Constructive Gelfand duality for non-unital commutative C*-algebras
aa r X i v : . [ m a t h . C T ] F e b Constructive Gelfand duality for non-unitalcommutative C ∗ -algebras Simon HenryJuly 3, 2018
Abstract
We prove constructive versions of various usual results related to theGelfand duality. Namely, that the constructive Gelfand duality extend toa duality between commutative nonunital C ∗ -algebras and locally com-pact completely regular locales, that ideals of a commutative C ∗ -algebrasare in order preserving bijection with the open sublocales of its spectrum,and a purely constructive result saying that a commutative C ∗ -algebrahas a continuous norm if and only its spectrum is open. We also extendall these results to the case of localic C ∗ -algebras. In order to do so we de-velop the notion of one point compactification of a locally compact regularlocale and of unitarization of a C ∗ -algebra in a constructive framework. Contents C ∗ -algebras 104 The non-unital Gelfand duality 155 Local positivity and continuity of the norm 186 Extension of the results to localic C ∗ -algebras 21 This paper has been written to provide two technical tools which were neededin the proof of the main theorems of [6] : the non-unital Gelfand duality, in-cluding the characterization of the spectrum given by proposition 4.4, and (onedirection) of the “possitivity” theorem 5.2. We took the opportunity to prove
Keywords.
Gelfand duality, one-point compactification, C ∗ -algebras, C ∗ -locales email: [email protected] N . The subobject classifier is denoted by Ω, and ⊤ and ⊥ denotes its top and bottom element, i.e. the proposition true and false.A frame is a complete Heyting algebra, a frame homomorphism is an orderpreserving map commuting to arbitrary supremums and finite infimums. Thecategory of locales is defined as the opposite of the category of frames, if X isa locale the corresponding frame is denoted by O ( X ). If f is a morphism oflocales, the corresponding frame homomorphism is denoted f ∗ .Elements of O ( X ) are called open sublocales of X . The top element of O ( X ) isdenoted by X , the bottom element by ∅ . When talking about open sublocales,“ V is bigger than U ” or “ U is smaller than V ” always means U V . Formore information on the theory of locale, the reader can consult [8] (which isunfortunately non constructive) or [7, C1].Supremums and finite infimums in O ( X ) are called unions and intersections andare denoted by the symbols ∪ and ∩ .If U is an open sublocale of X , then ¬ U denote the open sublocale U ⇒ ∅ and U c denote the closed complement of U , ie the locale such that O ( U c ) = { V ∈O ( X ) | U V } . In particular, ¬ U is the interior of U c .An (increasing) net of open sublocales of X is an inhabited family ( U i ) i ∈ I ofopen sublocales of X such that for each i, j ∈ I there exists k ∈ I such that U k is bigger than U i and U j .If U and V are two open sublocales of a locale X , we say that: • U ≪ V ( U is way below V ) if for each increasing net ( U i ) i ∈ I whosesupremum is bigger than V there exists i ∈ I such that U U i . • U ⊳V ( U is rather below V ) if there exists W ∈ O ( X ) such that V ∪ W = X and U ∩ W = ∅ . Or equivalently if ¬ U ∪ V = X . • U ⊳ CR V ( U is completely below V ) if there exists a “scale” ( U q ) q ∈ [0 , ∩ Q such that U = U , U = V and for each q ′ < q one has U q ′ ⊳ U q . This isalso equivalent to the existence of a function f from X to the locale [0 , f ∗ (]0 , ⊂ V (i.e. f restrictdto V c is 0) and f restricted to U is constant equal to 1. (see [8, V.5.7 andXIV.6.2]).One also says that X is locally compact (resp. regular, resp. completely regular)if any open sublocale V of X can be written as a supremum of open sublocale U such that U ≪ V (resp. U ⊳ V , resp.
U ⊳ CR V ). A locale X is said to becompact if X ≪ X .One has the following properties: We mean the formal locale of real number, which might be non spatial and hence differentfrom the topological space of real number in the absence of the law of excluded middle.
2. Each of the three relations ≪ , ⊳ and ⊳ CR satisfies the properties: if a b , b ≪ c and c d then a ≪ d ; and if a ≪ b and c ≪ d then a ∪ c ≪ b ∪ d .2. In a regular (resp. completely regular) locale U ≪ V implies U ⊳ V (resp.
U ⊳ CR V ).3. In a locally compact locale X , if a ≪ b then there exists c such that a ≪ c ≪ b .4. In a compact locale a ⊳ b imply a ≪ b , more generally in any locale X , if a ⊳ b and a ≪ X then a ≪ b .5. In particular, A compact regular locale is locally compact.6. And also, in a compact regular locale, ⊳ and ≪ are equivalent and in acompact completely regular locale these three relations are equivalent.If f : T → E is a geometric morphism between two toposes, and X a locale in theinternal logic of E , we denote by f ♯ ( X ) the pullback of X along f , in particular f ∗ ( O ( X )) is different from O ( f ♯ ( X )) but is still a basis of the topologies of f ♯ ( X ).We will frequently use expression of the form S u a where u is a proposition and a is an element of a frame which might seems strange for a reader unfamiliarwith this. This expression makes sense because, as a proposition, u is a subset ofthe singleton and a can be seen as a family of elements indexed by the singleton(and hence also by its subset u ). This is of course the same as a ∩ p ∗ ( u ) where p ∗ denotes the canonical frame homomorphism from the initial frame Ω. Butthe expression with a union allows to emphasize the fact that this is indeed aunion, and also it might happen that the expression defining “ a ” only makessense when u holds, in which case only the first expression makes sense.We conclude these preliminaries by the definition of real numbers. We will needtwo spaces of real numbers.The first one is the set of non-negative upper semi-continuous real numbers,where the norm function of C ∗ -algebras will take value. A upper-semi-continuousreal number is a subset x of the set