A Positive Operator-Valued Measure for an Iterated Function System
aa r X i v : . [ m a t h . F A ] M a r A POSITIVE OPERATOR-VALUED MEASURE FOR AN ITERATEDFUNCTION SYSTEM
Trubee Davison A BSTRACT . Given an iterated function system (IFS) on a complete and separable met-ric space Y , there exists a unique compact subset X ⊆ Y satisfying a fixed point rela-tion with respect to the IFS. This subset is called the attractor set, or fractal set, associ-ated to the IFS. The attractor set supports a specific Borel probability measure, calledthe Hutchinson measure, which itself satisfies a fixed point relation. P. Jorgensen gen-eralized the Hutchinson measure to a projection-valued measure, under the assumptionthat the IFS does not have essential overlap [13] [14]. In previous work, we developedan alternative approach to proving the existence of this projection-valued measure [6][7] [8]. The situation when the IFS exhibits essential overlap has been studied by Jor-gensen and colleagues in [10]. We build off their work to generalize the Hutchinsonmeasure to a positive-operator valued measure for an arbitrary IFS, that may exhibit es-sential overlap. This work hinges on using a generalized Kantorovich metric to definea distance between positive operator-valued measures. It is noteworthy to mention thatthis generalized metric, which we use in our previous work as well, was also introducedby R.F. Werner to study the position and momentum observables, which are central ob-jects of study in the area of quantum theory [18]. We conclude with a discussion ofNaimark’s dilation theorem with respect to this positive operator-valued measure, andat the beginning of the paper, we prove a metric space completion result regarding theclassical Kantorovich metric. C ONTENTS
1. Background: 22. Results: 102.1. Metric Space Completion of M loc ( Y ) : 102.2. Generalizing the Kantorovich Metric to Positive Operator-ValuedMeasures 152.3. A Fixed POVM Associated to an IFS 202.4. Dilation of the Fixed POVM 213. Acknowledgements: 23 Department of MathematicsUniversity of ColoradoCampus Box 395Boulder, CO [email protected]
References 24[Keywords: Kantorovich metric, Operator-valued measure, Cuntz algebra, Fixed point][Mathematics subject classification 2010: 46C99 - 46L05][Publication note: The final publication is available at Springer viahttp://dx.doi.org/10.1007/s10440-018-0161-6. This version has been updated with smallchanges to match the final publication.]1. B
ACKGROUND :In this opening section, we will provide relevant background information, providean overview of what is to come, and highlight two applications to quantum theory. Tobegin, let ( Y, d ) be a complete and separable metric space. Definition 1.1.
A Lipschitz contraction on Y is a map L : Y → Y such that d ( L ( x ) , L ( y )) ≤ rd ( x, y ) for all x, y, ∈ Y , where < r < . Let L : Y → Y be a Lipschitz contraction on Y . Since Y is a complete metricspace, it is well known that L admits a unique fixed point y ∈ Y , meaning that L ( y ) = y . This result is known as the Contraction Mapping Principle, or the Banach FixedPoint Theorem. In 1981, J. Hutchinson published a seminal paper (see [9]), where hegeneralized the Contraction Mapping Principle to a finite family, S = { σ , ..., σ N − } ,of Lipschitz contractions on Y , where N ∈ N is such that N ≥ . Indeed, one canassociate to S a unique compact subset X ⊆ Y which is invariant under the S , meaningthat X = N − [ i =0 σ i ( X ) . (1.1)A finite family of Lipschitz contractions on Y is called an iterated function system(IFS) on Y , and the compact invariant subset X described above is called the self-similar fractal set, or attractor set, associated to the IFS. The existence and uniqueness ofthe attractor set can be obtained in several ways. One way is the following: if A, B ⊆ Y ,the Hausdorff distance, δ , between A and B is defined by δ ( A, B ) = sup { d ( a, B ) , d ( b, A ) : a ∈ A, b ∈ B } . Denote by K the collection of compact subsets of Y . It is well known that the metricspace ( K , δ ) is complete. The following theorem guarantees the existence and unique-ness of X . Theorem 1.2. [9][2] [Hutchinson, Barnsley] The Hutchinson-Barnsley operator F : K → K given by K N − [ i =0 σ i ( K ) POSITIVE OPERATOR-VALUED MEASURE FOR AN ITERATED FUNCTION SYSTEM 3 is a Lipschitz contraction on the complete metric space ( K , δ ) . By the ContractionMapping Principle, there exists a unique compact subset X ⊆ Y such that F ( X ) = X, which is equation (1.1). In this paper, we will consider the attractor set from a measure theoretic perspec-tive. In particular, the attractor set can be realized as the support of a Borel probabilitymeasure on Y . This measure, which we denote by µ , satisfies the fixed point relation µ ( · ) = N − X i =0 N µ ( σ − i ( · )) , (1.2)and is often referred to as the Hutchinson measure. It is the unique fixed point of aLipschitz contraction, T , on an appropriate complete metric space of Borel probabilitymeasures on Y . Naturally, the map T is given by T ( ν ) = N − X i =0 N ν ( σ − i ( · )) , for ν a Borel probability measure on Y . The metric, H , is given by H ( µ, ν ) = sup f ∈ Lip ( Y ) (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)Z Y f dµ − Z Y f dν (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) , where Lip ( Y ) = { f : Y → R : | f ( x ) − f ( y ) | ≤ d ( x, y ) for all x, y ∈ Y } , and where µ and ν are Borel probability measures on Y . This metric is called the Kantorovichmetric.What is the appropriate complete metric space of Borel probability measures on Y?To answer this question, we make the following definitions: • Let Q ( Y ) be the collection of all Borel probability measures on Y . • Let M loc ( Y ) be the collection of Borel probability measures on Y that havebounded support. • Let M ( Y ) be the collection of Borel probability measures ν on Y such that R | f | dν < ∞ for all f ∈ Lip ( Y ) , where Lip ( Y ) is the collection of all real-valued Lipschitz functions on Y .We consider two cases:(1) In the case that ( Y, d ) is a compact (and therefore bounded) metric space, Q ( Y ) = M loc ( Y ) = M ( Y ) . Moreover, in this case, the Kantorovich metric is well-defined (finite) on Q ( Y ) . Indeed, it is well known that ( Q ( Y ) , H ) is a compactmetric space (and therefore complete). It is henceforth appropriate to considerthe Contraction Mapping Principle on ( Q ( Y ) , H ) , with respect to the Lipschitzcontraction T .(2) In the case that ( Y, d ) is an arbitrary complete and separable metric space, theKantorovich metric is not necessarily well-defined (finite) on Q ( Y ) . Accord-ingly, we must restrict the Kantorovich metric to a sub-collection of Borel prob-ability measures on Y , where it is well-defined. The intent is to find a sub-collection of measures such that the resulting metric space is complete, and A POSITIVE OPERATOR-VALUED MEASURE FOR AN ITERATED FUNCTION SYSTEM such that T restricts to a map on this sub-collection. To our knowledge, thereare two choices for a sub-collection which have been studied in the literature.Hutchinson suggested (see [9]) that ( M loc ( Y ) , H ) constitutes a complete metricspace. In a later paper, A. Kravchenko indicated this not to be true, and showedthat ( M ( Y ) , H ) is a complete metric space (see [15]).The following theorem assures the existence and uniqueness of a measure satisfyingequation (1.2). It can be shown that the support of this measure is the attractor set X . Theorem 1.3. [15] [Kravchenko] The map T : M ( Y ) → M ( Y ) given by T ( ν ) = N − X i =0 N ν ( σ − i ( · )) , is a Lipschitz contraction on the complete metric space ( M ( Y ) , H ) . By the ContractionMapping Principle, there exists a unique Borel probability measure µ ∈ M ( Y ) suchthat T ( µ ) = µ , which is equation (1.2). Although M loc ( Y ) does not constitute a complete metric space in the H metric, it isinteresting to investigate its relationship to M ( Y ) with respect to the H metric. Indeed,the first result of the present paper will be to show that the metric space completion of M loc ( Y ) is M ( Y ) . We will follow this result by examining a bounded version of theKantorovich metric, which we denote by M H , and is given by
M H ( µ, ν ) = sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)Z Y f dµ − Z Y f dν (cid:12)(cid:12)(cid:12)(cid:12) : f ∈ Lip ( Y ) and || f || ∞ ≤ (cid:27) . By conglomerating a result of Kravchenko with a previously known result about theweak topology on Q ( Y ) , we will be able to observe that ( Q ( Y ) , M H ) is a completemetric space. Moreover, we will show that ( Q ( Y ) , M H ) is the metric space completionof ( M loc ( Y ) , M H ) . Remark 1.4.
