aa r X i v : . [ m a t h . S T ] J un Image transformationson locally compact spaces
Gunnar Taraldsen
Trondheim, Norway. e-mail:
Abstract:
An image is here defined to be a set which is either open or closed in X and an image transformation q is structure preserving in the following sense: It corre-sponds to an algebra homorphism q : A ( a ) → A ( q ( a )) for each singly generated alge-bra A ( a ) = { φ ( a ) | φ ∈ C ( R , R ) , a ∈ C ( X, R ) } . We extend parts of J.F. Aarnes’ re-sults on quasi-measures, -states, -homomorphisms, and image-transformations fromthe setting compact Hausdorff spaces to locally compact Hausdorff spaces. Contents
1. Introduction and Definitions.
In the C ∗ -algebraic formulation of quantum mechanics the bounded real observables areidentified with selfadjoint elements in a C ∗ -algebra A . A physical state can be consideredto be [1, p.602], [10] an assignment of a probability measure µ a on the spectrum of eachselfadjoint a . The measure µ a is interpreted as the probability distribution which modelsthe outcome in an experiment where the observable a is measured. This interpretationforces the consistency condition µ φ ( a ) = µ a ◦ φ − , since a measurement of a is also ameasurement of any observable φ ( a ) being a function of a . The function φ is assumed tobe continuous since the observable φ ( a ) is supposed to be an element in the C ∗ -algebra.The end result is that a physical state may be identified with the functional µ given by µ ( a ) := µ a ( id ), which fullfils the fundamental equation µ ( φ ( a )) = µ a ( φ ) := Z φ ( t ) µ a ( dt ) . (1)Let A ( a ) = { φ ( a ) | φ ∈ C ( R , R ) } be the unital norm-closed real algebra of selfadjointelements generated by a . A functional defined on the selfadjoint elements of a C ∗ -algebra AMS Subject Classification (1991): 46J10, 28A25, 28C15, 46L30, 81P10. Keywords: Banach algebras of continuous functions, Integration with respect to measures and otherset functions, Set functions and measures on topological spaces, States, Logical foundations of quantummechanics. 1 araldsen/Image transformations which is linear on each A ( a ) is said to be quasi-linear. It follows from the fundamentalequation that µ is a quasi-linear functional.In 1991 [2] Aarnes presented the first example of a proper quasi-linear functional. Theidea is to extend the Riesz representation theorem from integrals to quasi-integrals toobtain a correspondence between quasi-integrals and quasi-measures. Let A = A ( X ) bethe class of sets in a Hausdorff space X which are either open or closed. A (compact-regular, additive, normalized) quasi-measure µ is a real valued function defined on A withproperties (i) µ ( X ) = 1, (ii) µ is additive (on disjoints); µ ( A ⊎ B ) = µ ( A ) + µ ( B ), and(iii) µ is compact-regular: The measure of an open set U equals the supremum of themeasures of compact sets K contained in U . A quasi-measure is said to be simple if itonly takes the values 0 and 1. Integration with respect to a quasi-measure µ is definedas in [2, p.46]: If a : X → R is continuous, then µ ◦ a − is the restriction of a measure µ a on R . This gives a consistent family of measures and ρ ( a ) := µ ( a ) := R t µ a ( dt ) is aquasi-integral: (i) ρ : C b ( X ) → R ; (ii) ρ : A ( a ) → R is linear; (iii) a ≥ ρ ( a ) ≥ ρ (1) = 1; (v) ρ ( a ) = sup k ≤ a ρ ( k ). ρ is simple if it is multiplicative on each A ( a ).A random variable X is a measurable function from a probability space Ω to a measur-able space E . The set function X − pulls measurable sets in E back to measurable setsin Ω in such a way that the measure P on Ω is pushed to the measure P X := P ◦ X − on E . The fundamental change of variable formula E ( φ ( X )) := Z φ ( X ( ω )) P ( dω ) = Z φ ( x ) P X ( dx ) (2)shows that the expectation value of any (measurable) function of X may be computedfrom the distribution P X of X . One may replace the set function X − with a set function ψ with properties: (i) ψ ( E ) = Ω, (ii) ψ ( A c ) = ψ ( A ) c , (iii) ψ ( A ∪ A ∪ · · · ) = ψ ( A ) ∪ ψ ( A ) ∪ · · · , and the measure P is again pushed to a measure ψ ∗ P := P ◦ ψ on E with aintegration result as above.These results can be generalized to the setting of quasi-measures. The family of mea-surable sets is replaced by the family A of images. An image is a set which is open orclosed. The measurable functions are replaced by continuous functions, and the result µ ( φ ( a )) = µ a ( φ ) is the generalization of the change of variable formula. The set func-tions ψ are replaced by image-transformations q . An image-transformation q from X to Y takes an image A in X to an image q ( A ) in Y , and has properties (i) q ( X ) = Y , (ii) q ( U ) is open when U is open, (iii) q is additive; q ( A ⊎ B ) = q ( A ) ⊎ q ( B ), and (iv) q iscompact-regular: Given an open set U and a compact set K ⊂ q ( U ), there is a compactset L ⊂ U such that K ⊂ q ( L ). A quasi-measure µ on Y is pulled to a quasi-measure q ∗ µ := µ ◦ q on X . The integral q ( a ) of a continuous bounded function a on X is thecontinuous bounded function q ( a ) on Y given by q ( a )( y ) :=( q ∗ δ y )( a ). The integral withrespect to an image-transformation is a (compact-regular) quasi-homomorphism from C b ( X ) to C b ( Y ): (i) q : A ( a ) → A ( q ( a )) is an algebra homomorphism for each a , and(ii) q ( a )( y ) = sup k q ( k )( y ), where k ≤ a has compact support. The integral