Q of rational such that: • ∃ q ∈ x • If q ∈ x and q < q ′ then q ′ ∈ x • For all q ∈ x there exists a q ′ < q such that q ′ ∈ x .It is said to be non-negative if it is included in the set Q ∗ + of positive ratio-nal numbers. Of course q ∈ x has to be interpreted as x < q (and will bedenoted this way). Upper semi-continuous real number have good order prop-erty (every bounded set has a supremum) but poor algebraic property: even if This means that f ∗ ( O ( X )) generate O ( f ♯ ( X )) under arbitrary join R of continuous real numbers, which will play the role of the“scalars field” for C ∗ -algebras. A continuous real number is a pair x = ( L, U )of subsets of the set of Q of rational numbers such that: • U is upper semi-continuous real number, and L is a lower semi-continuousnumber (i.e. satisfy the same three axioms but for the reverse order rela-tion). • L ∩ U = ∅ . • for all q < q ′ either q ∈ L or q ′ ∈ U .Of course q ∈ L mean q < x and q ∈ U means x < q . The continuous realnumber have good algebraic properties (they form a locale ring) and topologicalproperties (they are complete, in fact they are exactly the completion of Q byCauchy filters) but no longer have supremums in general. The complex numbersare defined as R × R endowed with their usual product.Finally the map ( L, U ) U induce an injection of the continuous real numbersinto the semi-continuous real numbers, in particular it makes sense to wonderwhether a given semi-continuous real number is continuous or not. In this section we will define a constructive and pointfree version of the processof one point compactification of a locally compact separated topological space.2.1. Let X be a locally compact regular locale, and U ∈ O ( X ). We will denoteby ω ( U ) the proposition: ω ( U ) := “ ∃ W ∈ O ( X ) such that W ≪ X and U ∪ W = X ”i.e. ω ( U ) is the proposition “ U has a compact complement”. The underlyingidea is that in the one point compactification, the neighbourhoods of ∞ areexactly the open subspaces whose complement is compact, i.e. the U such that ω ( U ).The main result of this section is:2.2. Theorem :
Let X be a locally compact regular locale. Then there exists aunique (up to unique isomorphism) compact regular locale X ∞ with a (closed)point {∞} ⊂ X ∞ and an isomorphism between X and the open complement of {∞} .Moreover: O ( X ∞ ) ≃ { ( U, p ) ∈ O ( X ) × Ω | p ⇒ ω ( U ) } nd the two projections from O ( X ∞ ) to O ( X ) and Ω are the frame homomor-phisms corresponding to the injections of X and {∞} into X ∞ . The proof will be completed in 2.7. One can also note that this is a special caseof Artin gluing of a closed point to X . This will be extremly apparent in 2.4and in 2.7.For the rest of this section, we fix a locally compact regular locale X .2.3. Proposition :
The function ω : O ( X ) → Ω = O ( {∞} ) is cartesian, i.e.order preserving and satisfies ω ( X ) = ⊤ and ω ( a ∩ b ) = ω ( a ) ∧ ω ( b ) . Proof : if A B and ω ( A ), then ω ( B ) also holds with the same W . One has ω ( X )with W = ∅ . As ω is order preserving one has ω ( A ∩ B ) ω ( A ) ∧ ω ( B ). Forthe converse inequality, if one has ω ( A ) and ω ( B ) then there is W and W ′ suchthat W, W ′ ≪ X and W ∪ A = X , W ∪ B = X . Taking W ′′ = W ∪ W ′ onehas W ′′ ≪ X and W ′′ ∪ ( A ∩ B ) = X which proves ω ( A ∩ B ). (cid:3) Corollary :
There is a locale X ∞ such that O ( X ∞ ) = { ( U, p ) ∈ O ( X ) × Ω | p ⇒ ω ( U ) } as in theorem 2.2. Moreover the two projections from O ( X ∞ ) to O ( X ) and Ω = O ( {∞} ) corresponds to an open inclusion of X into X ∞ and the complementaryclosed inclusion. Proof :
From the fact that ω is cartesian one deduces that { ( U, p ) ∈ O ( X ) × Ω | p ⇒ ω ( U ) } is stable under arbitrary joins and finite meets in O ( X ) × Ω hence itis a frame and the two projections are frame homomorphisms. Consider theelement X := ( X, ⊥ ) ∈ O ( X ∞ ) then the elements of O ( X ∞ ) smaller than X are the ( U, ⊥ ) for U ∈ O ( X ) hence the open sublocale X is isomorphic to X .Conversely, the element of O ( X ∞ ) bigger than X are exactly the ( X, p ) hencethe closed complement of X is just a point, denoted ∞ and corresponding to ∞ ∗ ( X, p ) = p . (cid:3) For now on, the open sublocale X of X ∞ will be identified with X (and inparticular denoted X ). We mean by that the localic form of construction like [7, A2.1.12, A4.1.12 and A4.5.6] This is exactly the general construction of an Artin gluing.
Lemma : X ∞ is compact. Proof :
Let ( X i ) i ∈ I = ( U i , p i ) i ∈ I be a covering net of open sublocales of X ∞ . Assupremum in O ( X ∞ ) are computed componentwise one has in particular W p i = ⊤ i.e. there exists i ∈ I such that p i holds. As p i ⇒ ω ( U i ) one also has ω ( U i )i.e. there exists a W such that W ≪ X and W S U i = X . As the U i form acovering net of X one also has a i such that W U i and a i bigger than i and i . Hence U i > W ∪ U i = X and p i > p i = ⊤ , hence X i = X ∞ whichconcludes the proof. (cid:3) Lemma : X ∞ is regular. Proof :
Let A = ( U, p ) be any open sublocale of X ∞ . We can first see that: A = ( U, ⊥ ) ∪ [ pW ≪ X,U ∪ W = X ( ¬ W, ⊤ )The term in the union makes sense because if W ≪ X then there exists a W ′ such that W ≪ W ′ ≪ X hence W ⊳ W ′ and ¬ W ∪ W ′ = X hence ω ( ¬ W ). A is bigger than this union because when W ∪ U = X one has ¬ W U (andwhen p holds then ⊤ p ). Conversely, using the fact that unions in O ( X ∞ )are computed componentwise one easily checks that the right hand side unionis indeed smaller than ( U, p ) = A .Now, for any V ≪ U in X one has ω ( ¬ V ) = ⊤ (using a W such that V ≪ W ≪ X ). Hence ( ¬ V, ⊤ ) ∈ O ( X ∞ ) is an open such that ( V, ⊥ ) ∩ ( ¬ V, ⊤ ) = ∅ and ( ¬ V, ⊤ ) ∪ ( U, ⊥ ) = ( X, ⊤ ) hence ( V, ⊥ ) ⊳ ( U, ⊥ ) and( U, ⊥ ) = [ V ≪ U ( V, ⊥ )because X is locally compact.Moreover, if we assume p , then for any W such that W ≪ X and ( U ∪ W ) = ⊤ one has ( ¬ W, ⊤ ) ⊳ ( U, ⊤ ) = ( U, p ) = A as attested by ( W, ⊥ ). Hence, if onewrites: A = [ V ≪ U ( V, ⊥ ) ! ∪ [ pW ≪ X,U ∪ W = ⊤ ( ¬ W, ⊤ ) , then all the terms of the union are rather below A ( ⊳A ) which proves that X ∞ is regular. (cid:3) X ∞ stated in 2.2 are proved, all that remains to do is to prove theuniqueness, and that is what we will do now: Proof :