The advantage of the
M H metric is that it is well defined on Q ( Y ) ; that is,there is no need to restrict the metric to a sub-collection of Borel probability measureson Y . However, the reason that the M H metric is not used in the study of iteratedfunction systems is that the condition || f || ∞ ≤ prevents T from being a Lipschitzcontraction in the M H metric.
We now proceed to the functional analytic setting, with the goal of discussing a gen-eralization of the Hutchinson measure to an operator-valued measure. Consider theHilbert space L ( X, µ ) , where X ⊆ Y is the attractor set of the IFS S = { σ , ..., σ N − } ,and µ is the unique Borel probability measure on Y satisfying equation (1.2), whosesupport is X . We first consider the case that equation (1.1) is a disjoint union. Wefurther assume that there exists a Borel measurable function σ : X → X such that σ ◦ σ i = id X , for all ≤ i ≤ N − . We provide a standard example for the abovescenario: • Let X = Cantor Set ⊆ [0 , , with the standard metric on R . • Let σ ( x ) = x and σ ( x ) = x + . • Let σ ( x ) = 3 x mod . POSITIVE OPERATOR-VALUED MEASURE FOR AN ITERATED FUNCTION SYSTEM 5
Under these assumptions, define S i : L ( X, µ ) → L ( X, µ ) by φ ( φ ◦ σ ) √ N σ i ( X ) for all i = 0 , ..., N − , and its adjoint S ∗ i : L ( X, µ ) → L ( X, µ ) by φ √ N ( φ ◦ σ i ) for all i = 0 , ..., N − . This leads to the following result due to P. Jorgensen. Theorem 1.5. [11] [Jorgensen] The maps { S i : 0 ≤ i ≤ N − } are isometries, andthe maps { S ∗ i : 0 ≤ i ≤ N − } are their adjoints. Moreover, these maps and theiradjoints satisfy the Cuntz relations: (1) N − X i =0 S i S ∗ i = H (2) S ∗ i S j = δ i,j H where ≤ i, j ≤ N − . Remark 1.6.
Another way to rephrase the above theorem is to say that the Hilbertspace L ( X, µ ) admits a representation of the Cuntz algebra, O N , on N generators. Let Γ N = { , ..., N − } . For k ∈ Z + , let Γ kN = Γ N × ... × Γ N , where the product is k times. If a = ( a , ..., a k ) ∈ Γ kN , where a j ∈ { , , ..., N − } for ≤ j ≤ k , define A k ( a ) = σ a ◦ ... ◦ σ a k ( X ) .Using that equation (1.1) is a disjoint union, we conclude that { A k ( a ) } a ∈ Γ kN partitions X for all k ∈ Z + . For k ∈ Z + and a = ( a , ..., a k ) ∈ Γ kN define P k ( a ) = S a S ∗ a ,where S a = S a ◦ ... ◦ S a k . Remark 1.7.
One can show that P k ( a ) = M Ak ( a ) , where M Ak ( a ) : L ( X, µ ) → L ( X, µ ) is given by f ∈ L ( X, µ ) A k ( a ) f. We now recall two definitions.
Definition 1.8.
Let H be a Hilbert space. If B ( H ) denotes the C ∗ -algebra of boundedoperators on H , a projection P ∈ B ( H ) satisfies P ∗ = P (self-adjoint) and P = P (idempotent). In view of Remark 1.7, note that P k ( a ) is a projection in B ( L ( X, µ )) . For the nextdefinition, denote the empty set by ∅ . Definition 1.9.
Let ( X, B ( X )) be a measure space, and let H be a Hilbert space. Aprojection-valued measure with respect to the pair ( X, H ) is a map F : B ( X ) → B ( H ) such that: • F (∆) is a projection in B ( H ) for all ∆ ∈ B ( X ) ; • F ( ∅ ) = 0 and F ( X ) = id H (the identity operator on H ); • F (∆ ∩ ∆ ) = F (∆ ) F (∆ ) for all ∆ , ∆ ∈ B ( X ) (where the product oper-ation F (∆ ) F (∆ ) is operator composition in B ( H )) ; • If { ∆ n } ∞ n =1 is a sequence of pairwise disjoint sets in B ( X ) , and if g, h ∈ H ,then * F ∞ [ n =1 ∆ n ! g, h + = ∞ X n =1 h F (∆ n ) g, h i . A POSITIVE OPERATOR-VALUED MEASURE FOR AN ITERATED FUNCTION SYSTEM
With this definition, we have the following result of Jorgensen.
Theorem 1.10. [13] [14] [Jorgensen] There exists a unique projection-valued measure, E ( · ) , defined on the Borel subsets of X , B ( X ) , taking values in the projections on L ( X, µ ) such that, (1) E ( · ) = P N − i =0 S i E ( σ − i ( · )) S ∗ i , and (2) E ( A k ( a )) = P k ( a ) for all k ∈ Z + and a ∈ Γ kN . Remark 1.11.
