Constructive Aspects of Gelfand Duality Christopher Mulvey - - PowerPoint PPT Presentation

Constructive Aspects of Gelfand Duality

Christopher Mulvey University of Sussex

www.maths.sussex.ac.uk/Staff/CJM c.j.mulvey@cantab.net

Background

In this talk, I will tell you something about Gelfand duality, seen through the eyes of the constructive world. The work that I'll be outlining dates from almost exactly thirty years ago, when it began to be realised that the construction of locales out of geometric theories could be used to provide constructive versions of mathematical results that classically had depended upon, indeed often been almost equivalent to, the Axiom of Choice. The beginnings of this work in fact lay in events of almost exactly ten years earlier, in the earliest days of topos theory. At the International Congress of Mathematicians in Nice in 1970, Lawvere had given an impressive talk outlining the significant ideas that were then unfolding and the contributions that they were likely to make. In talking about these developments he had wanted to emphasise the unity of the opposites of geometry and logic, by indicating an aspect of each to which the emerging theory applied. The first was to the internalisation of the construction of the spectrum of a commutative ring, which had been analysed by Monique Hakim to provide an important part of Grothendieck's contributions to algebraic geometry. The second was to the independence of the continuum hypothesis, that applied the topos theoretic context to yield a categorical approach to the proof.

Background

Although the insights needed in each case were already present, the detail needed further

• development. In each case, this eventually came with the realisation of the centrality and

importance of the theory of locales, of which we have already heard certain aspects, to the theory of toposes. In the case of the spectrum of a commutative ring A , Lawvere had correctly noted that the points intended were not the prime ideals of the ring, but their complements, which I shall refer to as the primes of the ring. In the constructive context of a topos, these had their own simple axiomatisation, as those subsets P satisfying the following conditions:

true d 1 c P 0 c P d false a + b c P d a c P - b c P ab c P dw a c P . b c P

for all .

a,b c A

That we have written the axioms in this form will have significance in a moment.

Background

The assertion made at that time, that the spectrum Spec A of a commutative ring A in a topos was exactly its space of primes, was misguided. The correct approach to the concept of spectrum, although not yet its correct context, had already been introduced by Joyal prior to a talk given at Oberwolfach in 1972 in which he described the spectrum as being the universal support of the commutative ring, in the sense of a mapping D : A d L from A to a distributive lattice L satisfying universally the conditions:

1L [ D(1) D(0) [ 0L D(a + b) [ D(a) - D(b) D(ab) = D(a) . D(b)

for all . It may be noted that these are just the conditions satisfied classically by the

a,b c A

basis for Zariski topology on the spectrum. It may also be noted that the axioms for a prime are just a transcription of these conditions into another notation.

Background

With the realisation of the importance of the theory of locales, the concept of spectrum of a commutative ring A introduced by Joyal could be rephrased as a mapping D : A d L from A to a locale L satisfying universally with respect to homomorphisms the conditions:

1L [ D(1) D(0) [ 0L D(a + b) [ D(a) - D(b) D(ab) = D(a) . D(b)

for all .

a,b c A

In the case of a commutative ring A in the topos of sets, the locale created is exactly that of the Zariski topology on the set of primes (or equivalently, prime ideals) of the ring. More importantly, it may be shown that in any topos the locale plays the role of the spectrum of A .

Background

Although in this case the concept of the spectrum Spec A as the universal support of the commutative ring A was arrived at pragmatically, it may equally be obtained canonically as the Lindenbaum locale of the propositional geometric theory that introduces a proposition a c P for each together with the axioms:

a c A true d 1 c P 0 c P d false a + b c P d a c P - b c P ab c P dw a c P . b c P

for all that we have already met. In other words, once one has the constructive form

a,b c A

• f the classical points of the spectrum, then forming the constructive version of the classical

space is straightforward: one just takes the Lindenbaum locale of the theory of its constructive points, namely the locale obtained by taking the propositions generated by the theory, modulo provable equivalence within the theory, ordered by provable entailment within the theory.