Let Y be a compact regular locale with a (closed) point denoted ∞ such that Y −{∞} is identified with X . Let i be the inclusion of X into Y , open sublocalesof X will be identified with the corresponding open sublocales of Y included in X . We will first show that for any U ∈ O ( X ) one has ∞ ∈ i ∗ ( U ) if and onlyif ω ( U ) (i.e. ω is the unique “gluing function” giving rise to a compact regularArtin gluing).Indeed, assume ω ( U ), i.e. that there exists a W ∈ O ( X ) such that W ≪ X and W ∪ U = X .Now as W ≪ X in O ( X ) one also have W ≪ X in Y , hence (as Y is regular)there exists W ′ ∈ O ( Y ) such that X ∪ W ′ = Y , (i.e. ∞ ∈ W ′ ) and W ∩ W ′ = ∅ .In particular i ∗ ( W ′ ) ∩ W = i ∗ ( W ′ ) ∩ i ∗ ( W ) = ∅ hence as W ∪ U = X one has i ∗ ( W ′ ) ⊂ U hence W ′ ⊂ i ∗ U , which proves that ∞ ∈ i ∗ U .Conversely, assume that ∞ ∈ i ∗ ( U ), in particular, i ∗ ( U ) ∪ X = Y , hence bylocale compactness of X : Y = [ V ≪ X V ∪ i ∗ ( U )as Y is compact, there exists V ≪ X such that V ∪ i ∗ ( U ) = Y in particular X = i ∗ ( Y ) = i ∗ ( V ∪ i ∗ ( U ) = V ∪ i ∗ i ∗ ( U ) V ∪ U which proves ω ( U ).The end of the proof is then a general fact about Artin gluing: consider thenatural map p : X ` {∞} → Y . This a surjection because X is the opencomplement of {∞} , hence O ( Y ) can be identified with the set of open sublocale A of X ` {∞} such that p ∗ p ∗ A = A .An open sublocale of X ` {∞} is exactly a pair ( U ∈ O ( X ) , m ∈ O ( {∞} ) = Ω)and from the first part of the proof one can deduce that p ∗ p ∗ ( U, m ) = (
U, m ∩ ω ( U )), which show that O ( Y ) is canonically identified with O ( X ∞ ) and thereis a unique identification which is compatible to the inclusion of X and {∞} . (cid:3) f from X ∞ to any locale Y is the same thing as a map f from X to Y and a point f ( ∞ ) ∈ Y such that for any open sublocale U ⊂ Y which contains f ( ∞ ) one has ω ( f ∗ ( U )). Indeed this follows directly from thedecomposition of f ∗ : O ( Y ) → O ( X ∞ ) in the expression of O ( X ∞ ) as a subsetof O ( X ) × Ω.This allows to define:
Definition :
Let C ( X ) be the set of functions f from X to the locale C suchthat for any positive ǫ ∈ Q one has ω ( f ∗ ( B ǫ where B ǫ denote the ball of adius ǫ and of center or, equivalently, the set of functions from X ∞ to C which send ∞ to . Proposition : X ∞ is completely regular if and only if X is. Proof : If X ∞ is completely regular, then any of its sublocales, in particular X , iscompletely regular.Conversely assume that X is completely regular. Let A = ( U, p ) ∈ O ( X ∞ ).Consider first a V ≪ U then there exists W such that V ⊳ CR W ≪ U ⊂ X andany function on X which is zero outside of W can be extended by f ( ∞ ) = 0and hence one has ( V, ⊥ ) ⊳ CR A .Assume now p , then one also has ω ( U ) hence there exists a W such that W ≪ X and U ∪ W = X . Consider any W ′ such that W ≪ W ′ ≪ X , and (as X iscompletely regular) f a function from X to the locale [0 ,
1] such that f is zeroon W and 1 outside of W ′ . As W ′ ≪ X , this function extend to a function from X ∞ to [0 ,
1] which satisfies f ∗ (]0 , ⊂ ( U, ⊤ ) = ( U, p ) and f ( ∞ ) = 1. Let F ∞ the set of such functions, one can write that: A = [ V ≪ U ( V, ⊥ ) ! ∪ [ p,f ∈ F ∞ f ∗ (]1 / , which concludes the proof, as, assuming p , one has f ∗ (]1 / , ⊳ CR A for any f ∈ F ∞ . (cid:3) X and X ∞ : Proposition :
Let X and Y be two regular locally compact locales and X ∞ and Y ∞ their one point compactifications. Let f : X → Y the following conditionsare equivalent:1. For any U ∈ O ( Y ) such that U ≪ Y one has f ∗ ( U ) ≪ X
2. There is an extension f ∞ : X ∞ → Y ∞ of f such that ( f ∞ ) ∗ ( Y ) = X .3. f is proper (see [7, C.3.2.5]).Moreover in this case the extension f ∞ is unique and this induces a bijectionbetween proper maps from X to Y and maps from X ∞ to Y ∞ such that f ∗ ( Y ) = X . Proof : . ⇒ . We define f ∞ by: ( f ∞ ) ∗ ( U, p ) = ( f ∗ ( U ) , p ) . Assuming 1 . if ω ( U ) for U ∈ O ( Y ) then there exists a W such that W ≪ Y and W ∪ U = Y , and hence f ∗ ( W ) ≪ X and f ∗ ( U ) ∪ f ∗ ( W ) = X hence ω ( f ∗ ( U )). This proves that if ( U, p ) ∈ O ( Y ∞ ) , i.e. if p ⇒ ω ( U ) thenone also has p ⇒ ω ( f ∗ ( U )) hence ( f ∗ ( U ) , p ) ∈ Ω( X ∞ ). Moreover asintersections and unions in O ( X ∞ ) are computed componentwise, f ∞ isindeed a morphism of locales. As X and Y correspond to the elements( X, ⊥ ) and ( Y, ⊥ ) of O ( X ∞ ) and O ( Y ∞ ) one also has ( f ∞ ) ∗ ( Y ) = X .2 . ⇒ . In the situation of 2 . , the map f from X to Y is a pullback of the map f ∞ along the open inclusion of Y into Y ∞ . But any map between two compactregular locales is proper (see [7, C.3.2.10 (i) and (ii)]) and a pullback of aproper map is again a proper map (see [7, C.3.2.6]).3 . ⇒ . If f is a proper then f ∗ commutes to directed joins. So if we assume that U ≪ Y , if X is covered by some net V i then Y f ∗ [ i V i ! = [ i f ∗ ( V i )hence there exists a j such that U f ∗ ( V j ) and hence f ∗ ( U ) V j forsome j which concludes the proof of the equivalence.The uniqueness of the extension is immediate because f ∞ is defined both on X and on its closed complement, and hence this indeed induce a bijection asstated in the proposition. (cid:3) Definition :
Let X and Y be two locally compact regular locales. A partialproper map from X to Y is the data of an open sublocale Dom ( f ) ⊂ X and aproper map (denoted f ) from Dom ( f ) to Y . Partial proper maps can be composed (by restricting the domain of definition asmuch as neccessary) and as a pullback of a proper map is proper the compositeof two proper partial maps is again a proper partial map, hence one has acategory of proper partial map.
