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Institutional Members: CEPR, NBER and Università Bocconi WORKING PAPER SERIES ilbert A-Modules S. Cerreia Vioglio, F. Maccheroni, and M. Marinacci Working Paper n. 544 This Version: March, 2015 IGIER Università
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Institutional Members: CEPR, NBER and Università Bocconi WORKING PAPER SERIES ilbert A-Modules S. Cerreia Vioglio, F. Maccheroni, and M. Marinacci Working Paper n. 544 This Version: March, 2015 IGIER Università Bocconi, Via Guglielmo Röntgen 1, Milano Italy The opinions expressed in the working papers are those of the authors alone, and not those of the Institute, which takes non institutional policy position, nor those of CEPR, NBER or Università Bocconi. ilbert A-Modules S. Cerreia-Vioglio, F. Maccheroni, and M. Marinacci Department of Decision Sciences and IGIER, Università Bocconi March 2015 Abstract We consider real pre-ilbert modules on Archimedean f-algebras A with unit e. We provide conditions on A and such that a Riesz representation theorem for bounded/continuous A-linear operators holds. 1 Introduction Let A be an Archimedean f-algebra with (multiplicative) unit e. It is well known that Archimedean f-algebras are commutative. We next proceed by de ning the objects we study in this paper. De nition 1 An abelian group (; +) is an A-module if and only if an outer product : A! is well de ned with the following properties, for each a; b 2 A and for each x; y 2 : (1) a (x + y) = a x + a y; (2) (a + b) x = a x + b x; (3) a (b x) = (ab) x; (4) e x = x. An A-module is a pre-ilbert A-module if and only if an inner product h ; i :! A is well de ned with the following properties, for each a 2 A and for each x; y; z 2 : We thank Lars ansen for very useful discussions and comments. Simone Cerreia-Vioglio and Fabio Maccheroni gratefully acknowledge the nancial support of MIUR (PRIN grant 20103S5RN3_005). Massimo Marinacci gratefully acknowledges the nancial support of the AXA Research Fund. 1 (5) hx; xi 0, with equality if and only if x = 0; (6) hx; yi = hy; xi ; (7) hx + y; zi = hx; zi + hy; zi ; (8) ha x; yi = a hx; yi. For A = R conditions (1)-(4) de ne vector spaces, while (5)-(8) de ne pre-ilbert spaces. We will use Latin letters a; b; c to denote elements of A, Latin letters x; y; z to denote elements of, and Greek letters ; to denote elements of R. It is well known that 1 hx; yi 2 hx; xi hy; yi 8x; y 2 : We can thus conclude that each z 2 induces a map f :! A, via the formula f (x) = hx; zi 8x 2 ; with the following properties: - A-linearity f (a x + b y) = af (x) + bf (y) for all a; b 2 A and for all x; y 2 ; - Boundedness There exists c 2 A + such that f (x) 2 c hx; xi for all x 2. In light of this fact, we give the following de nition: De nition 2 Let A be an Archimedean f-algebra with unit e and a pre-ilbert A- module. We say that is self-dual if and only if for each f :! A which is A-linear and bounded there exists y 2 such that f (x) = hx; yi 8x 2 : The goal of this paper is to provide conditions on A and that will allow us to conclude that a pre-ilbert A-module is self-dual. Our initial motivation comes from Finance. There, ilbert modules are the extension of the notion of ilbert spaces that the analysis of conditional information requires, as rst shown by ansen and Richard [19]. In particular, self-duality is key to represent price operators through traded stochastic discount factors. Our results provide the general mathematical framework where conditional asset pricing can be performed. 1 See uijsmans and de Pagter [23, Theorem 3.4] and also Proposition 4 below. 2 Examples Consider a probability space (; F; P ) and assume that G is a sub- -algebra of F. Denote by L 0 (F) = L 0 (; F; P ) and L 1 (F) = L 1 (; F; P ), respectively, the space of F-measurable functions and the space of F-measurable and essentially bounded functions. Similarly, de ne L 0 (G) and L 1 (G). De ne also L 2;0 (; G; F; P ) = f 2 L 0 (F) : E f 2 jjg 2 L 0 (G) and L 2;1 (; G; F; P ) = f 2 L 0 (F) : E f 2 jjg 2 L 1 (G) : The inner product, in both cases, can be de ned by (f; g) 7! E (fgjjg). In Section 6, we show that L 2;0 (; G; F; P ) is a pre-ilbert L 0 (G)-module and L 2;1 (; G; F; P ) is a pre-ilbert L 1 (G)-module. Both spaces are of particular interest in Finance. The rst space is the one originally used in the seminal paper of ansen and Richard [19]. On the other hand, in Filipovic, Kupper, and Vogelpoth [14] (see also [30]), the second space has been shown to represent the family of all continuous and L 1 (G)-linear operators from L 2 (F) to L 2 (G). 