Background

The second application outlined by Lawvere was a sketch of a proof of the independence of the continuum hypothesis by topos theoretic means. The dénouement of this approach, in terms of the detail, also came at the Oberwolfach meeting in 1972, when it was pointed out that the concept of real number on which it was based was not quite that intended. That described by Tierney in his talk defined in the topos of sheaves on a space not the sheaf of continuous real functions, but instead the sheaf of upper semi-continuous real functions on the space. On this occasion, inadequate attention had been given to the importance constructively of defining a real number by describing both its lower cut and its upper cut on the

x c ‘

• rationals. Explicitly, a real number

needed to be defined as a pair

• f subsets of the

x c ‘ ( - x,

.

x)

rationals satisfying the conditions:

≥p p c - x ≥q q c

.

x p c - x . p∏ < p d p∏ c - x q c

.

x . q < q∏ d q∏ c

.

x p c - x d ≥p ∏>p p∏ c - x q c

.

x d ≥q∏<q q∏ c

.

x

.

p c - x . q c

.

x d p < q p < q d p c - x - q c

.

x

Background

With the constructive axiomatisation of the real numbers in place, the same approach as in the case of the spectrum may be applied to construct the locale of real numbers by writing down in any topos with natural numbers a propositional geometric theory of which the real numbers in the topos are the canonical models, hence the points of the locale described by that theory. The propositional geometric theory of real numbers is defined by introducing for each pair r ,s

• f rational numbers in the topos a proposition

,

x c (r,s)

together with the following axioms: whenever ;

x c (r,s) d false (r,s) [ 0

;

true d -(r,s) x c (r,s)

whenever ;

x c (r,s) d x c (p,q) - x c (p∏,q∏) (r,s) ü (p,q) - (p∏,q∏)

whenever ;

x c (p,q) . x c (p∏,q∏) d x c (r,s) (p,q) . (p∏,q∏) ü (r,s)

.

x c (r,s) dw -(r∏,s∏)ü(r,s) x c (r∏,s∏)

The locale of real numbers in the topos is then the Lindenbaum locale of this theory.

‘

Background

Similarly, consider the propositional geometric theory defined by introducing a proposition

z c (r,s)

for each pair

• f complex rationals, together with the following axioms:

r,s

;

z c (r,s) d false whenever (r,s) [ 0

;

true d -(r,s) z c (r,s)

;

z c (r,s) d z c (p,q) - z c (p∏,q∏) whenever (r,s) ü (p,q) - (p∏,q∏)

;

z c (p,q) . z c (p∏,q∏) d z c (r,s) whenever (p,q) . (p∏,q∏) ü (r,s)

,

z c (r,s) dw -(r∏,s∏)ü(r,s) z c (r∏,s∏)

in which the conditions are to be interpreted in the geometry of the complex rational plane, expressed algebraically. Then the Lindenbaum locale of the theory is the locale of complex numbers. In each case,

Š

the points of the relevant locale are respectively the real numbers and the complex numbers of the topos. Equivalently, these are exactly the canonical models of the respective theory.

Background

The Gelfand duality theorem in its constructive form was proved in ten days of intensive work with Banaschewski over the summer of 1980, following similarly intense work with Wick-Pelletier on the constructive form of the Hahn-Banach theorem, which developed many

• f the techniques used. The motivation and the outcome were linked talks at the International

Meeting on Categorical Topology held in Ottawa that summer. The ten days of work led to a widely circulated preprint of some 180 pages detailing the result, which was deemed too lengthy for publication. Over the years that followed, it was divided into four papers, inevitably coming to rather more than 180 pages, which were published individually, under the unifying title of The Spectral Theory of Commutative C*-Algebras:

Background

The Gelfand duality theorem in its constructive form was proved in ten days of intensive work with Banaschewski over the summer of 1980, following similarly intense work with Wick-Pelletier on the constructive form of the Hahn-Banach theorem, which developed many

• f the techniques used. The motivation and the outcome were linked talks at the International