Proposition :
The category of pointed compact (completely) regular localesis equivalent to the category of locally compact (completely) regular locales andproper partial maps between them.
Proof :
The functor are the same as those of proposition 2.10, they just apply to alarger category: to a map f : X ∞ → Y ∞ of pointed compact regular locale9ne associate the partial map f ′ : X → Y whose domain is f ∗ ( Y ), and f ′ isproper because it is the pullback of f along the inclusion of Y into Y ∞ (see theproof of 2 . ⇒ . in 2.10). Conversely, if f is a partial proper map from X to Y then it extend into a map from U ∞ to Y ∞ by 2.10, composing it to the map r U of the next lemma yields the desired map from X ∞ to Y ∞ and these twoconstructions are clearly inverse of each other. (cid:3) Lemma :
Let X be a locally compact regular locale, and U ⊂ X an opensublocale of X then their exists a (unique) map r U : X ∞ → U ∞ such that r U isthe identity on U and ( r U ) ∗ ( U ) = U . Proof : r U is defined on U ⊂ X ∞ and on its closed complement (as the constant equalto ∞ ). Hence one has a map f U from U ` U c to U ∞ . It is a general fact thatthe canonical map U ` U c → X ∞ is a surjection of locale (hence correspondsto an injection of frame). So all we have to do to proves that f U factors intoa map r U on X ∞ is to check that for any open sublocale ( V, p ) ∈ O ( X ∞ ), theopen sublocal ( f U ) ∗ ( V, p ) of U ` U c comes from an open sublocale of U ∞ .By definition, ( f U ) ∗ ( V, p ) is V on the U part and s ∗ ( p ) on the U c part (where s is the canonical map U c → ∗ ). If we assume p then there exists a W ≪ U suchthat W ∪ V = U , and as W ≪ U , there exists a D ⊂ X ∞ such that W ∩ D = ∅ and U ∪ D = X ∞ . We define: V ′ = V ∪ a pD D where the coproduct is on the set of D such that p holds and D satisfy theproperties just describe. If f denotes the map U ` U c → X ∞ then p ∗ ( V ′ ) =( V ′ ∩ U, V ′ ∩ U c ). On one hand V ′ ∩ U = V because as D ∩ W = ∅ and W ∪ V = U one has D ∩ U ⊂ V , and on the other hand, U c ∩ V ′ = U c ∩ ` p,D D , and as D ∪ U = X ∞ , one has U c ⊂ D hence, U c ∩ V ′ = s ∗ ( p ), and this concludes theproof. (cid:3) C ∗ -algebras C ∗ -algebras as for example in [1]. Inparticular the norm of an element is only assumed to be a upper semi-continuousreal number (the definition using rational ball of [1] is equivalent to a normfunction with value into the non-negative upper semi-continuous real numbers).Of course contrary to [1], we do not assume the algebras to be unital.10.2. If C is a a C ∗ -algebra we define C + as the as the set of couples ( c, z ) with c ∈ C and z ∈ C , We endow C + with the componentwise addition and themultiplication ( c, z )( c ′ , z ′ ) = ( cc ′ + cz ′ + zc ′ , zz ′ ). It is a unital algebra, withunit (0 , C + with the anti-linear involution ( c, z ) ∗ = ( c ∗ , z ).If ( c, z ) ∈ C + we define: k ( c, z ) k = max( | z | , sup c ′ ∈ C k c ′ c + c ′ z k )Where C denotes the set of elements of C of norm
1, and the supremum isto be considered in the set of upper semicontinuous real numbers.Remark: Classically, it is usual to define the norm on C + to be simply sup c ′ k c ′ c + c ′ z k . This works perfectly when C is indeed non-unital, but when C is unitalthis gives a norm 0 for the element (1 , −
1) and hence (after taking the quotientby the ideal of norm zero elements) with this definition C + will be isomorphicto C when C is unital. This is not what we want because if X is a compactlocale, then its one point compactification X ∞ is not X itself but X ` {∞} .Classically this difference is harmless but in intuitionist logic the question ofbeing compact/unital might be non decidable and hence it is important to havea uniform treatment on both side.3.3. Before proving that C + is indeed a C ∗ -algebra we need a few lemmas whichare immediate in classical mathematics but require to be slightly more carefulin intuitionist mathematics. Lemma :
Let x be a nonnegative upper semicontinuous real number and q bea nonnegative rational number then if x qx one has x q . Proof :
Let e be a rational number such that x q + e .One has: x q + qe ( q + e hence x q + ( e )by induction one obtains that for all k > x q + e k and hence that x q which concludes the proof. (cid:3) Lemma :
Let C be a C ∗ -algebra and c ∈ C then: k c k = sup b ∈ C k bc k = sup b ∈ C k cb k Proof :
We start by the first equality. It is immediate that k bc k k c k for any b of norm b ∈ C k bc k k c k We only need to prove the reverse inequality. let q be a rational number suchthat sup b ∈ C k bc k < q , i.e. there exists a q ′ < q such that for all b , k bc k < q ′ .Let α be a rational number such that k c k < α . One has k c ∗ /α k < α k c ∗ c k < q ′ q ′ k c ∗ c k < α, and as this holds for any α such that k c k < α this proves that:1 q ′ k c ∗ c k k c k Using k c ∗ c k = k c k one obtains k c k q ′ k c k and hence by lemma 3.3 this provesthat k c k q ′ < q which concludes the proof of the first equality.The second equality follows either by exactly the same proof, or by applyingthe first equality using k c ∗ k = k c k and ( cb ) ∗ = b ∗ c ∗ . (cid:3) Lemma :
For any x = ( c, z ) ∈ C + one has: sup b ∈ C k cb + bz k = sup b ∈ C k bc + bz k , and, k x k = k x ∗ k . Proof :
By lemma 3.4 one has:sup b ∈ C k cb + bz k = sup b ∈ C sup b ′ ∈ C k b ′ cb + b ′ bz k But the two supremums can be exchanged and as:sup b ∈ C k bc + bz k = sup b ∈ C sup b ′ ∈ C k bcb ′ + zbb ′ k k x k = k x ∗ k follows immediately: k x ∗ k = max( | z | , sup b ∈ C k bc ∗ + bz k )and: sup b ∈ C k bc ∗ + bz k = sup b ∈ C k cb ∗ + zb ∗ k = sup b ∈ C k cb + zb k which concludes the proof. (cid:3) Proposition : C + is a C ∗ -algebra. Proof :
The fact that k . k is a norm of algebra (i.e. such that k xy k k x kk y k ) is easyand exactly as in the classical case. Thanks to the term | z | in the definition itis an actual norm and not a semi-norm. C + is complete for this norm because it is complete for the norm | z | + k c k as aproduct of two Banach spaces, and these two norms are equivalent, indeed, onehas in one direction: k ( c, z ) k | z | + k c k and in the other, one clearly have: | z | k ( c, z ) kk c k − | z | k ( c, z ) k hence: | z | + k c k k ( c, z ) k All we have to do to conclude is to prove the C ∗ -equality k x ∗ x k = k x k . Let x = ( c, z ) and element of C + then: x ∗ x = ( c ∗ c + zc ∗ + zc, | z | )Hence, k x ∗ x k = max( | z | , sup b ∈ C k c ∗ cb + zc ∗ b + zcb k )as, k c ∗ cb + zc ∗ b + zcb k > k b ∗ c ∗ cb + zb ∗ c ∗ b + zb ∗ cb k = k ( cb + zb ) ∗ ( cb + zb ) k = k cb + zb k One obtains that k x ∗ x k > k x k . The other inequality follow from lemma 3.5together with the fact that k x ∗ x k k x ∗ kk x k . (cid:3) C with the two sidded ideal of C + of elements of the form( c, χ ∞ the character of C + defined by χ ∞ ( c, z ) = z .3.8. Proposition :