above gives1-1 correspondence between image-transformations and quasi-homomorphisms for locallycompact normal spaces. The generalization of the change of variable formula is( q ∗ µ )( a ) = µ ( q ( a )) . (3) araldsen/Image transformations The main result in this work is the characterization of all image-transformations in termsof the Aarnes factorization theorem [3]: To any image-transformation q from X to Y thereexists a continuous function w : Y → X ∗ such that q = w − ◦ [ ∗ ]. The [ ∗ ] is the canonicalimage-transformation from X to X ∗ given by A [ ∗ ]( A ) := A ∗ := { σ ∈ X ∗ | σ ( A ) = 1 } .The space X ∗ is the set of simple quasi-measures σ ( σ ( A ) equals 0 or 1) equipped with theweak topology: σ σ ( a ) is continuous for all bounded continuous a . In terms of quasi-homomorphisms the factorization is as above with ( w − a )( y ) = a ( w ( y )), and ([ ∗ ] b )( σ ) = σ ( b ) is the quasi-linear Gelfand transformation. This means in particular that [ ∗ ] is aquasi-homomorphism from C b ( X ) to C b ( X ∗ ), and w − is a (quasi-)homomorphism from C b ( X ∗ ) to C b ( Y ). The function w is unique and given by w = q ∗ ◦ ι Y , where ι Y : Y → Y ∗ is the inclusion which maps y to δ y , and q ∗ maps Y ∗ into X ∗ . The factorization result issummarized by the following commutative diagrams: C b ( X ∗ ) w − % % ❏❏❏❏❏ ( q ∗ ) − / / C b ( Y ∗ ) ι − Y (cid:15) (cid:15) C b ( X ) [ ∗ ] O O q / / C b ( Y ) A ( X ∗ ) w − $ $ ■■■■■ ( q ∗ ) − / / A ( Y ∗ ) ι − Y (cid:15) (cid:15) A ( X ) [ ∗ ] O O q / / A ( Y ) X ∗ Y ∗ q ∗ o o Y ι Y O O w a a ❉❉❉❉❉❉❉❉ As an indication of another possible application of this theory we quote Aarnes [3, p.1]:
Once defined, image-transformations take on a life of their own. In some sense they seemto be better vehicles for the litteral transfering of an “image” or a message than ordinaryfunctions, since they allow for the possibility that images of “small” sets will vanish, i.e.equal the empty set.
In the following we include normalization, compact-regularity and additivity in the defi-nitions of quasi-measures, quasi-integrals, image-transformations, and quasi-homomorphismsin order to simplify the language. We follow the notational conventions:
K, L, M are com-pact sets;
F, G, H are closed sets;
U, V, W are open sets;
A, B, C are images; A = A ( X ) isthe set of images in a Hausdorff space X ; k, l, m are real valued continuous functions withcompact support; a, b, c are real valued bounded continuous functions; and C b = C b ( X )is the set of real valued bounded continuous functions on X .The first version of this work was a result of a seminar based on [3] in the springof 1995. We acknowledge comments from Andenæs, Knudsen, Rustad, and Aarnes whoparticipated in the seminar. The results here corresponds to generalizations of the firstpart of [3] and are approximately unchanged from the seminar, but the organization ofthe proofs is different. Rustad [9] refers to an earlier version of this work. Aarnes andGrubb [5] treat image transformations in completely regular spaces and their resultscomplements the results in the following.
2. Integration and the Riesz Representation Theorem.
The aim in this section is to give the ingredients in the proof of the Riesz representationtheorem.
Theorem 2.1.
Let X be a locally compact normal space. A one-one correspondencebetween quasi-measures µ and quasi-integrals ρ on C b is given by ρ ( a ) = Z a ( x ) µ ( dx ) , µ ( U ) = sup k ≤ U ρ ( k ) . araldsen/Image transformations The simple quasi-measures corresponds to the simple quasi-integrals.
We start with the development of an integration theory based on quasi-measures in aHausdorff space. Some properties of quasi-measures are summarized by: A quasi-measureis monotone: A ⊂ B ⇒ µ ( A ) ≤ µ ( B ) . It is continuous in the following sense: U λ ↑ U ⇒ µ ( U λ ) ↑ µ ( U ), and F λ ↓ F ⇒ µ ( F λ ) ↓ µ ( F ). If X is locally compact, then µ ( U ) = sup { µ ( V ) | V = K ⊂ U } . The proof of these statements are similar to the proofof the corresponding statements for image-transformations. The main difference betweena measure and a quasi-measure is that the latter is not defined on an algebra of sets: Theunion and intersection of two images need not be an image. In certain cases it turns outthat quasi-measures may be identified with measures, and in particular Proposition 1.
A quasi-measure µ on R is the restriction of a unique Borel measure ν .Proof. Put F ( t ) := µ ( −∞ , t ]. F is right continuous F ( t ) = lim t i ↓ t µ ( −∞ , t i ] since µ iscontinuous. Monotonity of µ ensures that F is the distribution function of a unique Borelmeasure ν . Any open set is the disjoint union of a countable family of open intervalls, sothe restriction claim follows from ν ( a, b ) = lim b i ↑ b ν ( a, b i ] = lim b i ↑ b F ( b i ) − F ( a ) = lim b i ↑ b µ ( −∞ , b i ] − µ ( −∞ , a ]= lim b i ↑ b µ ( −∞ , b i ) − µ ( −∞ , a ] = µ ( −∞ , b ) − µ ( −∞ , a ] = µ ( a, b ) . This Proposition is a special case of a more general recent result [11, p.4]: EveryBaire quasi-measure on a Tychonoff space with Lebesgue covering dimension ≤ µ be a quasi-measure on X . If f : X → Y is continuous, then f − : A ( Y ) → A ( X )is an image-transformation. It follows in particular that µ ◦ f − is a quasi-measure on Y , as will be proven in the next section. The particular case Y = R together with theprevious Proposition gives us integration: Definition 2.1.