Meeting on Categorical Topology held in Ottawa that summer. The ten days of work led to a widely circulated preprint of some 180 pages detailing the result, which was deemed too lengthy for publication. Over the years that followed, it was divided into four papers, inevitably coming to rather more than 180 pages, which were published individually, under the unifying title of The Spectral Theory of Commutative C*-Algebras: The Constructive Spectrum

Background

The Gelfand duality theorem in its constructive form was proved in ten days of intensive work with Banaschewski over the summer of 1980, following similarly intense work with Wick-Pelletier on the constructive form of the Hahn-Banach theorem, which developed many

• f the techniques used. The motivation and the outcome were linked talks at the International

Meeting on Categorical Topology held in Ottawa that summer. The ten days of work led to a widely circulated preprint of some 180 pages detailing the result, which was deemed too lengthy for publication. Over the years that followed, it was divided into four papers, inevitably coming to rather more than 180 pages, which were published individually, under the unifying title of The Spectral Theory of Commutative C*-Algebras: The Constructive Spectrum The Constructive Gelfand-Mazur Theorem

Background

The Gelfand duality theorem in its constructive form was proved in ten days of intensive work with Banaschewski over the summer of 1980, following similarly intense work with Wick-Pelletier on the constructive form of the Hahn-Banach theorem, which developed many

• f the techniques used. The motivation and the outcome were linked talks at the International

Meeting on Categorical Topology held in Ottawa that summer. The ten days of work led to a widely circulated preprint of some 180 pages detailing the result, which was deemed too lengthy for publication. Over the years that followed, it was divided into four papers, inevitably coming to rather more than 180 pages, which were published individually, under the unifying title of The Spectral Theory of Commutative C*-Algebras: The Constructive Spectrum The Constructive Gelfand-Mazur Theorem A Constructive Proof of the Stone-Weierstrass Theorem

Background

The Gelfand duality theorem in its constructive form was proved in ten days of intensive work with Banaschewski over the summer of 1980, following similarly intense work with Wick-Pelletier on the constructive form of the Hahn-Banach theorem, which developed many

• f the techniques used. The motivation and the outcome were linked talks at the International

Meeting on Categorical Topology held in Ottawa that summer. The ten days of work led to a widely circulated preprint of some 180 pages detailing the result, which was deemed too lengthy for publication. Over the years that followed, it was divided into four papers, inevitably coming to rather more than 180 pages, which were published individually, under the unifying title of The Spectral Theory of Commutative C*-Algebras: The Constructive Spectrum The Constructive Gelfand-Mazur Theorem A Constructive Proof of the Stone-Weierstrass Theorem A Globalisation of the Gelfand Duality Theorem

Background

The first of these identified the appropriate constructive form of the maximal spectrum of a commutative C*-algebra A , in fact two appropriate constructive forms: the locale MFn A of multiplicative linear functionals on A , depending heavily on the work with Wick-Pelletier, and the locale Max A of maximal ideals on A , more properly of open primes of A . The second provided the constructive form of the Gelfand-Mazur theorem, in this case showing that the canonical map

MFn A d Max A

• f locales arising, corresponding classically to taking the kernel of a multiplicative linear

functional, was an isomorphism of locales. The third provided the constructive result showing that the resulting Gelfand representation

A d Š(Max A)

into the C*-algebra of continuous complex functions on Max A was in fact an isomorphism. The final paper tied these together into the form of Gelfand duality that holds constructively in any appropriate topos.

Gelfand Duality

Classically, Gelfand duality is a categorical duality between the category of commutative C*-algebras and the category of compact Hausdorff topological spaces. Precisely,

• THEOREM. Consider the category of commutative C*-algebras and homomorphisms and the

dual of category of compact Hausdorff topological spaces and continuous maps. Then the functors that assign to each commutative C*-algebra A the topological space Max A given by its maximal spectrum, and to each compact Hausdorff topological space X the algebra Š(X)

• f continuous complex functions on X determine an adjoint equivalence of categories.