Let C be a (possibly non unital) C ∗ -algebra and B be aunital C ∗ -algebra. Any morphism from C to B extend uniquely into a unitalmorphism of C ∗ -algebra from C + to B . Proof :
The extension is necessary defined by f ( c, z ) = f ( c ) + z , it is clearly a morphismof ∗ -algebra. It is continuous because: k f ( c, z ) k k f ( c ) k + | z | k f kk c k + | z | and as we observed in the proof of proposition 3.6 the norm k c, z k is equivalentto the norm k c k + | z | this proves that this extension is continuous. (cid:3) Proposition :
Let X be a locally compact completely regular locale, then C ( X ∞ ) ≃ ( C ( X )) + Proof : C ( X ) is identified with the set of functions on X ∞ which send ∞ ∈ X ∞ to 0,hence this induce a morphism from ( C ( X )) + to C ( X ∞ ) by proposition 3.8. Afunction f ∈ C ( X ∞ ) can be written in a unique way h + c with h ∈ C ( X ) and c a constant: c has to be f ( ∞ ) and h = f − f ( ∞ ) hence the map from ( C ( X )) + to C ( X ∞ ) is a bijection. One easily check that it is isometric, either by generaltheorems on C ∗ -algebras (which should of course be proved constructively first)or directly: If f is a function on X ∞ then k f k < q if and only if both f ( ∞ ) < q (which imply that f < q on some neighbourhood of ∞ ) and for each U ⊳ CR X the function f is strictly smaller than a q ′ < q on U , which is equivalent tothe fact that k f h k < q ′ < q for every h ∈ C ( X ) of norm
1. And these twoconditions are equivalent to the fact that k ( f − f ( ∞ ) , f ( ∞ )) k < q . (cid:3) C is a C ∗ -algebra in a topos E and f : T → E is a geometric morphism, then f ∗ ( C ) is in general not a C ∗ -algebra: it still satisfies all the “algebraic” axioms.But it might not be complete and separated (in the sense that k x k = 0 ⇒ x = 0).Fortunately, the separated completion of f ∗ C is complete and separated andhence is a C ∗ -algebra which we denote by f ♯ ( C ). Proposition :
For any C ∗ -algebra C one has a natural isomorphism: This also corresponds to the pullback of the localic completion of C , hence this is essentiallycompatible with the notation for pullback of locales ♯ ( C + ) ≃ f ♯ ( C ) + Proof : In T , the canonical morphisms of pre- C ∗ -algebras f ∗ C → f ♯ ( C ) and f ∗ C → C extend into a map f ∗ C × f ∗ C → f ♯ ( C ) + . One can check that the semi-normand the pre- C ∗ -algebra structure induced on C × f ∗ C by this map are exactlythose of f ∗ ( C + ), hence f ♯ ( C + ) is exactly the closure of f ∗ ( C + ) in f ♯ ( C ) + . But f ∗ ( C + ) is clearly dense (because each component is dense) hence this concludesthe proof. (cid:3) Definition : If C is a C ∗ -algebra we denote by Spec ∞ C the spectrum ofthe unital C ∗ -algebra C + and by Spec C the locally compact completely regularlocale obtained by removing the point ∞ of Spec ∞ C . Of course by the uniqueness property in theorem 2.2, Spec ∞ C is the one pointcompactification of Spec C . Also, if C is unital then C + is isomorphic to C × C hence Spec ∞ C is isomorphic to Spec C ` {∞} and the two definitions of Spec C (by considering C as a unital or general C ∗ -algebra) agree and there is no possibleconfusion.At this point, the following theorem is immediate:4.2. Theorem :
The category of commutative C ∗ -algebras and arbitrary mor-phisms between them is anti-equivalent to the category of locally compact com-pletely regular locales and partial proper maps between them. The equivalence isgiven on object by the constructions C and Spec . Proof :
The process of unitarization produce an equivalence between the category ofcommutative C ∗ -algebra and arbitrary morphism, and the category of unital C ∗ -algebras endowed with a character χ ∞ and unital morphism compatible tothe character. Applying the Gelfand duality for unital C ∗ -algebra this categoryis in turn anti-equivalent to the category of pointed compact completely regularlocales, which by proposition 2.11 is equivalent to the category of locally com-pact completely regular locale and partial proper map between them. Underthese composed equivalences, a commutative C ∗ -algebra C is associated to thespectrum of C + minus the point at infinity, i.e. exactly Spec C and a locallycompact completely regular locale X is associated to the algebra of functionson X ∞ which vanish at ∞ , which is C ( X ). (cid:3)
15n the rest of this section, we will give interpretation of Spec and Spec ∞ interm of classifying space of characters (proposition 4.4), we will show that theopen sublocales of Spec C correspond to the closed ideals of C (theorem 4.7) andthat total proper maps of locales correspond to non-degenerate morphisms of C ∗ -algebras (theorem 4.8).4.3. We recall that when C is a unital commutative C ∗ -algera, then Spec C denotes the classyfing space of the theory of characters of C . A precise geometricformulation of this theory can be found in [1] or in [4], but this can also beinterpreted as the fact that for any locale Y and p : Y → {∗} the canonical map,functions from Y to Spec C correspond to morphisms from p ♯ ( C ) (or equivalentlyfrom p ∗ ( C )) to C internally in Sh ( Y ).4.4. Proposition :
The locale Spec ∞ C classifies “nonunital characters” of C ,i.e. possibly nonunital morphism of C ∗ -algebras from C to C .The locale Spec C classifies “nonzero characters” of C , i.e. characters whichsatisfy the additional axiom ∃ c ∈ C , | χ ( c ) | > Proof :
The important observation is that the process of unitarization of C ∗ -algebrascommute with pullback along geometric morphisms. Hence points of Spec ∞ C over any locale L (with p its canonical morphism to the point) are the charactersof p ♯ ( C ) + which are in bijection with the non unital morphisms from p ♯ ( C ) to C which proves the first part of the result.For the second part, let us denote by D ( f ) the (biggest) open sublocale ofSpec ∞ C on which | f | >