The projection-valued measure E ( · ) is the canonical projection-valuedmeasure on the measure space ( X, B ( X ) , µ ) , meaning that if ∆ ∈ B ( X ) , E (∆) = M ∆ , where M ∆ is multiplication by ∆ . Previously, we presented an alternative approach to proving the above theorem (see[6][7] [8]). This is summarized below. Let P ( X ) be the collection of all projection-valuedmeasures from B ( X ) into the projections on L ( X, µ ) . Define a metric ρ on P ( X ) by ρ ( E, F ) = sup f ∈ Lip ( X ) (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z f dE − Z f dF (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:27) , (1.3)where || · || denotes the operator norm in B ( H ) , and E and F are arbitrary members of P ( X ) . This is called the generalized Kantorovich metric.At this juncture, we would like to note that the generalized Kantorovich metric de-fined above in equation (see (1.3)) has been previously defined by R.F. Werner in the set-ting of mathematical physics, namely in the area of quantum theory (see [18]). Indeed,a projection-valued measures is a more specific instance of a positive operator-valuedmeasure (POVM), which is also called an observable in physics. Werner introducesthe generalized Kantorovich metric as a tool for studying the position and momentumobservables, which are central objects of study in quantum theory.In the present paper, and in our related paper ([6]), we develop new properties of thegeneralized Kantorovich metric, and discuss its application to iterated function systems. Theorem 1.12. [6] [8] [Davison] ( P ( X ) , ρ ) is a complete metric space. Theorem 1.13. [6] [7] [8] [Davison] The map U : P ( X ) → P ( X ) given by F ( · ) N − X i =0 S i F ( σ − i ( · )) S ∗ i is a Lipschitz contraction on the ( P ( X ) , ρ ) metric space. By the Contraction MappingPrinciple, there exists a unique projection-valued measure, E ∈ P ( X ) , satisfying part (1) of Theorem 1.10. Part (2) of Theorem 1.10 follows as a consequence. The fact that U ( F ) ∈ P ( X ) for F ∈ P ( X ) depends on the Cuntz relations. Forinstance, the computation that U ( F ) is an idempotent relies on part (2) of Theorem 1.5.Indeed, if ∆ ∈ B ( X ) POSITIVE OPERATOR-VALUED MEASURE FOR AN ITERATED FUNCTION SYSTEM 7 ( U ( F )(∆)) = N − X i =0 S i F ( σ − i (∆)) S ∗ i ! = N − X i =0 S i F ( σ − i (∆)) S ∗ i N − X j =0 S j F ( σ − j (∆)) S ∗ j = N − X i =0 S i F ( σ − i (∆)) S ∗ i = N − X i =0 S i F ( σ − i (∆)) S ∗ i = U ( F )(∆) . More generally, the fact that U ( F )(∆ ∩ ∆ ) = U ( F )(∆ ) U ( F )(∆ ) similarly relieson part (2) of Theorem 1.5. The fact that U ( F )( X ) = id H relies on part (1) of Theorem1.5. That is, U ( F )( X ) = N − X i =0 S i F ( σ − i ( X )) S ∗ i = N − X i =0 S i F ( X ) S ∗ i = N − X i =0 S i id H S ∗ i = N − X i =0 S i S ∗ i = id H . Remark 1.14.
We see by the above computations that part (2) of Theorem 1.5 is usedto show that U ( F ) takes values in the projections on L ( X, µ ) . However, if we hypo-thetically obtain a family of operators { S i } N − i =0 satisfying part (1) of Theorem 1.5, andnot part (2) of Theorem 1.5, the map U will still carry structure. In particular, if F is aPOVM, U ( F ) will also be a POVM. This situation appears when our iterated functionsystem has essential overlap, which we describe below. The construction of the isometries { S i } N − i =0 that satisfy parts (1) and (2) of Theorem1.5 depends on the fact that equation (1.1) is a disjoint union, and on the existence ofa map σ : X → X satisfying σ ◦ σ i = id X for all ≤ i ≤ N − . An IFS exhibitinga disjoint union in equation (1.1) is an example of a broader class of iterated functionsystems called iterated function systems with non-essential overlap, meaning that µ ( σ i ( X ) ∩ σ j ( X )) = 0 , (1.4) A POSITIVE OPERATOR-VALUED MEASURE FOR AN ITERATED FUNCTION SYSTEM when i = j , and where µ is the Hutchinson measure. Like an iterated function systemwhich is disjoint, an iterated function system with non-essential overlap also admits arepresentation of the Cuntz algebra, assuming that each member of the IFS is of finitetype. This is due to Jorgensen and his collaborators K. Kornelson, and K. Shuman, andis stated below [10]. Definition 1.15. [10]
A measurable endomorphism τ : X → X is said to be of finitetype if there is a finite partition E , ..., E k of τ ( X ) , and measurable mappings σ i : E i → X, i = 1 , ..., k such that σ i ◦ τ | E i = id E i . Theorem 1.16. [10] [Jorgensen, Kornelson, Shuman] Let S = { σ , ..., σ N − } be anIFS with attractor set X , and let µ be the corresponding Hutchinson measure. Furthersuppose that each member of the IFS is of finite type. For each i = 0 , ..., N − , define F i : L ( X, µ ) → L ( X, µ ) given by φ √ N ( φ ◦ σ i ) . The family of operators { S i := F ∗ i } N − i =0 define a representation of the Cuntz algebra ifand only if the IFS has non-essential overlap. In view of this result, an additional question to ask is what can be said about anIFS , S = { σ , ..., σ N − } , that has essential overlap, meaning that equation µ ( σ i ( X ) ∩ σ j ( X )) > , for some i = j . More generally, what can be said about an arbitrary IFS,that may or may not exhibit non-essential overlap? This situation was also studied byJorgensen and his collaborators in [10]. In particular, we can still define the operators F i : L ( X, µ ) → L ( X, µ ) given by φ √ N ( φ ◦ σ i ) for all i = 0 , ..., N − . Theorem 1.17. [10] [Jorgensen, Kornelson, Shuman] The family of operators { F i } N − i =0 satisfy the operator identity N − X i =0 F ∗ i F i = id H . Proof.
We include a proof of this result because this result is fundamental to the mainpurpose of the present paper. Note that the the Hutchinson measure µ associated to thisIFS has the property that N N − X i =0 Z X | f | ◦ σ i dµ = Z X | f | dµ = || f || for all f ∈ L ( X, µ ) . Since the operator P N − i =0 F ∗ i F i is self-adjoint, in order to showthat P N − i =0 F ∗ i F i = id H , it is enough to show that POSITIVE OPERATOR-VALUED MEASURE FOR AN ITERATED FUNCTION SYSTEM 9 * N − X i =0 F ∗ i F i ! f, f + = h f, f i = || f || , for all f ∈ L ( X, µ ) . Accordingly, let f ∈ L ( X, µ ) , and observe that for all ≤ i ≤ N − , || F i f || = h F i f, F i f i = Z X √ N ( f ◦ σ i ) 1 √ N ( f ◦ σ i ) dµ = 1 N Z X | f | ◦ σ i dµ. Therefore, * N − X i =0 F ∗ i F i ! f, f + = N − X i =0 h F ∗ i F i f, f, i = N − X i =0 || F i f || = 1 N N − X i =0 Z X | f | ◦ σ i dµ = || f || , which proves the result. (cid:3) Consequently, if we define S i = F ∗ i , we can rewrite the above operator identity as N − X i =0 S i S ∗ i = id H , (1.5)which is exactly part (1) of Theorem 1.5.Referring back to Remark 1.14, our intent is to generalize the map U from Theorem1.13 to the case of an arbitrary IFS; that is, to the case that we have a family of oper-ators { S i } N − i =0 that satisfy part (1) of Theorem 1.5. Toward this end, let S ( X ) be thecollection of all positive operator-valued measures from B ( X ) into the positive oper-ators on L ( X, µ ) . We will show that ( S ( X ) , ρ ) is a complete metric space, therebygeneralizing Theorem 1.12 to positive operator-valued measures. Additionally, we willshow that the map U extends to a map on S ( X ) , and is a Lipschitz contraction in the ρ metric. As a consequence, there will exist a unique POVM A ∈ S ( X ) satisfying A ( · ) = N − X i =0 S i A ( σ − i ( · )) S ∗ i . (1.6)This will generalize part (1) of Theorem 1.10 to the case of an arbitrary IFS. We wouldlike to note that Jorgensen proved the existence and uniqueness of a POVM that satisfiesequation (1.6) in a special case, using Kolmogorov’s extension theorem. We refer thereader to Lemma I.3 in [12].A family of operators { S i } N − i =0 defined on a Hilbert space satisfying equation (1.5) isone of the starting points for our results below. To provide broader context, it is worth-while to note that equation (1.5) also appears in quantum information theory. Indeed,a measurement of a quantum system on a Hilbert space H is composed of a familyof such operators, as described in papers by D.W. Kribs and colleagues (see [16] and [17]). The number of operators in the family corresponds to the number of measure-ment outcomes of an experiment. If the state of the system before the experiment is theunit vector h ∈ H , then the probability that measurement outcome i occurs is given by p h ( i ) = h S i S ∗ i h, h i . Using equation (1.5), we obtain P N − i =0 p h ( i ) = 1 , which impliesthat p h ( · ) is a probability measure on the measurement outcomes. This connection toquantum information theory was made aware to the author by Jorgensen and colleaguesin their paper [10].To conclude the introductory section, we mention an important result in operatortheory, which is Naimark’s dilation theorem. Theorem 1.18. [Naimark’s Dilation Theorem] Let F be a POVM with respect to thepair ( X, H ) . There exists a Hilbert space K , a bounded operator V : K → H , and aprojection-valued measure P with respect to the pair ( X, K ) , such that F ( · ) = V P ( · ) V ∗ . We call P a dilation of the POVM F . In the results section, we will build off an existingresult in [10] to identify an explicit Hilbert space that supports such a dilation of thePOVM A in equation (1.6). 2. R ESULTS :2.1.