Gelfand Duality

Classically, Gelfand duality is a categorical duality between the category of commutative C*-algebras and the category of compact Hausdorff topological spaces. Precisely,

• THEOREM. Consider the category of commutative C*-algebras and homomorphisms and the

dual of category of compact Hausdorff topological spaces and continuous maps. Then the functors that assign to each commutative C*-algebra A the topological space Max A given by its maximal spectrum, and to each compact Hausdorff topological space X the algebra Š(X)

• f continuous complex functions on X determine an adjoint equivalence of categories.

Constructively, we have instead the following:

• THEOREM. Consider the category of commutative C*-algebras and homomorphisms and the

dual of category of compact completely regular locales and maps of locales. Then the functors that assign to each commutative C*-algebra A the locale Max A given by its maximal spectrum, and to each compact completely regular locale X the algebra

• f continuous

Š(X)

complex functions on X determine an adjoint equivalence of categories.

Gelfand Duality

Classically, Gelfand duality is a categorical duality between the category of commutative C*-algebras and the category of compact Hausdorff topological spaces. Precisely,

• THEOREM. Consider the category of commutative C*-algebras and homomorphisms and the

dual of category of compact Hausdorff topological spaces and continuous maps. Then the functors that assign to each commutative C*-algebra A the topological space Max A given by its maximal spectrum, and to each compact Hausdorff topological space X the algebra Š(X)

• f continuous complex functions on X determine an adjoint equivalence of categories.

Constructively, we have instead the following:

• THEOREM. Consider the category of commutative C*-algebras and homomorphisms and the

dual of category of compact completely regular locales and maps of locales. Then the functors that assign to each commutative C*-algebra A the locale Max A given by its maximal spectrum, and to each compact completely regular locale X the algebra

• f continuous

Š(X)

complex functions on X determine an adjoint equivalence of categories. Of course, there is a little bit of work to do to explain the terms involved.

Commutative C*-Algebras

To define the notion of a seminorm we need to make allowances for the constructive context.

• DEFINITION. By a seminormed involutive algebra A is meant an involutive algebra over the

complex rational numbers together with a map Q

N :

+ d ✡A

that assigns to each positive rational q a subset

• f A satisfying the following conditions:

N(q) ≥q a c N(q) a c N(q) . a∏ c N(q∏) d aa∏ c N(qq∏) a c N(q) . a∏ c N(q∏) d a + a∏ c N(q + q∏) a c N(q) d a& c N(q)

whenever whenever

a c N(q∏) d ✍a c N(qq∏) ✍ c N(q) 1 c N(q) q > 1

,

0 c N(q) a c N(q) f ≥q∏ < q a c N(q∏)

for all , for each complex rational , and for all positive rationals .

a,a∏ c A ✍ q,q∏

The seminorm on the algebra therefore has properties which are expressed in terms of the open ball N(q) of radius q about the zero element of A for each positive rational q .

Commutative C*-Algebras

In order to define completeness, once again allowance must be made for the constructive context in which we are working, in this case, for the absence of countable choice:

• DEFINITION. By a Cauchy approximation on a seminormed *-algebra A will be meant a

mapping C : N d ✡A which satisfies the following conditions:

N

, N N .

≤n c ≥a c A a c Cn ≤k c ≥m c ≤n,n∏ m m a c Cn . a∏ c Cn∏ d a − a∏ c N(1/k)

A Cauchy approximation C on a seminormed *-algebra A will be said to be convergent to an element provided that N N .

b c A ≤k c ≥m c ≤n m m a c Cn d a − b c N(1/k)

Then the seminormed *-algebra A will be said to be complete provided that for each Cauchy approximation C on the algebra A there exists a unique element to which C converges.

b c A

It may be noted that the uniqueness implies that the seminorm on A satisfies the condition that Q+ for each that defines a norm.

(≤q c a c N(q) ) d a = 0 a c A

Commutative C*-Algebras

It may be remarked immediately that in the case of completeness the seminormed algebra that we have defined over the complex rationals is necessarily an algebra over the completion of the algebra of complex rationals, and that this is exactly the algebra of complex numbers. The reason that we have not insisted on this in defining seminormed involutive algebras is that this notion is preserved under inverse image functors of geometric maps between toposes, of which we shall have need to make use.