0, where f is an element of C . Using the completeregularity of Spec ∞ C it appears that Spec C is the union of the D ( f ) for f ∈ C .For each f ∈ C , the open sublocale D ( f ) classifies characters of C such that | χ ( f ) | >
0. Hence points of Spec C are the characters such that ∃ c ∈ p ∗ ( C ) with | χ ( c ) | > ∃ c ∈ C ” in the statement of the proposition is unambiguousbecause as p ∗ ( C ) is dense in p ♯ ( C ) it is equivalent to says that ∃ c ∈ p ∗ ( C ) , | χ ( c ) | > ∃ c ∈ p ♯ ( C ) , | χ ( c ) | > (cid:3) Lemma :
Let X be a locally compact completely regular locale and U ⊂ X an open sublocale, then the restriction to U of functions in C ( X ) which vanishoutside of U are exactly the functions in C ( U ) . Proof :
Let f be a function in C ( X ) which vanish outside of U , we will show that therestriction of f to U is in C ( U ). Let ǫ be any positive rational number, let V be the open sublocale of X on which | f | > ǫ . As f ∈ C ( X ), V ≪ X and as f vanish outside of U on has V ⊳ CR U and in particular V ⊳ U . As mentioned in16he preliminaries, this two properties together imply V ≪ U . This being truefor any ǫ this proves that f ∈ C ( U ).Conversely, assume that f ∈ C ( U ), then because of the map r U : X ∞ → U ∞ constructed in 2.12 one can define a map f ◦ r U on X ∞ which vanish at infinityand outside of U and which coincide with f on U . (cid:3) Lemma :
Let I ⊂ C be an ideal of a commutative C ∗ -algebra. and let χ bea non-zero character of I (in the sense that ∃ i ∈ I, | χ ( i ) | > ). Then χ admit aunique extension as a character of C . Proof :
The proof is exactly as in the classical case : any extension of χ to C has tosatisfy χ ( c ) = χ ( ci ) /χ ( i ) for any i ∈ I such that | χ ( i ) | > f, g two elements of I such that | χ ( f ) | > | χ ( g ) | >
0, and c any element of C then, as: χ ( gf c ) = χ ( g ) χ ( f c ) = χ ( f ) χ ( gc )one has: χ ( f c ) χ ( f ) = χ ( gc ) χ ( g ) . This proves that we can define χ ( c ) = χ ( f c ) /χ ( f ) for any f ∈ I such that χ ( f )is invertible and as χ ( c ) χ ( c ′ ) = χ ( f c ) χ ( f c ′ ) /χ ( f ) = χ ( f cc ′ ) /χ ( f ) = χ ( cc ′ ),the extension of χ is a character of C . (cid:3) Theorem :
The constructions C and Spec induce for any commutative C ∗ -algebra C an order preserving bijection between the open sublocales of Spec C and the closed ideals of C .Moreover, if f : C → C ′ is a morphism between commutative C ∗ -algebra thepull-back of open sublocales along the corresponding (partial) continuous mapcorresponds under this bijection to the map which send an ideal I of C to theclosure of the ideal spammed by f ( I ) . Proof :
Because C is an ideal of C + it suffices to prove these results for unital algebrasand a unital morphism.If U ⊂ Spec C is an open sublocale, then by lemma 4.5 one can identify C ( U )with an ideal of C whose spectrum is U . Conversely, if I ⊂ C is an idealthen an application of lemma 4.6 internally in Spec I give rise to a map fromSpec I to Spec C and a map from an arbitrary locale L to Spec C factor intoSpec I if and only the corresponding character of C in the logic of L satisfies Except that we need to be more careful on the “non-zero” hypothesis which is unessentialin the classical case. i ∈ I, | χ ( i ) | > L ). HenceSpec I is identified precisely with the open sublocales of C defined by S i ∈ I D ( i ).And as C (Spec I ) = I this proves that this two constructions are inverse of eachother, and they clearly preserve the order.For the second part, if f : C → C ′ is a unital morphism of C ∗ -algebra, g thecorresponding continuous map Spec C ′ → Spec C , and if I ⊂ C and I ′ ⊂ C ′ aretwo closed ideals, then f ( I ) ⊂ I ′ if for any function h on Spec C which vanishoutside of Spec I its composite with g vanish outside of Spec I ′ . This will bethe case if and only if g ∗ (Spec I ) ⊂ Spec I ′ . Hence the ideal corresponding to g ∗ (Spec I ) is indeed the smallest closed ideal containing f ( I ). (cid:3) C ∗ -algebra is said to be non-degenerate it its image spama dense ideal. A morphism between unital C ∗ -algebra is non-degenerate if andonly if it is unital. Theorem :
The equivalence of category of theorem 4.2 restrict to a (contravari-ant) equivalence of category between the category of commutative C ∗ -algebrasand non degenerate morphisms and the category of locally compact completelyregular locales and proper maps between them. Proof :
A morphism f : C → C ′ will corresponds to a total map on the spectrum if andonly if the corresponding partial proper map g satisfy g ∗ (Spec C ) = Spec C ′ ,i.e., applying the previous theorem, if and only if f is non-degenerate. (cid:3) X is said to be positive if when X = S i ∈ I U i then ∃ i ∈ I . This is a “positive” way of saying that X is non-zero. We will say thata locale X is locally positive if every open sublocale of X can be written as aunion of positive open sublocales. Assuming the law of excluded middle, a localeis positive if and only if it is non-zero and any locale is locally positive, but inan intuitionist framework locale positivity is an extremly important properties:A locale X is locally positive if and only if the map from X to the terminallocale is an open map (see [7, C3.1.17]) for this reason locally positive localeare aslo called open locale (but this cause a confusion with open sublocales) orsometimes overt.5.2. Theorem :
Let C be a commutative C ∗ -algebra, then the following condi-tions are equivalent: • For any c ∈ C , the norm of c is a continuous real number. There is a dense family of elements of C whose norms are continuous realnumbers. • Spec C is locally positive. (i.e. is open or overt). It appears that this result was already known for unital algebras and due toT.Coquand in [3, section 5]. We do needed the result for non-unital algebrasin [6], but one could also deduce the non-unital case from the unital one usingthe unitarization process developed in section 3. This being said, we were notaware of Coquand’s paper at the time the first version of this paper has beenwritten, and as the following proof is more complete than the original one wedecided to leave it here.