Let µ a be the extension of µ ◦ a − to a Borel measure on R . The integral µ ( a ) of a with respect to µ is µ ( a ) := Z a ( x ) µ ( dx ) := Z t µ a ( dt ) := µ a ( id ) . We remark that this definition is consistent with the conventional for ordinary measuresdue to the change of variable formula. Borel measures on the real line are uniquely givenby their values on open sets, so the family { µ a } is a consistent family of measures: µ φ ( a ) ( U ) = µ (( φ ( a )) − ( U )) = µ a ◦ φ − ( U ) . This gives that a quasi-measure on a Hausdorff space X gives a quasi-linear functional: µ ( φ ( a )) = Z t µ φ ( a ) ( dt ) = Z t µ a ◦ φ − ( dt ) = Z φ ( s ) µ a ( ds ) = µ a ( φ ) . The following Staircase Lemma is fundamental. araldsen/Image transformations Lemma 2.1.
Let a ≤ b . For each δ > we have the decomposition a = a + · · · + a n , b = b + · · · + b n , a i ∈ A ( a ) ∩ A ( a i + b i ) , b i ∈ A ( b ) ∩ A ( a i + b i ) , and a i ≤ b i + δ/n .Proof. [2, p.54] Choose a constant M such that ˜ a := a + M , ˜ b := b + M + δ obeys 0 ≤ ˜ a ≤ ˜ b − δ . Choose 0 = β < · · · < β n = β := (cid:13)(cid:13)(cid:13) ˜ b (cid:13)(cid:13)(cid:13) , β i +1 − β i < δ , and define φ ( x ) := x ≤ x ≤ x ≤ ββ x ≥ β , φ i ( x ) := x ≤ β i − x − β i − β i − ≤ x ≤ β i β i − β i − x ≥ β i . With a i := φ i (˜ a ) − M/n , b i := φ i (˜ b ) − ( M + δ ) /n , and the observation φ = P i φ i , weconclude a = P i a i , b = P i b i , and a i ∈ A ( a ) , b i ∈ A ( b ). We prove a i , b i ∈ A ( a i + b i ), orequivalently ˜ a i := φ i (˜ a ) , ˜ b i := φ i (˜ b ) ∈ A (˜ a i + ˜ b i ). From ˜ a ≤ ˜ b − δ we conclude ˜ b i ( x ) = β i − β i − when ˜ a i ( x ) >
0. This gives ˜ a i · ( β i − β i − − ˜ b i ) = 0, ˜ a i , β i − β i − − ˜ b i ∈ A (˜ a i − β i + β i − +˜ b i ),and finally ˜ a i , ˜ b i ∈ A (˜ a i + ˜ b i ). Proposition 2. If ρ : C b ( X ) → R is positive and quasi-linear, then a ≤ b ⇒ ρ ( a ) ≤ ρ ( b ) ,and | ρ ( a ) − ρ ( b ) | ≤ ρ (1 X ) k a − b k .Proof. The staircase Lemma gives ρ ( a ) = P i ρ ( a i ) ≤ δρ (1 X ) + P i ρ ( b i ) = δρ (1 X ) + ρ ( b )from which we conclude ρ ( a ) ≤ ρ ( b ). a ≤ b + k a − b k gives ρ [ a ] ≤ ρ [ b ] + ρ [1 X ] k a − b k anda switch of b and a gives | ρ [ a ] − ρ [ b ] | ≤ ρ [1 X ] k a − b k .It follows in particular that a quasi-integral is monotone and continuous. In [2] it isproven that the integral with respect to a quasi-measure on a compact Hausdorff spaceis a quasi-linear functional. The following Proposition is a generalization to the case oflocally compact Hausdorff spaces. Proposition 3.
Let µ be a quasi-measure on a locally compact Hausdorff space. Thequasi-integral with respect to µ is a quasi-integral, and the change of variable identity µ ( φ ( a )) = µ a ( φ ) is valid for all continuous φ : R → R . The quasi-integral from a simple quasi-measure isa simple quasi-integral.Proof. (i) The identity was proven above for a general Hausdorff space. Every element in A ( a ) is on the form φ ( a ), so linearity on A ( a ) follows from the above identity and linear-ity of µ a . Positivity follows from the change of variable identity applied to the function φ ( t ) = t . Finally µ (1) = µ (1 R ( a )) = µ a (1 R ) = 1.(ii) If a λ ↑ a , then µ ◦ a − λ ( t, ∞ ) ↑ µ ◦ a − ( t, ∞ ) from the continuity of µ and a − λ ( t, ∞ ) ↑ a − ( t, ∞ ). Helly’s second theorem [8, p.53] applied to the distribution functions F ( t ) := µ ◦ a − ( −∞ , t ] and F λ ( t ) := µ ◦ a − λ ( −∞ , t ] gives µ ( a λ ) ↑ µ ( a ).(iii) Now we need local compactness. The set Λ := { k | k (cid:22) X } is directed by the conven-tional ≤ for real-valued functions. Given a ∈ C b ( X ) it follows that a k := a · k ↑ a , when X is locally compact. This, together with (ii), imply the regularity µ ( a ) = sup k ≤ a µ ( k ).