• DEFINITION. By a C*-algebra A is meant a complete seminormed involutive algebra over the

complex rationals satisfying the condition that:

a c N(q) f aa& c N(q2)

for each and each positive rational q .

a c A

The C*-algebras A with which we shall be concerned are those that are commutative, in the sense that the underlying algebra is commutative.

Compact, Completely Regular Locales

• DEFINITION. A locale L is said to be compact provided that any open covering

1L = -i ui

• f its identity element admits a finite subcovering.
• DEFINITION. For any locale L , the rather below relation is defined for

by

u,v c L v ü u

provided that there exists for which

w c L

,

v . w = 0L and u - w = 1L

and the completely below relation is defined for by

u,v c L v üü u

provided that there exists a family of elements indexed by the rationals for

vq c L 0 [ q [ 1

which

.

v = v0, vp ü vq whenever p < q , and v1 = u

Compact, Completely Regular Locales

It may be noted that the interpolation between encountered in defining the

v c L and u c L

completely below relation is the information required in order to specify a continuous real function which separates v and u in the sense of Urysohn's Lemma. Indeed, this is a point to which we shall later explicitly return.

• DEFINITION. A locale L is said to be regular (respectively, completely regular ) provided that

(respectively, )

u = -v ü u v u = -v üü u v

for each .

u c L

It may be noted that in a compact, regular locale L , given elements with , it is

v,u c L v ü u

always possible to show the existence of an element satisfying . However, this

v∏ c L v ü v∏ ü u

cannot be extended to the existence of an interpolation showing that without

(vq) v üü u

appealing to the principle of countable dependent choice. The notions of compact, regular locale and of compact, completely regular locale are therefore distinct in most constructive contexts.

The C*-Algebra C C C C(M)

Consider any compact, completely regular locale M and let C(M) denote the set of all maps

• f locales

. The locale of complex numbers may be shown straightforwardly to

✍ : M d Š Š

have the usual algebraic operations of addition, negation, and zero, and of multiplication, identity and conjugation, expressed as maps in the category of locales. In turn, these allow the set C(M) to inherit these operations, indeed to become a commutative involutive algebra over the complex rationals. The algebra C(M) may be given a seminorm by assigning to each positive rational q the subset ,

N(q) = { ✍ c Š(M)| 1M [ ✍&(N(q)) }

• btained by taking those continuous complex functions for which the inverse image of the
• pen subset
• f the complex plane is the top element of the locale M.

N(q)

• THEOREM. The assignment to each compact, completely regular locale M of the commutative

seminormed involutive algebra C(M) yields a functor from the dual of the category of compact, completely regular locales to the category of commutative C*-algebras.

The Spectrum of a C*-Algebra

Classically, there are two ways of approaching the construction of the spectrum of a commutative C*-algebra A , linked by a fundamental theorem, the Gelfand-Mazur theorem, which yields that they give the same result, which we shall call the maximal spectrum, Max A ,

• f the C*-algebra.

The first of these, and the one of perhaps the most provocative interest to this audience, is that it is the topological space of multiplicative linear functionals on the commutative C*-algebra A , namely of those linear mappings

✩ : A d Š

that are also multiplicative and unital, with the topology generated by taking as subbasic open sets the subsets

{ ✩ | ✩(a) c (r,s) }

for each and rational open rectangle

• f the complex plane.

a c A (r,s)

Denoting this subbasic open set by , we begin to see a constructive way forward, as

a c (r,s)

well as an idea that the spectrum might be a way of seeking information about observables.