2 In other words, L 2;1 (; G; F; P ) can be interpreted as the space of all conditional stochastic discount factors (state price densities). Related literature The literature on self-dual modules can be roughly divided in two main streams. The rst one introduced the notion of ilbert A-modules and considers complex C -algebras A. In particular, it considers algebras that admit a concrete representation as a space of continuous functions over a compact space. The second focuses on a particular algebra of functions, namely, L 0 (G) = L 0 (; G; P ). The notion of pre-ilbert A-modules was introduced by Kaplansky [25]. Kaplansky [25] considers modules over commutative (complex) AW -algebras A with unit and shows that a pre-ilbert A-module is self-dual if and only if satis es some extra algebraic property (De nition 9). Paschke [33] investigates the properties of self-dual modules de ned over complex B -algebras. Two other related papers are Frank [15] and [16] (see also [29], for a textbook exposition). In both papers, when A is assumed to be a W complex algebra, a pre-ilbert A-module is shown to be self-dual if and only if the unit ball (properly de ned) of is complete with respect to some linear topology. On the other hand, Guo, in [17] and [18], studies pre-ilbert L 0 (G)-modules and shows that they are self-dual if and only if is complete with respect to a particular metrizable topology. 3 2 L 2 (F) = L 2 (; F; P ) is the space of F-measurable and square integrable functions. 3 In this paper, we focus on ilbert modules. For the Banach case, we refer to Cerreia-Vioglio, Kupper, Maccheroni, Marinacci, and Vogelpoth [11] and the references therein. A pioneer work on the subject is aydon, Levy, and Raynaud [20]. 3 Our Contributions We provide (topological) conditions on A and that will allow us to conclude that a pre-ilbert A-module is self-dual. We start by considering A to be an algebra of L 1 type (Subsection 2.1). In this case, can be suitably topologized with several norm topologies. In particular, two norms stand out: k k and k k m (Subsection 3.1). When A is of L 1 type and is a pre-ilbert A-module, in Theorem 3, we show that the following conditions are equivalent: (i) is self-dual; (ii) B is weakly compact (where B is the unit ball induced by k k ); (iii) is weakly sequentially complete; (iv) B is complete with respect to k k m. Conditions (ii) and (iii) are novel conditions. On the other hand, a condition of completeness, similar to Condition (iv), has been found also in the complex case by Frank [15] (see the proof of [29, Theorem 3.5.1]). When A = R, it is easy to show that k k and k k m are equivalent (Proposition 9). Thus, in this case, properties (i)-(iv) are well known to be equivalent and we can conclude that our Theorem 3 is a generalization of the classical Riesz representation theorem for ilbert spaces. We then move to consider A to be an f-algebra of L 0 type (Subsection 5.3). In this case, can be topologized with an invariant metric d. When A is of L 0 type and is a pre-ilbert A-module, in Theorem 5, we show that the following conditions are equivalent: (i ) is self-dual; (ii ) is complete with respect to d. We are thus able to obtain Guo s self-duality result ([17] and [18]). The contribution to the literature of our Theorem 5 is to show the connection with the self-duality result for modules on algebras of L 1 type. In fact, we show that each pre-ilbert L 0 -module contains a dense pre-ilbert L 1 -module e. Thus, verifying the self-duality of amounts to verify the self-duality of e, which can then be extended to via a density argument. Outline of the paper Section 2 introduces the two kinds of algebras A we will consider in studying the self-duality problem. Subsection 2.1 deals with Arens algebras, that is, real Banach algebras which admit a concrete representation as a 4 space of continuous functions. Algebras of L 1 type will belong to this class (De nition 5). Instead, Subsection 2.2 deals with f-algebras of L 0 type (De nition 6). In Section 3, we show how a pre-ilbert A-module naturally turns out to be a vector space that can also be topologized in several di erent and useful ways. In Subsection 3.2, we study the corresponding topological duals. Section 4 deals with the study of the dual module, that is, the set of all A-linear and bounded operators from to A. The set turns out to be an A-module which can also be topologized and its study is key in dealing with the self-duality problem. From a topological point of view, the structure of di ers depending if A is of L 1 type or of L 0 type. In Subsection 4.1, we study the rst case. In Subsection 4.2, we study the second case. Finally, in Subsection 4.3, we show that can be identi ed with the norm dual of some Banach space when A is of L 1 type. Section 5 contains our results on self-duality. First, we discuss the case when A is of L 1 type. An important subcase is when A is nite dimensional, which we discuss right after. We conclude the section by discussing the case in which A is an f-algebra of L 0 type. In Section 6 we discuss ve examples of pre-ilbert A-modules that, given our results, turn out to be self-dual. We relegate to the Appendix the proofs of some ancillary facts. 