(iv) If µ = σ is simple, then σ a = δ σ ( a ) (4)since σ a = δ z is a simple Borel measure, and σ ( a ) = σ a ( id ) = z . Multiplicativity followsfrom σ ( φ ( a )) = σ a ( φ ) = φ ( σ ( a )) applied to the function φ = φ φ . araldsen/Image transformations In the above we proved quasi-linearity and normality ( a λ ↑ a ⇒ µ ( a λ ) ↑ µ ( a )) inthe case of a quasi-measure on a general Hausdorff space. This monotone convergencetheorem for nets holds in a general Hausdorff space for a general quasi-integral ρ due tocompact-regularity: Let k (cid:22) a with ρ [ a ] ≤ ρ [ k ] + ǫ , which is possible due to regularity.Monotone convergence a λ ↑ a gives uniform convergence on supp k (Dini’s Lemma), andthen a λ k such that k ≤ a λ + ǫ whenever λ ≥ λ k . Quasi-linearity and monotonity give ρ [ a ] ≤ ρ [ a λ ] + 2 ǫ if λ ≥ λ k , which proves that ρ is normal.If K ≤ a ≤ U , then µ ( K ) ≤ µ ◦ a − { } = R { } t µ a ( dt ) ≤ µ ( a ) = R (0 , t µ a ( dt ) ≤ R (0 , µ a ( dt ) = µ ◦ a − (0 , ≤ µ ( U ). If X is locally compact, this gives µ ( U ) = sup k (cid:22) U µ ( k ) , µ ( K ) = inf K ≤ k µ ( k ) . (5)This gives in particular that the integrals corresponding to different quasi-measures aredifferent, and that the quasi-measure is determined by its corresponding integral as statedin the Riesz representation theorem. We will now sketch the proof of the second part ofthe Riesz representation theorem. Let a quasi-integral ρ : C b ( X ) → R be given. Define µ ( U ) := sup k (cid:22) U ρ ( k ). It follows that µ is additive and can be extended to A by µ ( F ) := 1 − µ ( F c ). The regularity gives µ ( U ) = sup a ≤ U ρ ( a ), and therefore µ ( F ) = inf F ≤ a ρ ( a ), whichgives additivity on the closed sets from normality. Monotonity of ρ gives that U ⊂ F implies µ ( U ) ≤ µ ( F ). Normality and Urysohn gives F ≤ a ≤ U from which µ ( F ) ≤ µ ( U )follows. The set function µ is therefore additive and monotone. Regularity follows fromconsideration of k ≤ supp k ≤ V ≤ K = V ≤ U , so µ is a quasi-measure. We prove ρ ( a ) = µ ( a ). The representation theorem applied to C ( Sp a ) gives a Borel measure ν a determined by φ ρ [ φ ( a )] = R φ ( t ) ν a ( dt ). The claim follows if we prove ν a ( α, β ) = µ a ( α, β ) for an arbitrary intervall ( α, β ). Let φ n ( t ) := n ( t − α ) α ≤ t < α + 1 /n α + 1 /n ≤ t < β − /n − n ( t − β ) β − /n ≤ t < β , so φ n ↑ ( α, β ) and φ n ( a ) ≤ U := a − ( α, β ). The monotone convergence theorem gives ν a ( α, β ) = lim ρ ( φ n ( a )) and from µ a ( α, β ) = sup k ≤ U ρ ( k ) we conclude µ a = ν a . Finally weprove that µ is simple if ρ is a simple quasi-integral. From ρ ( φ ( a )) = φ ( ρ ( a )) we conclude µ a = δ ρ ( a ) . Let K be compact with µ ( K ) >
0. From the regularity of µ we can find k and U with K (cid:22) k (cid:22) U and µ ( U ) ≤ µ ( K ) + ǫ . From 0 < µ ( K ) ≤ µ ( k − { } ) = 1 ≤ µ ( U ) ≤ µ ( K ) + ǫ , we conclude µ ( K ) = 1. From this and regularity it follow that µ is simple.
3. Image-Transformations and the Aarnes Factorization Theorem.Proposition 4. If q is an image-transformation, then ( i ) q ( A c ) = q ( A ) c ; ( ii ) A ⊂ B ⇒ q ( A ) ⊂ q ( B ); ( iii ) A ∩ B = ∅ ⇒ q ( A ) ∩ q ( B ) = ∅ ;( iv ) q ( U ) = [ K ⊂ U q ( K ); ( v ) U λ ↑ U ⇒ q ( U λ ) ↑ q ( U ) . araldsen/Image transformations Proof. (i) Y = q ( A ∪ A c )= q ( A ) ⊎ q ( A c ). (ii) and (iv) Consider F ⊂ U , so q ( U ) = q ( F ⊎ ( U ∩ F c ))= q ( F ) ∪ q ( U ∩ F c ) ⊃ q ( F ). Since compacts are closed we find q ( U ) ⊃ S K ⊂ U q ( K ).Let y ∈ q ( U ). Since { y } is compact, the regularity gives a K ⊂ U with y ∈ q ( K ) and(iv) follows. U ⊂ V and (iv) gives q ( U ) ⊂ q ( V ). U ⊂ F gives q ( F ) = q ( U ⊎ ( F ∩ U c ))= q ( U ) ∪ q ( F ∩ U c ) ⊃ q ( U ), and finally F ⊂ G gives q ( F c ) ⊃ q ( G c ) and q ( F ) ⊂ q ( G )from (i). (iii) Because of additivity it is sufficient to consider the case U ∩ F = ∅ , but then q ( U ) ⊂ q ( F c )= q ( F ) c , so q ( U ) ∩ q ( F ) = ∅ . (v) Monotonity gives ∪ λ q ( U λ ) ⊂ q ( U ), so only ⊃ remains. Let y ∈ q ( U ). Regularity gives a compact K ⊂ U = ∪ λ U λ with y ∈ q ( K ), butthen K ⊂ U λ ∪ · · · ∪ U λ n ⊂ U ˆ λ . We conclude y ∈ q ( U ˆ λ ) from monotonity.The image-transformations are arrows in a category Image with objects A ( X ). Theidentity arrow id : A ( X ) → A ( X ) is an image-transformation and: Proposition 5.
The composition of two image-transformations is an image-transformation.Proof.