Proof :
The first two conditions are clearly equivalent because a semi-continuous realnumber which can approximated arbitrarily closed by continuous real numbersis also continuous.Assume first that the first two conditions hold.We recall that if f ∈ C then D ( f ) denotes the largest open sublocale of Spec C on which | f | >
0. Let also p denotes the canonical map from Spec C to theterminal locale.We will first show that: D ( f ) ⊂ p ∗ (“ k f k > . Indeed, in the logic of Spec C , D ( f ) is the proposition χ ( | f | ) >
0, which implythat ∃ ǫχ ( | f | ) > ǫ >
0. But, as k f k is continuous, one has (still internally inSpec C ) k f k < ǫ or k f k > k f k < ǫ is in contradiction with χ ( | f | ) > ǫ , hence k f k > D ( f )) f ∈C form a basis of the topology of Spec C , hence as: D ( f ) = D ( f ) ∩ p ∗ (“ k f k > [ k f k > D ( f )the D ( f ) for k f k > C . We will nowprove that the D ( f ) for k f k > k f k > ǫ > k f k > ǫ . Let U i be a famillyof open sublocales of D ( f ) such that: D ( f ) = [ i ∈ I U i Let W be the open sublocale on which | f | is greater than ǫ/
2. One has
W ⊳ CR D ( f ) by definition and W ≪ Spec C because f ∈ C ( C ), hence, as mentioned inthe preliminaries, one has W ≪ D ( f ), hence there exists a finite subset J ⊂ I such that: W ⊂ [ j ∈ J U j As J is finite, it is either empty or inhabited, but if J is empty then W isempty hence f is smaller than ǫ/ C and hence k f k < ǫ J is inhabited and hence that I is inhabited, which concludes the proof of the first implication.We now assume that Spec C is a locally positive locale. For any h ∈ C , wedenote ( | h | > q ) the biggest open sublocale of Spec C on which | h | > q holds(where h is seen as a function on Spec C ). We fix an element h ∈ C , and we willprove that k h k is a continuous real number. We define: L = { q ∈ Q | q < | h | > q ) is positive } one has: • If q ∈ L and q ′ < q then q ′ ∈ L • L is inhabited (it contains all the negative rational numbers). • if q ∈ L then there exists q ′ ∈ L such that q < q ′ , indeed, if q < | h | > q ) is positive then it is the union for q ′ > q of the( | h | > q ′ ), as Spec C is assumed to be locally positive ( | h | > q ) is also theunion of the ( | h | > q ′ ) which are positive and hence there exists a q ′ suchthat ( | h | > q ′ ) is positive.This shows that L is a lower semi-continuous real number. We will show that( L, k h k ) form a continuous real number, which means that k h k is a continuousreal number. • Let q such that q ∈ L and k h k < q , this means that | h | is both smallerthan q everywhere and bigger than q on some positive sublocale which isimpossible. Hence L ∩ k h k = ∅ . • Let q < q ′ be two rational numbers. Internally in Spec C one has | h | < q ′ or q < | h | . Hence Spec C is the union of the open sublocales ( | h | < q ′ )and ( q < | h | ), moreover as h ∈ C (Spec C ) one has ( q < | h | ) ≪ Spec C .By locale positivity of Spec C , the open sublocale ( q < | h | ) can be writtenas a union positive open sublocales ( u i ) for i ∈ I . In particular:Spec C = ( | h | < q ′ ) ∪ [ i ∈ I u i hence, as ( q < | h | ) ≪ Spec C , there exists a finite subset J ⊂ I such that:( q < | h | ) ⊂ ( | h | < q ′ ) ∪ [ j ∈ J u j As J is finite, it is either empty or inhabited. If J is empty, then ( q < | h | ) ⊂ ( | h | < q ′ ) hence ( | h | < q ′ ) = ([ h | < q ′ ) ∪ ( q < | h | ) = Spec C hence k h k < q ′ . On the other hand, if J is inhabited then ( q < | h | ) contains apositive open sublocale, hence it is positive and hence q ∈ L . This provesthat either k h k < q ′ or q ∈ L .This two conditions together show that ( L, k h k ) form a continuous real numberand this concludes the proof. (cid:3) Extension of the results to localic C ∗ -algebras C ∗ -algebras” and proved (aspreviously conjectured by C.J.Mulvey and B.Banachewski in [2]) that the (con-structive) Gelfand duality can be extended into a duality between compactregular locales and localic commutative unital C ∗ -algebras. The goal of thislast section is to explain how the methods developed in [5] allow to extend theresults of the present paper to the localic framework (we have in mind theorems4.2, 4.7 and 5.2 and proposition 4.4). In particular, this section is not meant tobe read independently of [5].6.2. Let us start with the construction of the spectrum of a localic C ∗ -algebra. Proposition : If C is a (possibly non-unital) commutative C ∗ -locale then thereexist locales Spec ∞ C and Spec C such that Spec ∞ ( C ) classifies the morphisms χ : C → C of C ∗ -locales, and Spec C classifies those which satisfy additionally“ χ − ( C − { } ) is positive”. Moreover Spec C is a locally compact regular localeand Spec ∞ ( C ) is its one point compactification. Finally, these constructions arecompatible with pullback along geometric morphisms. Proof :
In section 3 . C to C denoted [ C , C ] , one can then construct Spec ∞ C as a sublocale of[ C , C ] using the same kind of co-equalizer as in 4 . . χ − ( C −{ } ) is positive ” is an open subspace of [ C , C ] (it is even one of the basic opensubspace) hence Spec C will be an open subspace of Spec ∞ C .The compatibility with pullback along geometric morphisms follows immediatelyfrom this definition as classifying space.For the rest of the proposition we can use descent theory: from proposition 2 . . L , with p : L → {∗} the canonical map, such that p ♯ ( C )is weakly spatial, hence is the localic completion of an ordinary C ∗ -algebra. Inparticular, as character of a C ∗ -algebra and of its localic completion are thesame, p ♯ (Spec C ) and p ♯ (Spec ∞ C ) are the spectrums of an ordinary C ∗ -algebra,hence they are respectively locally compact completely regular and compactcompletely regular and the second is isomorphic to the one point compactifica-tion of the first. (Complete) regularity alone is not a property that descend wellalong open surjections, but it is proved in [7, Lemma C3.2.10] that for compactlocale regularity is equivalent to begin Hausdorff and [7 , C . .