The Spectrum of a C*-Algebra

For any commutative C*-algebra A , consider the locale MFn A of multiplicative linear functionals on A determined by taking the propositional geometric theory obtained by introducing, for each and each complex rational open rectangle , a proposition

a c A (r,s) a c (r,s)

together with the following axioms:

true d 0 c (r,s) whenever 0 c (r,s), and 0 c (r,s) d false

• therwise;

a c (r,s) d ta c (tr,ts) whenever t > 0, and a c (r,s) d ia c i(r,s);

;

a c (r,s) . a∏ c (r∏,s∏) d a + a∏ c (r + r∏,s + s∏)

;

true d a c N(1) whenever a c N(1)

;

a c (r,s) d a c (p,q) - a c (p∏,q∏) whenever (r,s) ü (p,q) - (p∏,q∏)

;

a c (r, s) dw -(r∏,s∏)ü(r,s) a c (r∏, s∏) true d 1 c (r,s) whenever 1 c (r,s), and 1 c (r,s) d false

• therwise;

;

a c (r,s) d a& c (r,s) aa∏ c (r,s) dw -i a c (pi,qi) . a∏ c (pi

∏,qi ∏)

.

whenever

• i(pi,qi) % (pi

∏,qi ∏) = ✙&(r,s)

The Spectrum of a C*-Algebra

It may be noted that the theory described is obtained by firstly, for each , introducing

a c A

axioms that specialise to the proposition

a c (r,s)

the axioms ;

a c (r,s) d false whenever (r,s) [ 0

;

true d -(r,s) a c (r,s)

;

a c (r,s) d a c (p,q) - a c (p∏,q∏) whenever (r,s) ü (p,q) - (p∏,q∏)

;

a c (p,q) . a c (p∏,q∏) d a c (r,s) whenever (p,q) . (p∏,q∏) ü (r,s)

,

a c (r,s) dw -(r∏,s∏)ü(r,s) a c (r∏,s∏)

describing the complex number assigned to .

a c A

To these are then added the axioms stating that this assignment is linear and multiplicative, followed by the elimination or amendment of certain axioms to remove redundancy.

The Spectrum of a C*-Algebra

The important consequence of the way in which this axiomatisation is arrived at is that firstly it allows it to be proved that the locale MFn A is a compact, completely regular locale, by showing that it in fact a closed sublocale of the locale Fn A of linear functionals of norm [ 1

• n A , which by the Alaoglu theorem proved with Wick-Pelletier, is known to be compact and

to inherit its complete regularity from that of the locale of complex numbers. That this latter is the case follows from what in these papers became known as the continuity principle, indicating that writing for any open subset U of the locale of complex numbers for the disjunction was notationally consistent and allowed the

a c U

• (r,s)ü U a c (r,s)

description, by mapping each the map of locales obtained by the assignment

a c A

,

(r,s) h a c (r,s)

• f the Gelfand representation into the commutative C*-algebra of continuous complex

functions

. : A d Š(Max A)

• n what we shall shortly call the maximal spectrum of the commutative C*-algebra A .

The Spectrum of a C*-Algebra

The second idea, classically, is that it is the topological space of maximal ideals of the commutative C*-algebra A in the Zariski topology, to which one should immediately add that the maximal ideals of a commutative C*-algebra are characterised as the closed prime ideals of A . In particular, each prime ideal of A is contained in a unique maximal ideal, an observation which means that the Zariski topology on the maximal ideal space is Hausdorff, as well as compact. Classically, there is a canonical map from the space of multiplicative linear functionals to the space of maximal ideals that maps each multiplicative functional, ,

✩ : A d Š

necessarily a surjective homomorphism, to its kernel, necessarily a maximal ideal of A . THEOREM (GELFAND-MAZUR). Any commutative C*-algebra which is a field is necessarily canonically isomorphic to the complex numbers.

• COROLLARY. Every maximal ideal of a commutative C*-algebra A is the kernel of a unique

multiplicative linear functional on A . In particular, the canonical map from the space of multiplicative linear functionals to the space of maximal ideals of A is a homeomorphism.

The Spectrum of a C*-Algebra

• DEFINITION. Given a commutative C*-algebra A , consider the propositional geometric theory

defined by introducing, for each and each non-negative rational q , a proposition

a c A

,

a c A(q)

together with the following axioms: ;

true d 1 c A(q) whenever q < 1

;

a c A(q) d false whenever a c N(q)

;

a c A(q) d a& c A(q)

;

a + b c A(r + s) d a c A(r) - b c A(s)

;

a c A(r) . b c A(s) d ab c A(rs)

;

ab c A(rs) d a c A(r) - b c A(s)

;

a c A(r) . b c A(s) d aa& + bb& c A(r2 + s2)

.

a c A(q) dw -q∏>q a c A(q∏)

Then the maximal spectrum Max A of the commutative C*-algebra A is given by the Lindenbaum locale of this theory.