Let q : A ( X ) → A ( Y ) and p : A ( Y ) → A ( Z ) be image-transformations. p ◦ q : A ( X ) → A ( Z ) is well defined, and: p ◦ q ( X ) = p ( Y ) = Z ; p ◦ q ( U ) = p ( q ( U ) open ) = p ( q ( U )) open ; and p ◦ q ( A ⊎ B ) = p ( q ( A ) ⊎ q ( B )) = p ( q ( A )) ⊎ p ( q ( B )). It remains to proveregularity. Assume K ⊂ p ◦ q ( U ). q ( U ) is open, so the regularity of p gives L ⊂ q ( U )such that K ⊂ p ( L ). The regularity of q gives M ⊂ U such that L ⊂ q ( M ). We conclude K ⊂ p ( L ) ⊂ p ( q ( M )) from the monotonity of p .Let X ∧ denote the set of functions σ : C b ( X ) → R with the property σ ( φ ( a )) = φ ( σ ( a )).In the proof of the quasi-multiplicativity of the quasi-integral from a simple quasi-measurewe proved that X ∗ ⊂ X ∧ . We identify X with a subset of X ∗ by the injection x δ x . Amore abstract characterization of X as a subset of X ∗ is given by Proposition 6. If µ ∈ X ∗ is subadditive on open sets; µ ( U ∪ V ) ≤ µ ( U ) + µ ( V ) , then µ = δ x . In particular R ∗ ≃ R .Proof. The family F µ := { F | µ ( F ) = 1 } is closed under intersection from complemen-tation of the subadditivity on open sets. The continuity of µ gives measure one to theset F := ∩ G ∈F µ G , and in particular F = ∅ . Assume that F contains two different points x, y . The additivity contradicts µ { x } = µ { y } = 1, so we can assume µ { y } = 0. From µ { y } = inf y ∈ U µ ( U ) it follows that there exists an open set U ∋ y with µ ( U ) = 0. Then U c ∈ F µ , which contradicts y ∈ F . We have proven µ { x } = 1. The claim µ = δ x follows.Since µ ∈ R ∗ is the restriction of a Borel measure, it is subadditive on open sets. Proposition 7.
The spectrum of a is Sp a = a ( X ) = X ∗ ( a ) = X ∧ ( a ) . Proof.
We define
Sp a = a ( X ) in agreement with the more general C ∗ definition. Assume σ ∈ X ∧ . We prove σ ( a ) ∈ Sp a by a contradiction argument. Assume σ [ a ] Sp a .Urysohn’s Lemma gives
Sp a (cid:22) φ (cid:22) { σ ( a ) } c , and 0 = φ ( σ ( a )) = σ ( φ ( a )) = σ (1 X ) = 1is a contradiction. The inclusion X ⊂ X ∗ ⊂ X ∧ together with σ ( a ) ∈ Sp a gives
Sp a = a ( X ) = X ∗ ( a ) = X ∧ ( a ). Below we identify X ∧ with a compact Hausdorff space, and thecontinuity of σ σ ( a ) gives that X ∧ ( a ) = X ∧ ( a ) is compact.Each Sp a is a compact Hausdorff space and Tychonoff gives us the compact Hausdorffproduct space Q a ∈ C b ( X ) Sp a , which is the family of real valued functions ψ with ψ ( a ) ∈ araldsen/Image transformations Sp a and initial topology from the functions ψ ψ ( a ). This gives the inclusions X ⊂ X ∗ ⊂ X ∧ ⊂ Y a ∈ C b ( X ) Sp a, (6)and corresponding relative topologies on
X, X ∗ , and X ∧ . The original topology on X equals the relative topology on X if X is a Tychonoff space, and in particular if X is alocally compact Hausdorff space. X ∧ is compact since it is closed: σ [ φ ( a )] = lim σ λ [ φ ( a )] = φ (lim σ λ [ a ]) = φ ( σ [ a ]). X ∗ is a Hausdorff space as a subset of a Hausdorff space. It is tobe expected that X ∗ is locally compact when X is locally compact, but this is an openquestion. Proposition 8.
Let X be a locally compact Hausdorff space. The canonical image-transformation [ ∗ ] : A ( X ) → A ( X ∗ ) given by [ ∗ ]( A ) := A ∗ := { σ ∈ X ∗ | σ ( A ) = 1 } is an image-transformation.Proof. Elements in X ∗ are quasi-integrals, so U ∗ = { σ | σ ( U ) = 1 } = { σ | ∃ a ≤ U σ ( a ) > } = S a ≤ U { σ | σ ( a ) > } is open, and F ∗ = { σ | σ ( F ) = 0 } c = { σ | σ ( F c ) =1 } c = (( F c ) ∗ ) c is closed. Together with X ∗ = { σ | σ ( X ) = 1 } we have verified the firsttwo axioms. Additivity follows from ( F ⊎ G ) ∗ = { σ | σ ( F ) + σ ( G ) } = F ∗ ⊎ G ∗ . Put O c ( U ) := { V | U ⊃ V is compact } . Compact-regularity of σ and local compactness gives U ∗ = { σ | ∃ V ∈ O c ( U ) σ ( V ) = 1 } = S V ∈O c ( U ) V ∗ , since [ ∗ ] is monotone. Let K ⊂ U ∗ .Since O c ( U ) is closed under unions, the above gives V ∈ O c ( U ) with K ⊂ V ∗ , and L = V gives K ⊂ L ∗ , so [ ∗ ] is regular. Proposition 9.