Metric Space Completion of M loc ( Y ) : Let ( Y, d ) be a complete and separablemetric space. As mentioned earlier, it was first claimed in [9] that ( M loc ( Y ) , H ) is acomplete metric space. However, we will briefly outline an example, presented in [15],which shows this not to be true. Claim 2.1. [15] [Kravchenko] Let ( Y, d ) be an unbounded metric space. Then ( M loc ( Y ) , H ) is not complete.Proof. Choose a sequence of points x k ∈ Y for k = 0 , , , ... , such that d ( x , x k ) ≤ k for all k , and d ( x k , x ) → ∞ . For a point x ∈ Y , define the delta measure at x by δ x ( A ) = ( if x ∈ A if x / ∈ A. For n = 1 , , , ... , define the sequence of measures ν n = 2 − n δ x + Σ nk =1 − k δ x k ∈ M loc ( Y ) . This sequence is Cauchy in ( M loc ( Y ) , H ) . However, it can be shown that itdoes not converge to a measure in ( M loc ( Y ) , H ) . (cid:3) Since ( M loc ( Y ) , H ) is not a complete metric space (when Y is unbounded), we con-sider the larger sub-collection of measures, M ( Y ) , equipped with the H metric. Indeed,we will review that M loc ( Y ) ⊆ M ( Y ) . Definition 2.2.
A measure µ on the metric space Y is said to be regular if for everyBorel subset A ⊆ Y , and every ǫ > , there exists a closed set F and an open set G such that F ⊆ A ⊆ G and µ ( G \ F ) < ǫ . Definition 2.3.
A measure µ on the metric space Y is said to be tight if for every ǫ > ,there exists a compact set K such that µ ( Y \ K ) < ǫ . POSITIVE OPERATOR-VALUED MEASURE FOR AN ITERATED FUNCTION SYSTEM 11
Remark 2.4.
Since Y is a complete and separable metric space, every Borel probabilitymeasure on Y is regular and tight (see Ch. 1, Section 1 in [3] ). In particular, themeasures in M ( Y ) and M loc ( Y ) are all regular and tight. Lemma 2.5. [3] [Ch. 1, Section 1 in Billingsley] A Borel probability measure µ is tighton the metric space Y if and only if for each Borel subset A ⊆ Y , µ ( A ) = sup { µ ( K ) : K ⊆ A and K compact } . Corollary 2.6. If µ is a Borel probability measure which is tight on the metric space Y ,then µ ( Y \ supp ( µ )) = 0 .Proof. Note that Y \ supp ( µ ) = ∪{ A ⊆ Y : A is open and µ ( A ) = 0 } which isa Borel set in Y . Therefore by Lemma 2.5, µ ( Y \ supp ( µ )) = sup { µ ( K ) : K ⊆ Y \ supp ( µ ) and K compact } . Now if K ⊆ Y \ supp ( µ ) , then since K is compact, ithas a finite subcovering by µ -measure zero open sets. Hence, µ ( K ) = 0 , and therefore µ ( Y \ supp ( µ )) = 0 . (cid:3) Proposition 2.7. M loc ( Y ) ⊆ M ( Y ) .Proof. Let µ ∈ M loc ( Y ) . To show that µ ∈ M ( Y ) , we need to show that R Y | f | dµ < ∞ for all f ∈ Lip ( Y ) . Choose f ∈ Lip ( Y ) with Lipschitz constant γ , and choose a point x ∈ Y . Since µ has bounded support, we can assume that there exists a K ≥ suchthat supp ( µ ) ⊆ B K ( x ) , where B K ( x ) = { x ∈ Y : d ( x, x ) ≤ K } . Moreover, µ ( Y \ B K ( x )) = 0 by Corollary 2.6. This implies that Z Y | f | dµ = Z B K ( x ) | f | dµ + Z Y \ B K ( x ) | f | dµ = Z B K ( x ) | f | dµ. Continuing, observe that Z B K ( x ) | f ( x ) | dµ ( x ) ≤ Z B K ( x ) | f ( x ) − f ( x ) | dµ ( x ) + Z B K ( x ) | f ( x ) | dµ ( x ) ≤ Z B K ( x ) γd ( x, x ) dµ ( x ) + Z B K ( x ) | f ( x ) | dµ ( x ) ≤ γKµ ( B K ( x )) + | f ( x ) | µ ( B K ( x )) < ∞ . This shows that M loc ( Y ) ⊆ M ( Y ) . (cid:3) As mentioned in the introduction, it was proved by Kravchenko that ( M ( Y ) , H ) is acomplete metric space (see [15]). It is worth noting that C. Akerlund-Bistrom provedthe special case that if Y = R n , then ( M ( Y ) , H ) is a complete metric space (see[1]). During a seminar talk that the author presented at the University of Colorado,A. Gorokhovsky posed the question: is ( M ( Y ) , H ) the metric space completion of ( M loc ( Y ) , H ) ? This question is answered in the affirmative below. Theorem 2.8. [8] [Davison] ( M ( Y ) , H ) is the completion of the metric space ( M loc ( Y ) , H ) .Proof. Suppose that µ is a Borel probability measure in M ( Y ) . We need to find asequence of measures { µ n } ∞ n =1 ⊆ M loc ( Y ) such that µ n → µ in the H metric. Weknow from earlier, namely Lemma 2.5, that there exists a sequence of compact subsets { K n } ∞ n =1 of Y such that lim n →∞ µ ( K n ) = 1 . We can choose this sequence of compactsets such that K ⊆ K ⊆ K ⊆ ... , because the union of finitely many compactsets is compact, and because measures are monotone. Next, choose some x ∈ K .Since each K n is compact, it is bounded so there exists a positive integer k n such that K n ⊆ B k n ( x ) , where B k n ( x ) = { x ∈ Y : d ( x, x ) ≤ k n } . For each n = 1 , ..., ∞ ,define a Borel measure µ n on Y by µ n (∆) = µ (∆ ∩ K n ) µ ( K n ) for all Borel subsets ∆ ⊆ Y .Furthermore for f ∈ Lip ( Y ) , note that Z Y f dµ n = 1 µ ( K n ) Z Y f K n dµ. We claim that each µ n has bounded support. Consider the open set Y \ K n . µ n ( Y \ K n ) = µ (( Y \ K n ) ∩ K n ) µ ( K n ) = 0 , and hence the support of µ n is contained within the bounded set K n . Also, observe that µ n ( Y ) = µ ( Y ∩ K n ) µ ( K n ) = µ ( K n ) µ ( K n ) = 1 , so that µ n is a Borel probability measure on Y . We have shown that for all n = 1 , , ... , µ n ∈ M loc ( Y ) . It remains to show that µ n → µ in the H metric. For this we use thealternate formulation for the H metric which is shown in [1]; namely H ( µ n , µ ) = sup f ∈ Lip ( x ) (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)Z Y f dµ n − Z Y f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) , where Lip ( x ) are the Lip ( Y ) functions which vanish at x . Let ǫ > . Choose some f ∈ Lip ( x ) . Then (cid:12)(cid:12)(cid:12)(cid:12)Z Y f dµ n − Z Y f dµ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) µ ( K n ) Z Y f K n dµ − Z Y f dµ (cid:12)(cid:12)(cid:12)(cid:12) = 1 µ ( K n ) (cid:12)(cid:12)(cid:12)(cid:12)Z Y f K n − µ ( K n ) f dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ ( K n ) (cid:12)(cid:12)(cid:12)(cid:12)Z K n ( f K n − µ ( K n ) f ) dµ (cid:12)(cid:12)(cid:12)(cid:12) + 1 µ ( K n ) (cid:12)(cid:12)(cid:12)(cid:12)Z Y \ K n ( f K n − µ ( K n ) f ) dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) − µ ( K n ) µ ( K n ) Z K n | f | dµ (cid:19) + Z Y \ K n | f | dµ POSITIVE OPERATOR-VALUED MEASURE FOR AN ITERATED FUNCTION SYSTEM 13 ≤ (cid:18) − µ ( K n ) µ ( K n ) Z K n d ( x, x ) dµ (cid:19) + Z Y \ K n d ( x, x ) dµ := I ( n ) , where the last inequality is because | f ( x ) | = | f ( x ) − f ( x ) | ≤ d ( x, x ) .Since µ ∈ M ( Y ) and d ( x, x ) ∈ Lip ( Y ) ⊆ Lip ( Y )0 ≤ Z Y d ( x, x ) dµ := L < ∞ . Because d ( x, x ) is a non-negative function, we note that for all n , ≤ R K n d ( x, x ) dµ ≤ L < ∞ and ≤ R Y \ K n d ( x, x ) dµ ≤ L < ∞ . Since lim n →∞ µ ( K n ) = µ ( Y ) = 1 , and K ⊆ K ⊆ ... , observe that Y \ K n d ( x, x ) decreases pointwise to µ -almost everywhere. By the dominated convergence theorem, lim n →∞ Z Y \ K n d ( x, x ) dµ = lim n →∞ Z Y Y \ K n d ( x, x ) dµ = Z Y lim n →∞ Y \ K n d ( x, x ) dµ = 0 . Also, lim n →∞ (cid:18) − µ ( K n ) µ ( K n ) (cid:19) = 0 . Choose an N such that for n ≥ N , (cid:18) − µ ( K n ) µ ( K n ) (cid:19) ≤ ǫ L , and Z Y Y \ K n d ( x, x ) dµ ≤ ǫ . For n ≥ N , I ( n ) ≤ ǫ L ( L ) + ǫ = ǫ . Since the choice of N is independent of thechoice of f ∈ Lip ( x ) , we can conclude that H ( µ n , µ ) ≤ I ( n ) ≤ ǫ . Therefore, wehave shown that M ( Y ) is the completion of the metric space M loc ( Y ) in the H metric. (cid:3) As we remarked previously, the Kantorovich metric is not well defined (finite) on allBorel probability measures on Y when Y is unbounded. In the discussion above, thisissue is overcome by restricting the Kantorovich metric to a sub-collection of measures.Another option is to consider a modified Kantorovich metric, M H , on Q ( Y ) defined asfollows: For µ, ν ∈ Q ( Y ) ,M H ( µ, ν ) = sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)Z Y f dµ − Z Y f dν (cid:12)(cid:12)(cid:12)(cid:12) : f ∈ Lip ( Y ) and || f || ∞ ≤ (cid:27) . (2.1)The condition || f || ∞ ≤ guarantees that M H will be finite on Q ( Y ) . Also, observethat we have the containments: M loc ( Y ) ⊆ M ( Y ) ⊆ Q ( Y ) . We can equip Q ( Y ) with the weak topology. Indeed, a net of measures { µ λ } λ ∈ Λ ⊆ Q ( Y ) converges weakly to a measure µ ∈ Q ( Y ) , if for all f ∈ C b ( Y ) , R Y f dµ λ → R Y f dµ , where C b ( Y ) is the set of all bounded continuous real-valued functions on Y .The following result can be found in Section 8.3 of [4]. Theorem 2.9. [4] [Section 8.3 in Bogachev] The weak topology on Q ( Y ) coincides withthe topology induced by the M H metric on Y . We now state a result recently proved by Kravchenko in [15] (which was crucial forshowing that ( M ( Y ) , H ) is complete). First, we put Lip b ( Y ) to be the collection ofreal-valued bounded Lipschitz functions on Y. Proposition 2.10. [15] [Kravchenko] Let { µ n } ∞ n =1 be a sequence of Borel measureson the complete and separable metric space Y such that µ n ( Y ) = K < ∞ for all n = 1 , , ... , and such that for all f ∈ Lip b ( Y ) , the sequence { R Y f dµ n } ∞ n =1 of realnumbers is Cauchy. Then there exists a Borel measure µ on Y such that µ ( Y ) = K ,and such that the sequence { µ n } ∞ n =1 converges in the weak topology to µ . We can combine the above two results to gain the following.
Theorem 2.11. ( Q ( Y ) , M H ) is a complete metric space.Proof. If { µ n } ∞ n =1 is a Cauchy sequence of measures in Q ( Y ) , one can show that for all f ∈ Lip b ( Y ) , the sequence { R Y f dµ n } ∞ n =1 of real numbers is Cauchy. Therefore, by theabove proposition there will exist a Borel probability measure µ ∈ Q ( Y ) such that µ n converges to µ in the weak topology, or equivalently, in the M H metric. (cid:3)
We now adapt Theorem 2.8 to this setting.