7] prove that as p ♯ (Spec ∞ C ) is compact and separated, Spec ∞ C is also compact and separated,hence compact regular. As Spec C is an open subspace of Spec ∞ C it is locallycompact and regular. Finally, as one point compactification is also compatiblewith pullback along geometric morphism the isomorphism between p ♯ (Spec ∞ C )and the one point compatification of p ♯ (Spec C ) descend into an isomorphismbetween Spec ∞ C and the one point compactification of Spec C (which of courseis compatible with the natural inclusion of Spec C ). (cid:3) Theorem :
There is an anti-equivalence of categories between the cate-gory of commutative (possibly non-unital) C ∗ -locales and the category of locallycompact regular locales and partial proper maps between them. Proof :
The proof given in [5, 4.2.5] that the ordinary Gelfand duality extend to thelocalic Gelfand duality applies to the non-unital case almost without any change:If X is a locally compact regular locale one can define the C ∗ -locale C ( X ) as thekernel of the evaluation at infinity on the C ∗ -locale C ( X ∞ ). C ( X ) is a C ∗ -locale:the only non trivial point to check is that it is locally positive but it follows fromthe fact that the map C ( X ∞ ) → C ( X ) which send f to f − f ( ∞ ) is a surjection.There is a canonical map from X to Spec C ( X ). By [5, 2.3.17 and 2.6] thereexists a positive locally positive locale L such that p ♯ ( X ∞ ) is completely regularin the internal logic of L (and hence also p ♯ ( X )), in particular the canonical map p ♯ ( X ) → p ♯ (Spec C ( X )) ≃ Spec C ( p ♯ ( X )) is an isomorphism because of theordinary non-unital gelfand duality applied to the C ∗ -algebra of points of C ( X ),and because open surjections are effective descent morphisms (and p is an opensurjection) this imply that X ≃ Spec C ( X ).The exact same argument also show that for any commutative C ∗ -locale C thecanonical map C → C (Spec C ) is an isomorphism, and the correspondence ofmorphisms is also obtains exactly in the same way (because partial maps descendwell : first apply descent to their domain and then to the map itself). (cid:3) Theorem :
Let C be a commutative C ∗ -locale, then there is an order pre-serving bijection between open sublocales of Spec C and locally positive fiberwiseclosed ideals of C . One might be surprised to obtain “fiberwise” closed ideals, and not just closedideals in this duality. But there is no reason to be, one should just notice thatthe usual notion of closeness ans density that we use in the constructive theoryof Banach space does not correspond to closeness and density but to fiberwisecloseness and fiberwise density.Indeed, a point x is in the closure of a subset S of a Banach space B if forall ǫ > s ∈ S such that k x − s k < ǫ , i.e. such that for allneighbourhood V of x , there is a point in V ∩ S , i.e. in localic terms, V ∩ S ispositive which exactly says that x is in the fiberwise closure of S . Proof :
Let C be a commutative C ∗ -locale, and let I be a locally positive fiberwise closedideal of C . In particular I is a C ∗ -locale and hence has a spectrum U = Spec I .Let L be a positive locally positive locale such that p ♯ ( I ) and p ♯ ( C ) are weaklyspatial, by 4.7 p ♯ ( U ) identify with a open sublocale of p ♯ (Spec C ). This openinjection is canonical hence compatible to the descent data and hence comesfrom a map from U to Spec C which also has to be an open injection (becauseits pullback along p is an open injection). Conversely, if U is an open sublocaleof Spec C then C ( U ) is a C ∗ -locale and identify by the same descent argumentwith a locally positive fiberwise closed ideal of C , and these two construction are22nverse of each other essentially because of the localic gelfand duality we justproved. (cid:3) And finally:6.5.
Theorem :
Let C be a commutative C ∗ -locale, then the following condi-tions are equivalent: • Spec C is locally positive. • the norm map from C to the locale of upper semi-continuous real numberfactors into the natural map from the locale of continuous real number tothe locale of upper-semi-continuous number. Proof :
Let L be a positive locally positive locale and p the canonical map p : L → {∗} such that p ♯ ( C ) is weakly spatial.Assume the first condition, then in the logic of L , the locale Spec p ♯ C ≃ p ♯ Spec C is still locally positive, hence by theorem 5.2 each point of p ♯ C has a continuousnorm. The subalgebra of points is fiberwise dense and endowed with a map tothe locale of continuous real number which factors the norm map. This mapfrom the locale of points to the locale of real numbers is clearly a metric mapand hence extend by completion to a map from the all of p ♯ C to the locale R which also factors the norm (by [5, 3.3 and 3.2.5]). By uniqueness of such afactorisation, it is compatible with the descent datas on p ♯ C and p ♯ R and henceinduces a map from C to R which also factor the norm and this concludes theproof of the first implication.We now assume the second condition. This factorisation of the norm impliesthat every point of p ♯ ( C ) has a continuous norm, hence Spec p ♯ ( C ) ≃ p ♯ (Spec C )is locally positive by 5.2 and hence Spec C is also locally positive by [7, C5.1.7]. (cid:3) References [1] Banaschewski, Bernhard and Mulvey, Christopher J. The spectral theoryof commutative c-algebras: the constructive spectrum.
Quaestiones Mathe-maticae , 23(4):425–464, 2000.[2] Banaschewski, Bernhard and Mulvey, Christopher J. A globalisation of theGelfand duality theorem.
Annals of Pure and Applied Logic , 137(1):62–103,2006. 233] Thierry Coquand. About stone’s notion of spectrum.
Journal of Pure andApplied Algebra , 197(1):141–158, 2005.[4] Coquand, Thierry and Spitters, Bas and others. Constructive Gelfand du-ality for C*-algebras. In
Mathematical Proceedings of the Cambridge Philo-sophical Society , volume 147, pages 323–337. Cambridge Univ Press, 2009.[5] Henry, Simon. Localic Metric spaces and the localic Gelfand duality. arXivpreprint arXiv:1411.0898 , 2014.[6] Henry, Simon. Toward a non-commutative Gelfand duality: Boolean locallyseparated toposes and Monoidal monotone complete C*-categories. arXivpreprint arXiv:1501.07045 , 2015.[7] Johnstone, P.T.
Sketches of an elephant: a topos theory compendium .Clarendon Press, 2002.[8] Picado, Jorge and Pultr, Ale´es.