Let X and Y be Hausdorff spaces. If f : Y → X is continuous, then f − : A ( X ) → A ( Y ) is an image-transformation.Proof. Only the regularity needs a proof. If K ⊂ f − ( U ), then L = f ( K ) ⊂ U , and K ⊂ f − ( L ).The special case X = R of the above Proposition was used when we defined integration.It turns out that the two above examples of image-transformations covers all cases, inthe sense that all image-transformation are on the form w − ◦ [ ∗ ] for some continuous w : Y → X ∗ . We need some other aspects of the theory in order to prove this. Our mainmotivation for the study of image-transformations is the following result. Proposition 10.
Let q be an image-transformation from X to Y . Any quasi-measure µ on Y is pulled back to a quasi-measure q ∗ µ := µ ◦ q on X . The adjoint q ∗ maps Y ∗ into X ∗ . The adjoint map is anti-multiplicative: ( p ◦ q ) ∗ = q ∗ ◦ p ∗ .Proof. ν := q ∗ µ is clearly additive: ν ( A ⊎ B ) = µ ( q ( A ) ⊎ q ( B )) = ν ( A ) + ν ( B ). K ⊂ U ⇒ µ ( q ( K )) ≤ µ ( q ( U )), so regularity will follow from µ ( q ( U )) ≤ sup K ⊂ U µ ( q ( K )). Let L ⊂ q ( U ). Since µ is regular we are left with the proof of µ ( L ) ≤ sup K ⊂ U µ ( q ( K )). q is regular, so there is a compact K ⊂ U with L ⊂ q ( K ). The monotonity of µ gives µ ( L ) ≤ µ ( q ( K )). q ∗ ( Y ∗ ) ⊂ X ∗ follows from (i) σ ( q ( X )) = σ ( Y ) = 1 and (ii) σ ( q ( A )) ∈ { , } . The finalclaim is ( p ◦ q ) ∗ µ ( A ) = µ ◦ p ◦ q ( A ) = q ∗ ( p ∗ µ )( A ). araldsen/Image transformations Integration with respect to a quasi-measure gives a quasi-integral. One may also inte-grate with respect to an image-transformation, and the result is a quasi-homomorphism.If a r ( a )( y ) is positive and quasi-linear for each y , then a r ( a ) is monotone and k r ( a ) − r ( b ) k ≤ k r (1 X ) k k a − b k . A quasi-homomorphism is therefore monotone andcontinuous. If r ( φ ( a )) = φ ( r ( a )) for all continuous φ : R → R , then clearly axiom (i) issatisfied. The condition r ( φ ( a )) = φ ( r ( a )) is also necessary, from uniform approximationof φ with polynomials. Proposition 11.
Let q be an image-transformation from a locally compact Hausdorffspace X to a Hausdorff space Y . The integral q ( a ) defined by q ( a )( y ) := r q ( a )( y ) :=( q ∗ δ y )( a ) is a quasi-homomorphism from C b ( X ) to C b ( Y ) , and q ◦ a − = q ( a ) − . If µ is a quasi-measure on Y , then ( q ∗ µ )( a ) = µ ( q ( a )) . If q and p are composable image-transformations,then r q ◦ p = r q ◦ r p .Proof. Recall that σ a = δ σ ( a ) holds for a simple quasi-measure σ . The equality q ◦ a − = q ( a ) − follows from the equivalence of the following statements: q [ a ]( y ) ∈ A ;1 = δ q [ a ]( y ) ( A ) = δ [ q ∗ δ y ]( a ) ( A ) = ( q ∗ δ y ) a ( A ) = q ∗ δ y ◦ a − ( A ); y ∈ q ( a − ( A )). Thisproves continuity of y q ( a )( y ) from the case A = U . The case A = ( −∞ , − k a k ) ∪ ( k a k , ∞ ) gives q ( a ) − ( A ) = q ◦ a − ( A ) = q ( ∅ ) = ∅ , so k q ( a ) k ≤ k a k . So far we haveproven that q : C b ( X ) → C b ( Y ) is well defined. The property q ( φ ( a )) = φ ( q ( a )) fol-lows from q ( φ ( a ))( y ) = ( q ∗ δ y )( φ ( a )) = ( q ∗ δ y ) a ( φ ) = δ ( q ∗ δ y )( a ) ( φ ) = φ ( q ( a )( y )), whichgives that q : A ( a ) → A ( q ( a )) is a surjective algebra homomorphism. We prove q ( a )( y ) =sup k ≤ a q ( k )( y ). Quasi-linearity gives ≥ . Assume q ( a )( y ) ∈ U , or equivalently y ∈ q ( a − ( U )).The claim follows if we can find k ≤ a with y ∈ q ( k − ( U )). The regularity of q gives K ⊂ a − ( U ) with y ∈ q ( K ). Urysohn gives us l with K ≤ l ≤ X . With k := al , weconclude k ≤ a , and K ⊂ k − ( k ( K )) = k − ( a ( K )) ⊂ k − ( U ). This gives y ∈ q ( k − ( U )).Let µ be a quasi-measure on Y . Since q ( a ) − = q ◦ a − , we get µ q ( a ) = ( µ ◦ q ) a , andthe change of variable formula µ ( q ( a )) = ( q ∗ µ )( a ) follows. If q, p are composable, then q ( p ( a ))( z ) = ( q ∗ δ z )( p ( a )) = ( δ z ◦ q ) p ( a ) ( id ) = ( δ z ◦ q ◦ p ) a ( id ) = [( q ◦ p ) ∗ δ z ]( a ).It should be observed that a q ( a )( y ) is a simple quasi-integral for each y . Analternative proof of the regularity claim for a q ( a ) follows from the correspondingstatement for quasi-integrals. The special case µ = δ y in the change of variable formula µ ( q ( a )) = ( q ∗ µ )( a ) gives back the definition q ( a )( y ) = ( q ∗ δ y )( a ). Theorem 3.1.