The Spectrum of a C*-Algebra

The motivation for this definition comes from multiple sources. The primary influence is that it is axiomatising a conorm on the commutative C*-algebra A , that is by attributing to each the extent to which the co-seminorm defined is greater than the positive

a c A a c A(q)

rational q . Looking back to the description of a seminorm in terms of attributes

• ne sees that

a c N(q)

this is just the dual definition of this concept, in terms of awayness from zero, in the

a c A(q)

sense equivalently that the complex numbers inherits constructively from the complex rationals both an equality relation and an apartness relation, an observation due to Scott that derives from the complex numbers constructively being both a field, in the sense that

¥ ( z invertible) d z = 0

for any , but also a local ring, in the sense that

z c Š z + z∏ invertible d z invertible - z∏ invertible

for any .

z,z∏ c Š

The Spectrum of a C*-Algebra

Recall that the Gelfand-Mazur theorem classically expresses that every maximal ideal of a commutative C*-algebra A is the kernel of a unique multiplicative linear functional on A . In the constructive context, this is exactly expressed by stating that the interpretation

a c A(q) h

• (r,s)üA(q) a c (r,s)
• f the theory of the maximal spectrum of A in the theory of the multiplicative linear

functionals on A is an equivalence of theories, hence determines an isomorphism of locales. The proof may be reduced to the equivalent result in the geometry of the complex plane, namely that in any bounded region of the complex plane, the topology induced by the rational

• pen codisks

z − ✘.1 c A(q)

for each complex rational and each non-negative rational q coincides with that induced by

✘

the rational open rectangles

z c (r,s)

for each pair , an obervation inherited from the well-behaved concept of awayness in .

r,s Š

The Spectrum of a C*-Algebra

The proof is given by the following diagram: together with an interpretation derived from the inverse assignment .

z c (r,s) h

• z c A(✘1

q,r1 q) . £ . z c A(✘4 q,r4 q)

The Gelfand Duality Theorem

With the insight that the maximal spectrum Max A of a commutative C*-algebra A has this dual aspect as the locale of equivalently multiplicative linear functionals on A and the constructive equivalent of maximal ideals of A , one arrives at the adjunctions needed to establish a dual equivalence of categories, namely on the one hand the Gelfand representation

A d Š(Max A)

• f any commutative C*-algebra A , and the canonical map

M d Max (Š(M))

• btained by assigning to each proposition

for any the inverse image

✍ c (r,s) ✍ c Š(M)

.

✍&(r,s)

The proof that the latter is an isomorphism of locales is a straightforward argument based on the compactness, yielding surjectivity, and complete regularity, yielding injectivity, by a constructive version of Urysohn's Lemma, of the locale M . That the former is an isometric *-isomorphism of commutative C*-algebras derives from the observation that it derives from the fact that the theory of multiplicative linear functionals on A yields a generic such functional from A into the complex numbers in the topos of sheaves on Max A .

The Gelfand Duality Theorem

Denoting the open subset of

• btained by taking the inverse image under the Gelfand

Max A

transform of each

• f the open subset P of the complex plane obtained by deleting zero by

a c A

,

D(a)

we make the following:

• DEFINITION. A subalgebra A of the commutative C*-algebra

is said to separate the

Š(M)

compact, completely regular locale M provided that each open set U of the locale may be expressed in the form

U = - D(a)

taken over those elements for which is contained in U.

a c A D(a)

THEOREM (STONE-WEIERSTRASS). Let M be a compact, completely regular locale. Then any closed involutive subalgebra A of the commutative C*-algebra which separates the

Š(M)

locale M is necessarily equal to .