Theorem 2.12. [8] [Davison] The completion of the metric space ( M loc ( Y ) , M H ) is ( Q ( Y ) , M H ) .Proof. The proof of this theorem is similar to the earlier proof of Theorem 2.8. Supposethat µ ∈ Q ( Y ) . We need to find a sequence of measures { µ n } ∞ n =1 ⊆ M loc ( Y ) suchthat µ n → µ in the M H metric. Define, exactly as before, a sequence of measures { µ n } ∞ n =1 ⊆ M loc ( Y ) . In particular, µ n satisfies R f dµ n = µ ( K n ) R Y f K n dµ for all POSITIVE OPERATOR-VALUED MEASURE FOR AN ITERATED FUNCTION SYSTEM 15 f ∈ Lip ( Y ) . Choose f ∈ Lip ( X ) such that || f || ∞ ≤ . Then (cid:12)(cid:12)(cid:12)(cid:12)Z Y f dµ n − Z Y f dµ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z Y f dµ n − Z Y f dµ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) µ ( K n ) Z Y f K n dµ − Z Y f dµ (cid:12)(cid:12)(cid:12)(cid:12) = 1 µ ( K n ) (cid:12)(cid:12)(cid:12)(cid:12)Z Y f K n − µ ( K n ) f dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ ( K n ) (cid:12)(cid:12)(cid:12)(cid:12)Z K n ( f K n − µ ( K n ) f ) dµ (cid:12)(cid:12)(cid:12)(cid:12) + 1 µ ( K n ) (cid:12)(cid:12)(cid:12)(cid:12)Z Y \ K n ( f K n − µ ( K n ) f ) dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) − µ ( K n ) µ ( K n ) Z K n | f | dµ (cid:19) + Z Y \ K n | f | dµ ≤ (cid:18) − µ ( K n ) µ ( K n ) Z K n dµ (cid:19) + Z Y \ K n dµ ≤ (1 − µ ( K n )) + µ ( Y \ K n )= 2 µ ( Y \ K n ) . The last line of the above expression is independent of the choice of f and goes to zeroas n goes to infinity. Hence, µ n → µ in the M H metric. (cid:3)
Generalizing the Kantorovich Metric to Positive Operator-Valued Measures.
In this sub-section, we will generalize the Kantorovich metric to the space of posi-tive operator-valued measures on a Hilbert space, which are operator-valued measureswhich take values in the positive operators. The positive operators on a Hilbert spacecontain the projections.Let ( X, d ) be a compact metric space, and let H be an arbitrary Hilbert space. In ourapplication in the next sub-section, X will be the attractor set associated to an IFS, and H = L ( X, µ ) , where µ is the Hutchinson measure.We begin with some preliminary definitions and facts. Definition 2.13.
A positive operator L ∈ B ( H ) satisfies h Lh, h i ≥ for all h ∈ H . Definition 2.14.
A positive operator-valued measure with respect to the pair ( X, H ) isa map A : B ( X ) → B ( H ) such that: • A (∆) is a positive operator in B ( H ) for all ∆ ∈ B ( X ) ; • A ( ∅ ) = 0 and A ( X ) = id H (the identity operator on H ); • If { ∆ n } ∞ n =1 is a sequence of pairwise disjoint sets in B ( X ) , and if g, h ∈ H ,then * A ∞ [ n =1 ∆ n ! g, h + = ∞ X n =1 h A (∆ n ) g, h i . Remark 2.15.
A projection-valued measure with respect to the pair ( X, H ) is a positiveoperator-valued measure because projections are positive operators. Remark 2.16.
Let A be a positive operator-valued measure with respect to the pair ( X, H ) . The map [ g, h ] ∈ H × H 7→ A g,h ( · ) is sesquilinear. This follows from the factthat the inner product on H is sesquilinear. Our below discussion will rely on the following two standard theorems of functionalanalysis, which are stated with the amount of generality we will need.
Theorem 2.17. [5] [Theorem III.5.7 in Conway] Let X be a compact metric space, and T : C ( X ) → C be a bounded linear functional. There exists a unique complex-valuedregular Borel finite measure µ on X such that Z X f dµ = T ( f ) , for all f ∈ C ( X ) , and such that || µ || = || T || (where || µ || denotes the total variationnorm of µ ). Theorem 2.18. [5] [Theorem II.2.2 in Conway] Let u : H × H → C be a boundedsesquilinear form with bound M . There exists a unique operator A ∈ B ( H ) such that u ( g, h ) = h Ag, h i for all g, h ∈ H , and such that || A || ≤ M . Let S ( X ) be the collection of all positive operator-valued measures with respect tothe pair ( X, H ) . Consider the metric ρ on S ( X ) . That is, ρ ( A, B ) = sup f ∈ Lip ( X ) (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z f dA − Z f dB (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:27) , (2.2)where || · || denotes the operator norm in B ( H ) , and A and B are arbitrary members of S ( X ) . Theorem 2.19. [8] [Davison] The metric space ( S ( X ) , ρ ) is complete.Proof. Let { A n } ∞ n =1 ⊆ S ( X ) be a Cauchy sequence in the ρ metric. We have thefollowing claim. Claim 2.20.
Let f ∈ C ( X ) . The sequence of operators { A n ( f ) := R f dA n } ∞ n =1 isCauchy in the operator norm. Proof of claim: Note that the proof of this claim is identical to the proof of Claim2.12 in [6]. Let ǫ > . Let f = f + if , where f , f ∈ C R ( X ) , where C R ( X ) is thecollection of real-valued continuous functions on R . Since X is compact, by the densityof Lipschitz functions in continuous functions we can choose g , g ∈ Lip ( X ) such that || f − g || ∞ ≤ ǫ and || f − g || ∞ ≤ ǫ .There is a K > such that K g ∈ Lip ( X ) and K g ∈ Lip ( X ) . Since { A n } ∞ n =1 is a Cauchy sequence in the ρ metric, the sequence { A n ( K g ) } ∞ n =1 is Cauchy in theoperator norm, and hence the sequence { A n ( g ) } ∞ n =1 is Cauchy in the operator norm.Similarly, { A n ( g ) } ∞ n =1 is Cauchy in the operator norm. Therefore, choose N such thatfor n, m ≥ N , || A n ( g ) − A m ( g ) || ≤ ǫ and || A n ( g ) − A m ( g ) || ≤ ǫ . POSITIVE OPERATOR-VALUED MEASURE FOR AN ITERATED FUNCTION SYSTEM 17 If m, n ≥ N , || A n ( f ) − A m ( f ) || ≤ || A n ( f ) − A n ( g ) || + || A n ( g ) − A m ( g ) || + || A m ( g ) − A m ( f ) ||≤ || A n ( f − g ) || + ǫ || A m ( f − g ) ||≤ ǫ , where the third inequality is because || A n ( f − g ) || ≤ || f − g || ∞ and || A m ( f − g ) || ≤|| f − g || ∞ . Similarly, || A n ( f ) − A m ( f ) || ≤ ǫ . Then if n, m ≥ N , || A n ( f ) − A m ( f ) || = || A n ( f + if ) − A m ( f + if ) || = || ( A n ( f ) − A m ( f )) + i ( A n ( f ) + A m ( f )) ||≤ || A n ( f ) − A m ( f ) || + || A n ( f ) − A m ( f ) ||≤ ǫ. This proves the claim.In particular, the fact that { R f dA n } ∞ n =1 is Cauchy in the operator norm implies thefollowing: if g, h ∈ H and f ∈ C ( X ) , the sequence of complex numbers { R X f dA n g,h } ∞ n =1 is Cauchy. This is because we have the bound (cid:12)(cid:12)(cid:12)(cid:12)Z X f dA n g,h − Z X f dA m g,h (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:28)(cid:18)Z f dA n − Z f dA m (cid:19) g, h (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z f dA n − Z f dA m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) || g |||| h || , and the last term goes to zero as m and n approach infinity.For g, h ∈ H , define µ g,h : C ( X ) → C by