Let X be a locally compact Hausdorff space and let Y be a Hausdorffspace. There is 1-1 correspondence between image-transformations q : A ( X ) → A ( Y ) andcontinuous functions w : Y → X ∗ given by q = w − ◦ [ ∗ ] and w = q ∗ ◦ ι Y , with ι Y ( y ) := δ y .Proof. We prove that w := q ∗ ◦ ι Y is continuous when q is an image-transformation. Fix a ∈ C b ( X ). It is sufficient to prove continuity of y w ( y )( a ) = q ∗ ◦ ι Y ( y )( a ) = ( q ∗ δ y )( a ) = q ( a )( y ), but this follows since q ( a ) ∈ C b ( Y ). Equality q = w − ◦ [ ∗ ] follows from thefollowing equivalent statements: y ∈ q ( U ); 1 = δ y ( q ( U )); w ( y ) = ( q ∗ ◦ ι Y )( y ) = q ∗ δ y ∈ U ∗ ; y ∈ w − ([ ∗ ]( U )) . Let w : Y → X ∗ be continuous. Since composition of image-transformations produceimage-transformations, it follows that q := w − ◦ [ ∗ ] is an image-transformation. Equality araldsen/Image transformations w = q ∗ ◦ ι Y follows by inspection of:1 = ( q ∗ ◦ ι Y ( y ))( A ) = δ y ( q ( A )); y ∈ q ( A ) = w − ( A ∗ ); w ( y )( A ) = 1 . Continuity of w = q ∗ ◦ ι Y follows also from continuity of q ∗ and ι Y , which follows easilyby consideration of convergent nets in Y ∗ respectively Y . The proof of the remainingstatements in the commutative diagrams in the introduction is now rather straightfor-ward, and left to the reader.If q is an image-transformation and K ≤ a ≤ U , then K ⊂ a − { } ⊂ a − (0 , ∞ ) ⊂ U ,so q ( K ) ⊂ q ( a ) − { } ⊂ q ( a ) − (0 , ∞ ) ⊂ q ( U ). When X is locally compact, Urysohn givesus q ( U ) = [ K ⊂ U q ( K ) = [ k (cid:22) U q ( k ) − { } = [ k (cid:22) U q ( k ) − (0 , ∞ ) . (7)The above indicates a 1-1 correspondence between image-transformations and quasi-homomorphisms. Theorem 3.2.
Let X be a locally compact normal space. If r : C b ( X ) → C b ( Y ) is a quasi-homomorphism, then there exists a unique image-transformation q such that r ( a ) = q ( a ) .Proof. Put q ( U ) := S k (cid:22) U r ( k ) − (0 , ∞ ). From a l := al ↑ a with l (cid:22) U and the monotoneconvergence theorem, we conclude q ( U ) = S a ≤ U r ( a ) − (0 , ∞ ).We prove additivity q ( U ⊎ V ) = q ( U ) ⊎ q ( V ) on open sets: a ≤ U and b ≤ V give ab = 0, a, b ∈ A ( a − b ), 0 = r ( ab ) = r ( a ) q ( b ), r ( a ) − (0 , ∞ ) ∩ r ( b ) − (0 , ∞ ) = ∅ , and finally q ( U ) ∩ q ( V ) = ∅ . ⊃ follows from monotonity on open sets. We prove ⊂ : Let y ∈ q ( U ⊎ V ).Then there exists c ≤ U ∪ V with r ( c )( y ) >
0. It follows that c = a + b with a ≤ U and b ≤ V . From a, b ∈ A ( a − b ) it follows that r ( a )( y ) + r ( b )( y ) = r ( c )( y ) > r ( a )( y ) > r ( b )( y ) >
0, and finally y ∈ q ( U ) ∪ q ( V ). q ( X ) = Y : Let y ∈ Y . The regularity gives 1 = r [1 X ]( y ) = sup a ≤ X r [ a ]( y ) so there exists a ≤ X such that r [ a ]( y ) >
0. Since q is additive on open sets we can extend q to A by q ( F ) = q ( F c ) c . It follows that q ( F ) = T a ≥ F r ( a ) − { } . This, and normality, provesadditivity on closed sets.It is clear that q is monotone on open sets and on closed sets. Monotonity of r givesthat U ⊂ F implies q ( U ) ⊂ q ( F ). Normality and Urysohn gives F ≤ a ≤ U from which q ( F ) ⊂ q ( U ) follows. The set function q is therefore additive and monotone. Regularityfollows from consideration of k ≤ supp k ≤ V ≤ K = V ≤ U , so q is an image-transformation.We prove r ( a )( y ) = q ( a )( y ): The map a r ( a )( y ) = σ ( a ) defines a simple quasi-integral σ with a corresponding simple quasi-measure also denoted by σ . Equality σ = q ∗ δ y followsfrom equivalence of the following statements:1 = σ ( U ) = sup k (cid:22) U σ ( k ); ∃ k (cid:22) U r ( k )( y ) > y ∈ q ( U ); 1 = δ y ( q ( U )) . araldsen/Image transformations