Š(M)

The Gelfand Duality Theorem

The proof of the Stone-Weierstrass theorem is by considering the subsheaf of the sheaf

• f

ŠM

continuous complex functions on the compact, completely regular locale M that is generated by the closed involutive subalgebra A of the commutative C*-algebra

• f its global

Š(M)

• sections. It may be proved, firstly, that A is exactly the algebra of global sections of this

subsheaf, by the existence of finite partitions of unity lying in A. Secondly, it may be shown that the subsheaf is actually closed in the sheaf , again by an argument involving these

ŠM

partitions of unity. And, then, finally, it may be remarked that the subsheaf is necessarily also dense in the sheaf , because it necessarily contains the sheaf of complex rationals in the

ŠM

topos of sheaves on M from which the result then clearly follows. It may be shown straightforwardly that the Gelfand representation

A d Š(Max A)

• f any commutative C*-algebra is isometric, by examining the precise form of the norm on the

algebra . It follows that its image is a closed involutive subalgebra of

Š(Max A) Š(Max A)

which may be seen to separate the algebra . In consequence, the Gelfand

Š(Max A)

representation is indeed an isometric *-isomorphism.

Conclusion

Thinking about the axiomatisation of the maximal spectrum, it may be remarked that writing

a c P

for the proposition , and applying the lattice operations on the positive cone of A , it

a c A(0)

may be shown that

a c A(q) dw (a − q.1)+ c P

is provable in the theory for any and any non-negative rational q .

a c A

It follows immediately that the theory is equivalent to that given by the following axioms:

true d 1 c P 0 c P d false a c P d a& c P a + b c P d a c P - b c P ab c P dw a c P . b c P a c P dw -q ( a − q.1)+ c P

for all . Hence, the theory of Max A may be considered as a retract of that of Spec A.

a,b c A

Conclusion

With a little more thought, it may be shown to be equivalent to that axiomatised by

true d 1 c P 0 c P d false a c P d a& c P a + b c P d a c P - b c P ab c P dw a c P . b c P

whenever

a c P dw -i ai c P

a c ✟ …ai

for all and any , in other words we are describing an open prime of A , hence

a,b c A ai c A

the constructive equivalent of the observation that the maximal ideals of a commutative C*-algebra A are exactly the closed prime ideals.

Conclusion

I cannot conclude without remarking that one little change to the theory:

true d 1 c P 0 c P d false a c P d a& c P a + b c P d a c P - b c P ab c P dw a c P . b c P

whenever ,

a c P dw -i ai c P

a c ✟ …ai

Conclusion

I cannot conclude without remarking that one little change to the theory:

true d 1 c P 0 c P d false a c P d a& c P a + b c P d a c P - b c P ab c P dw a c P & b c P

whenever ,

a c P dw -i ai c P

a c ✟ …ai

Conclusion

I cannot conclude without remarking that one little change to the theory:

true d 1 c P 0 c P d false a c P d a& c P a + b c P d a c P - b c P ab c P dw a c P & b c P

whenever ,

a c P dw -i ai c P

a c ✟ …ai

yields a theory within the non-commutative constructive context that leads to involutive quantales, instead of locales, and of which the maximal spectrum Max A in the case of not-necessarily commutative C*-algebra A is the quantale of closed linear subspaces of A , a quantal space of which the points are the equivalence classes of irreducible representations of the C*-algebra A and which admits a Gelfand representation by a not-necessarily commutative C*-algebra of continuous complex functions on Max A .

Conclusion

I cannot conclude without remarking that one little change to the theory:

true d 1 c P 0 c P d false a c P d a& c P a + b c P d a c P - b c P ab c P dw a c P & b c P

whenever ,

a c P dw -i ai c P

a c ✟ …ai

yields a theory within the non-commutative constructive context that leads to involutive quantales, instead of locales, and of which the maximal spectrum Max A in the case of not-necessarily commutative C*-algebra A is the quantale of closed linear subspaces of A , a quantal space of which the points are the equivalence classes of irreducible representations of the C*-algebra A and which admits a Gelfand representation by a not-necessarily commutative C*-algebra of continuous complex functions on Max A . But that takes us away from the orthodoxies of today's meeting.