f lim n →∞ R f dA n g,h , which is welldefined by the above discussion, and since C is complete. Observe that µ g,h is a boundedlinear functional. We will show that it is bounded, and leave the proof of linearity to thereader. Let f ∈ C ( X ) . Then | µ g,h ( f ) | = (cid:12)(cid:12)(cid:12)(cid:12) lim n →∞ Z X f dA n g,h (cid:12)(cid:12)(cid:12)(cid:12) = lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z X f dA n g,h (cid:12)(cid:12)(cid:12)(cid:12) . Now for all n (cid:12)(cid:12)(cid:12)(cid:12)Z X f dA n g,h (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z X | f | d | A n g,h | ≤ || f || ∞ || g |||| h || , and hence lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z X f dA n g,h (cid:12)(cid:12)(cid:12)(cid:12) ≤ || f || ∞ || g |||| h || . This shows that µ g,h is bounded by || g |||| h || . We can now invoke Theorem 2.17 toconclude that µ g,h is a measure.The map [ g, h ] ∈ H × H 7→ µ g,h is sesquilinear. Indeed, we will show that [ g, h ] µ g,h is linear in the first coordinate. The remaining properties of sesquilinearity areproved with a similar approach, and are left to the reader. Let g, h, k ∈ H , and let f ∈ C ( X ) . Then Z X f dµ g + h,k = lim n →∞ Z X f dA n g + h,k = lim n →∞ (cid:18)Z X f dA n g,k + Z X f dA n h,k (cid:19) = Z X f dµ g,k + Z X f dµ h,k , where the second equality is because of Remark 2.16.Consider a closed subset C ⊆ X , and choose a sequence of functions { f m } ∞ m =1 ⊆ C ( X ) such that f m ↓ C pointwise. For instance, we could let f m ( x ) = max { − md ( x, C ) , } . By the dominated convergence theorem, Z X C dµ g + h,k = lim m →∞ Z X f m dµ g + h,k = lim m →∞ (cid:18)Z X f m dµ g,k + Z X f m dµ h,k (cid:19) = Z X C dµ g,k + Z X C dµ h,k . Hence, for any closed C ⊆ Xµ g + h,k ( C ) = µ g,k ( C ) + µ h,k ( C ) . (2.3)By decomposing the measures µ g + h,k , µ g,k , µ h,k into their real and imaginary parts,we can show that (2.3) is equivalent to the following: Re µ g + h,k ( C ) = Re µ g,k ( C ) + Re µ h,k ( C ) , (2.4)and Im µ g + h,k ( C ) = Im µ g,k ( C ) + Im µ h,k ( C ) . (2.5)By further decomposing Re µ g + h,k , Re µ g,k , Re µ h,k into their positive and negativeparts (denoted Re µ + g + h,k and Re µ − g + h,k respectively), we can show, by rearranging terms,that (2.4) is equivalent to M ( C ) = M ( C ) , (2.6)where M = Re µ + g + h,k + Re µ − g,k + Re µ − h,k , and M = Re µ − g + h,k + Re µ + g,k + Re µ + h,k . Since M and M are positive Borel measures on a metric space, M and M areregular (see Remark 2.4). That is, we can conclude that M (∆) = M (∆) for anyBorel subset ∆ ∈ B ( X ) . By invoking the equivalence of (2.4) and (2.6), we have that(2.4) is true for all ∆ ∈ B ( X ) . A similar approach, will yield that (2.5) is true for all ∆ ∈ B ( X ) . Hence, (2.3) is true for all ∆ ∈ B ( X ) . This shows linearity in the firstcoordinate. As mentioned above, the following additional properties listed below areproved similarly: • Let g, h, k ∈ H . Then µ g,h + k = µ g,h + µ g,k . • Let α ∈ C and g, h ∈ H . Then µ αg,h = αµ g,h . • Let β ∈ C and g, h ∈ H . Then µ g,βh = βµ g,h . POSITIVE OPERATOR-VALUED MEASURE FOR AN ITERATED FUNCTION SYSTEM 19
Hence, the map [ g, h ] µ g,h is sesquilinear. We also note that µ g,h inherits the follow-ing three additional properties: • For h ∈ H , µ h,h is a positive Borel measure on X . • For g, h ∈ H , µ g,h has total variation less than or equal to || g |||| h || . • For g, h ∈ H , µ g,h = µ h,g .We will spend a short time justifying the second item in the above list. Supposethat ∆ , ..., ∆ n is a collection of disjoint subsets of B ( X ) . Then using a generalizedSchwarz inequality for positive sesquilinear forms we calculate that n X k =1 | µ g,h (∆ k ) | ≤ n X k =1 ( µ g,g (∆ k ) µ h,h (∆ k )) ≤ n X k =1 µ g,g (∆ k ) n X k =1 µ h,h (∆ k ) ! = ( µ g,g ( X ) µ h,h ( X )) = ( || g || || h || ) = || g |||| h || , which shows that the total variation of µ g,h is less than or equal to || g |||| h || .Let ∆ ∈ B ( X ) . The map [ g, h ] R X ∆ dµ g,h is a bounded sesquilinear form withbound . Indeed, | [ g, h ] | ≤ || ∆ || ∞ || g |||| h || = || g |||| h || . By Theorem 2.18, there exists a unique bounded operator, A (∆) ∈ B ( H ) , such thatfor all g, h ∈ H h A (∆) g, h i = Z X ∆ dµ g,h , with || A (∆) || ≤ . Accordingly, define A : B ( X ) → B ( H ) by ∆ A (∆) , and notethat for g, h ∈ H , A g,h = µ g,h . Claim 2.21. A is a positive operator-valued measure. Proof of claim:(1) Let ∆ ∈ B ( X ) , and h ∈ H . Then h A (∆) h, h i = Z X ∆ dµ h,h ≥ . Hence, A (∆) is a positive operator.(2) Let h ∈ H . Then h A ( X ) h, h i = Z X dµ h,h = µ h,h ( X ) = h h, h i , and h A ( ∅ ) h, h i = Z X ∅ dµ h,h = µ h,h ( ∅ ) = 0 . Hence, A ( X ) = id H and A ( ∅ ) = 0 . (3) If { ∆ n } ∞ n =1 are pairwise disjoint sets in B ( X ) , then for all g, h ∈ H , * A ∞ [ n =1 ∆ n ! g, h + = Z X S ∞ n =1 ∆ n dµ g,h = ∞ X n =1 µ g,h (∆ n ) = ∞ X n =1 Z X ∆ n dµ g,h = ∞ X n =1 h A (∆ n ) g, h i . This completes the proof of the claim.We will now show that A n → A in the ρ metric. Let ǫ > . Choose an N such thatfor n, m ≥ N , ρ ( A n , A m ) ≤ ǫ. Let f ∈ Lip ( X ) . If n ≥ N , and h ∈ H with || h || = 1 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:28)(cid:18)Z f dA n − Z f dA (cid:19) h, h (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z X f dA n h,h − Z X f dA h,h (cid:12)(cid:12)(cid:12)(cid:12) = lim m →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z X f dA n h,h − Z X f dA m h,h (cid:12)(cid:12)(cid:12)(cid:12) = lim m →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:28)(cid:18)Z f dA n − Z f dA m (cid:19) h, h (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) , where the second equality is because R f dA h,h = µ h,h ( f ) = lim m →∞ R f dA m h,h . For m ≥ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:28)(cid:18)Z f dA n − Z f dA m (cid:19) h, h (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z f dA n − Z f dA m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) || h || = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z f dA n − Z f dA m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ ( A n , A m ) ≤ ǫ. Hence lim m →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:28)(cid:18)Z f dA n − Z f dA m (cid:19) h, h (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ, and therefore (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z f dA n − Z f dA (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ. Since the choice of N is independent of f ∈ Lip ( X ) , ρ ( A n , A ) ≤ ǫ , which shows thatthe metric space ( S ( X ) , ρ ) is complete. (cid:3) A Fixed POVM Associated to an IFS.
