in , one has obtain a dense familly of elements of of continuous norm hence every element of has a continuous norm and by [8, 5.2] one can conclude that is locally positive.

#### 6.1.8

If one works internally in , one still has a representation of (by pulling it back from the one in ) and by 6.1.6 the -algebra of endomorphisms of generated by this representation is isomorphic to the pullback of , hence (by [8, proposition 4.4] ) we have at our disposal a character satisfying such that . We will now examine what can be done with such a character, without necessary assuming that we are working internally in .

Assume one has a character of the algebra satisfying . The map is a bounded linear form on , also denoted . Also there exists a such that .

Let be defined as .

Then if is another element of one has:

 ⟨h,eχ⟩=(χ⊗h∗)(Δ(g))=χ((IdH⊗h∗)(Δ(g)))

But, for any one has:

 ⟨w,(IdH⊗h∗)(Δ(g))⟩=(w∗⊗h∗)(Δ(g))=⟨Δ∗(w⊗h),g⟩=⟨mhw,g⟩

hence and as is a character of one obtain that . Finally:

 ⟨eχ,h⟩=χ(h)

In particular does not depend on , but only on . This proves:

Lemma : If is a character of then there exists a unique element such that for all one has .

#### 6.1.9

The vector attached to a character of satisfies additional properties:

Lemma :

Proof :

1. Let , then:

 ⟨Δ(eχ),a⊗b⟩