4. Examples.4.1. The Aarnes Measure on the Square.
The following example is abstracted from Aarnes [2]. A set A in the unit square X = [0 , is solid if A and its complement A c are both connected. Let ∂X denote the border of X .Put µ ( A ) = 1 if A contains the border or if A intersects both the border and { (1 / , / } ,and put µ ( A ) = 0 otherwise. This defines a 0-1 valued set function µ on the class of solidsets in X . If F is closed and connected in X , then F c is a countable disjoint union ofopen solid sets U , U , . . . , and we extend µ by µ ( F ) := 1 − µ ( U ) − µ ( U ) − · · · . The setfunction is extended to the class C of disjoint finite unions of connected closed sets by µ ( F ⊎ · · · ⊎ F n ) = µ ( F ) + · · · + µ ( F n ). If U is open, µ ( U ) is defined to be the supremumof µ ( K ) for K ⊂ U , K in C . Finally µ ( U c ) := 1 − µ ( U ) extends µ to the class A of setswhich are open or closed. The set function µ is a proper quasi-measure since it is notsubadditive. We refer to this quasi-measure as the Aarnes measure on the square. Let a be the “pyramidal” [2, p.65] function on X whose graph is given by the four planeswhich contain the point (1 / , / ,
1) in R and respectively the four sides of ∂X . Inparticular a − ( { } ) = ∂X which has Aarnes measure µ ( ∂X ) = 1. This gives µ a = δ , and µ ( a ) = 0.Let the graph of b ≥ F = { ( t, s, | ≤ t ≤ , ≤ s ≤ / } , and the plane containing the line segments { ( t, / , | ≤ t ≤ } and { ( t, , | ≤ t ≤ } . Since b = 0 on F , and µ ( F ) = 1, it follows that µ ( b ) = 1. Let G bethe closed triangle in X with corners at (0 , , (1 / , / , (1 , a + b equals1 on G , and µ ( G ) = 1, so the quasi-integral µ is nonlinear1 = µ ( a + b ) = µ ( a ) + µ ( b ) = 0 + 0 = 0 . (8) Consider again the unit square X . A (normalized!) quasi-measure µ is parliamentary[7] if µ ( A ) = 1 / A . An example is given by the ordinary 3 point measure µ p,q,r = ( δ p + δ q + δ r ) /
3. A simple quasi-measure ˜ µ is defined on solid sets K by ˜ µ ( K ) =1 (1 / , ( µ ( K )) and extended to A as in 4.1. The 3-point quasi-measure is ˜ µ p,q,r . It followseasily that the 3-point quasi-measure is not a measure. Let ˜ µ np,q,r be the 3-point quasi-measure on X n :=[ − n, n ] , let ι n be the natural injection of X n into the plane, and con-clude that µ n := ˜ µ np,q,r ◦ ι n − is a quasi-measure on the plane. Define µ ∞ ( A ) := lim µ n ( A ) forall images A in the plane. The set function µ ∞ is additive, but not compact-regular. Thisexample shows that non-regular quasi-measures arise quite naturally. There is a Riesz rep-resentation theorem also for such measures [6]. Consider next a continuous f n : X → R with f n ( X ) = [ − / , / + { n, } , and put ν ( A ) := P n ≥ ˜ µ p,q,r ◦ f n − ( A ) / n . It followsthat ν is a quasi-measure in the plane which takes all values in [0 , A P n p n ν n ( A ) is a quasi-measure if P n p n = 1, p n ≥
0, and each ν n is a quasi-measure. Let σ be a simple quasi-measure on X . An image-transformation q [3, p.10-11] from X to Y is defined by q ( A ) = Y if σ ( A ) = 1 and q ( A ) = ∅ if σ ( A ) = 0. Let Y be a subset of X ∗ , let w : Y → X ∗ be the inclusion map, and conclude that q = w − ◦ [ ∗ ] is an image-transformation. Observe in particular that we may choose a finite Y . More examples araldsen/Image transformations may be constructed and investigated by continuous parametrizations w of simple quasi-measures in X ∗ .
5. Comments on Previous Results.
In [2] Aarnes establishes a Riesz representation theorem for quasi-states in terms ofquasi-measures on compact Hausdorff spaces. This has been generalized to the case of alocally compact Hausdorff space X in two different directions. Aarnes [4] arrives at quasi-measures which are compact-regular, but not additive, by consideration of the one pointcompactification of X . Boardman [6] obtains a representation theorem for quasi-linear in-tegrals on C b ( X ) in terms of quasi-measures which are additive, but not compact-regular.We introduced quasi-measures which are additive and compact-regular. Integration withrespect to a quasi-measure µ is defined as in [2, p.46]: If a : X → R is continuous, then µ ◦ a − is the restriction of a measure µ a on R , and µ ( a ) := R tµ a ( dt ) is the integral. Anovelty in our work is the simplified proof of the quasi-linearity of a µ ( a ), which avoidsthe consideration of the Riemann-Stieltjes integral [2, p.46-52]. We remark also that thismethod of integration fails in Boardman’s more general case. Our proof of the corre-spondence between image-transformations and quasi-homomorphisms is different fromthe proof by Aarnes. Aarnes [2, p.13] used the fact that all homorphisms are on the form a a ◦ w in the case of compact Hausdorff spaces, but this is not available for locallycompact spaces. Differences like this are generically found on comparison with [2] whichdeals only with the compact case. araldsen/Image transformations References [1] J.F. Aarnes. Quasi-states on C ∗ -algebras. Trans. of the American Math. Soc. , 149,1970.[2] J.F. Aarnes. Quasi-states and quasi-measures.
Adv. in Math. , 86(1):41–67, 1991.[3] J.F. Aarnes. Image transformations, attractors, and invariant non-subadditive mea-sures. preprint Mathematics Trondheim , 2:1–28, 1994.[4] J.F. Aarnes. Quasi-measures in locally compact spaces. preprint Mathematics Trond-heim , 4:1–43, 1995.[5] J.F. Aarnes and D.J. Grubb. Quasi-measures and image transformations on com-pletely regular spaces.
Topology and its Applications , 135:33–46, 2004.[6] J.P. Boardman. Quasi-measures on completely regular spaces.
Rocky Mountain J.Math. , 27(2):447–470, 1997.[7] F.F. Knudsen. Topology and the construction of extreme quasi-meaures.
Advancesin mathematics , 120, 1996.[8] J. Lamperti.
Probability . Benjamin, 1966.[9] A.B. Rustad. Unbounded quasi-integrals.
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Ann. of Math. , 48(2), 1947.[11] D. Shakhmatov. Linearity of quasi-states on commutative C ∗ -algebras of stable rank1. preprintpreprint