Let X be the attractor set associated to anarbitrary IFS (with possibly essential overlap), and consider the Hilbert space L ( X, µ ) ,where µ is the Hutchinson measure. Recall that the maps F i : L ( X, µ ) → L ( X, µ ) given by φ √ N ( φ ◦ σ i ) for all i = 0 , ..., N − , satisfy the operator identity N − X i =0 F ∗ i F i = id H . This is Theorem 1.17 stated in the introduction. If we define S i = F ∗ i , we can rewritethe above operator identity as POSITIVE OPERATOR-VALUED MEASURE FOR AN ITERATED FUNCTION SYSTEM 21 N − X i =0 S i S ∗ i = id H . (2.7)As desired, the following theorem generalizes the map U in Theorem 1.13 to positiveoperator-valued measures, in the case of an arbitrary IFS. Theorem 2.22. [Davison] The map V : S ( X ) → S ( X ) given by B ( · ) N − X i =0 S i B ( σ − i ( · )) S ∗ i is a Lipschitz contraction in the ρ metric.Proof. The proof of this theorem follows essentially the same line of reasoning as in theproof of Theorem 1.13. The correct version of the proof of Theorem 1.13 can be foundin [7]. At present, the only item we need to show is that V is well defined as a map on S ( X ) . Indeed, let B ∈ S ( X ) , and ∆ ∈ B ( X ) . Then V ( B )(∆) is a positive operatoron H . That is, if h ∈ H , h B (∆) h, h i = * N − X i =0 S i B ( σ − i (∆)) S ∗ i ! h, h + = N − X i =0 (cid:10) S i B ( σ − i (∆)) S ∗ i h, h (cid:11) = N − X i =0 (cid:10) B ( σ − i (∆)) S ∗ i h, S ∗ i h (cid:11) ≥ , since B ( σ − i (∆)) is a positive operator for all ≤ i ≤ N − . We leave it to the readerto verify that B satisfies the remaining properties of a POVM. (cid:3) Corollary 2.23. [Davison] By the Contraction Mapping Theorem, there exists a uniquepositive operator-valued measure, A ∈ S ( X ) , such that A ( · ) = N − X i =0 S i A ( σ − i ( · )) S ∗ i . (2.8)2.4. Dilation of the Fixed POVM.
We begin with some preliminary facts and defini-tions. Let N ∈ N such that N ≥ , let S = { σ , ..., σ N − } be an IFS (with possiblyessential overlap) whose attractor set is X , and let F i : L ( X, µ ) → L ( X, µ ) be asdefined above. Recall that we previously defined Γ N = { , ..., N − } . If we let Ω = Q ∞ Γ N , it is well known that Ω is a compact metric space. The metric m on Ω isgiven by m ( α, β ) = 12 j where α, β ∈ Ω , and j ∈ N is the first entry at which α and β differ.We next define the shift maps on this compact metric space. Indeed, for ≤ i ≤ N − , let η i : Ω → Ω be given by η i (( α , α , ..., )) = ( i, α , α , , ..., ) , and define η : Ω → Ω given by η (( α , α , α , ... )) = ( α , α , ...., ) . • The maps η i are Lipschitz contractions on Ω in the m metric, and therefore, thefamily of maps T = { η , ..., η N − } constitutes an IFS on Ω . • The compact metric space Ω is itself the attractor set associated to the IFS T . • The Hutchinson measure P on Ω associated to the IFS T is called the Bernoullimeasure, and it satisfies P ( · ) = 1 N N − X i =0 P ( η − i ( · )) . • The map η is a left inverse for each η i , meaning that η ◦ η i = id Ω for each ≤ i ≤ N − . Since the IFS T is disjoint, for each ≤ i ≤ N − we can define T i : L (Ω , P ) → L (Ω , P ) by φ
7→ √ N ( φ ◦ η ) η i ( X ) , and its adjoint T ∗ i : L (Ω , P ) → L (Ω , P ) by φ √ N ( φ ◦ η i ) , such that the family of operators { T i } N − i =0 satisfies the Cuntz relations. Consequently,there exists a unique projection-valued measure, E , with respect to the pair (Ω , L (Ω , P )) such that E ( · ) = N − X i =0 T i E ( η − i ( · )) T ∗ i . (2.9)We also have from Corollary 2.23 that there exists a unique POVM, A, with respect tothe pair ( X, L ( X, µ )) such that A ( · ) = N − X i =0 S i A ( σ − i ( · ) S ∗ i , (2.10)where S i = F ∗ i . For each α ∈ Ω , define π ( α ) = ∩ ∞ n =1 σ α ◦ ... ◦ σ α n ( X ) , where α = ( α , α , ..., α n , ... ) .Since the maps σ i are all contractive, π ( α ) is a single point in X . Define the map π : Ω → X by α → π ( α ) as the coding map. Lemma 2.24. [10] [Jorgensen, Kornelson, Shuman] The coding map is continuous.Moreover, for all ≤ i ≤ N − , we have the relation π ◦ η i = σ i ◦ π. (2.11)We now are prepared to state a result from Jorgensen and his colleagues, which wewill use in our below discussion. POSITIVE OPERATOR-VALUED MEASURE FOR AN ITERATED FUNCTION SYSTEM 23
Theorem 2.25. [10] [Jorgensen, Kornelson, Shuman] (1)
The operator V : L ( X, µ ) → L (Ω , P ) given by V ( f ) = f ◦ π is isometric. (2) The following intertwining relations hold:
V F i = T ∗ i V, for all ≤ i ≤ N − . Consider now the projection-valued measure E ( π − ( · )) from the Borel subsets of X into the projections on L (Ω , P ) . We have the following result, which will showthat E ( π − ( · )) is indeed a dilation of the POVM A , in the sense of Naimark’s dilationtheorem. Theorem 2.26. [Davison] The projection-valued measure E ( π − ( · )) , and the positiveoperator-valued measure A are related as follows: V ∗ E ( π − ( · )) V = A ( · ) . Proof.
Define L = V ∗ E ( π − ( · )) V, and observe that L is a POVM with respect to thepair ( X, L ( X, µ )) . Our goal is to show that L = A . To this end, note that by theintertwining relations of Theorem 2.25, we have that N − X i =0 F ∗ i L ( σ − i ( · )) F i = N − X i =0 F ∗ i V ∗ E ( π − ( σ − i ( · ))) V F i = N − X i =0 V ∗ T i E ( π − ( σ − i ( · ))) T ∗ i V = N − X i =0 V ∗ T i E ( η − i ( π − ( · ))) T ∗ i V = V ∗ N − X i =0 T i E ( η − i ( π − ( · ))) T ∗ i ! V = V ∗ E ( π − ( · )) V = L ( · ) , where the third equality is by equation (2.11), and the fifth equality is by equation (2.9).Now A is the unique POVM that satisfies equation (2.10). By the above computation,we see that L also satisfies equation (2.10), and therefore, L = A . (cid:3)
3. A
CKNOWLEDGEMENTS :The author would like to thank Judith Packer (University of Colorado) for her carefulreview of this material, and her guidance on this research. R EFERENCES [1] Akerlund-Bistrom, C., “A generalization of Hutchinson distance and applications,” Random Compu-tational Dynamics, 5, No. 2-3, 159-176 (1997).[2] Barnsley M.,
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