2 Function algebras 2.1 Arens algebras Given a commutative real normed algebra A with multiplicative unit e, we denote by k k A the norm of A. We denote by A the norm dual of A and by h ; i the dual pairing of the algebra A, that is, ha; 'i = ' (a) for all a 2 A and ' 2 A. Unless otherwise speci ed, the norm dual A of A is endowed with the weak* topology and all of its subsets are endowed with the relative weak* topology. In the rst part of the paper, we will mostly consider commutative real Banach algebras A that admit a concrete representation. These real Banach algebras were rst studied by Arens [8] and Kelley and Vaught [26]. 4 De nition 3 A commutative real Banach algebra A with unit e such that kek A = 1 and kak 2 A a 2 + b 2 A 8a; b 2 A is called an Arens algebra. 4 For two more recent studies, see also Albiac and Kanton [2] and [3]. Recall that a Banach algebra is such that kabk A kak A kbk A for all a; b 2 A. 5 Given an Arens algebra A, de ne S = f' 2 A : k'k A = ' (e) = 1g K = f' 2 S : ' (ab) = ' (a) ' (b) 8a; b 2 Ag : The set K is compact and ausdor. Denote by C (K) the space of real valued continuous functions on K. We endow C (K) with the supnorm. It is well known that A admits a concrete representation, that is, the map T : A! C (K), de ned by T (a) (') = ha; 'i 8' 2 K; 8a 2 A; is an isometry and an algebra isomorphism (see [26], [2], and [3]). The cone generated by the squares of A induces a natural order relation on A itself: a b if and only if a b belongs to the norm closure of fc 2 : c 2 Ag. By using standard techniques, the above concrete representation of A implies that (A; ) is a Riesz space with strong order unit e and T is also a lattice isomorphism. In particular, K coincides with the set of all nonzero lattice homomorphisms and A is an Archimedean f-algebra with unit e. We also have that each element ' 2 K is positive. Finally, k k A is a lattice norm such that kak A = min f 0 : jaj eg and a 2 A = kak 2 A 8a 2 A: In light of these observations, note that for each a 0, there exists a unique b 0 such that b 2 = a. From now on, we will denote such an element by a 1 2 or p a. Note that if A admits a strictly positive linear functional ' : A! R, then we could also renorm A with the norm k k 1 : A! [0; 1), de ned by kak 1 = ' (jaj) for all a 2 A. It is immediate to see that kak 1 k 'k A kak A for all a 2 A, and so the k k A norm topology A is ner than the k k 1 norm topology 1 ; i.e., 1 A. We will denote the norm dual of A with respect to k k 1 by A 0. Finally, we have that A 0 A. If A admits a strictly positive linear functional ' : A! R, then we could also consider A endowed with the invariant metric d : AA! [0; 1), de ned by d (a; b) = ' (jb aj ^ e) for all a; b 2 A. It is immediate to see that d (a; b) = ' (jb aj ^ e) ' (jb aj) = kb ak 1 for all a; b 2 A, and so the k k 1 norm topology 1 is ner than the d metric topology d ; i.e., d 1. The existence of a strictly positive linear functional ' : A! R will play a key role in the rest of the paper. 5 We conclude the section by exploring the extent of this assumption and its relation with the existence of a measure m on K whose support separates the points of A. Before presenting the formal result, we provide a de nition: 5 Without loss of generality, ' can always be assumed to be such that k 'k A = 1. 6 De nition 4 Let A be an Arens algebra and m a nite measure on the Borel -algebra of K. The measure m separates points if and only if the support of m separates the points of A. Proposition 1 Let A be an Arens algebra and ' 2 A. The following statements are equivalent: (i) The functional ' is strictly positive and such that k 'k A = 1; (ii) There exists a (unique) probability measure m ' = m on the Borel -algebra of K such that suppm = K and Z ' (a) = ha; 'i dm (') 8a 2 A; (1) K (iii) There exists a probability measure m ' = m that separates points and satis es (1). Remark 1 From now on, when we will be dealing with an Arens algebra that admits a strictly positive linear functional ' on A such that k 'k A = 1, the measure m will be meant to be m '. Viceversa, if A admits a measure that separates points, then ' will be meant to be de ned as in (1). We conclude by de ning a particular class of Arens algebras which are isomorphic to some space L 1 (; G; P ) (see [1, Corollary 2.2]). De nition 5 Let A be an Arens algebra. We say that A is of L 1 type if and only if A is Dedekind complete and admits a strictly positive order continuous linear functional ' on A. 2.2 f-algebras Assume that A is an Archimedean f-algebra with unit e (see Aliprantis and Burkinshaw [6, De nition 2.53]). It is well known that e is a weak order unit. If A is Dedekind complete and a 1 e for some n 2 N, then there exists a unique b 2 A n + such that ab = e. We denote this element a 1. If a 0 is such that there exists a 1 and b 2 A, then we alternatively denote ba 1 by b=a. By [22, Theorem 3.9], if A is also Dedekind complete, for each a 0, there exists a unique b 0 such that b 2 = a. Also in this case, we will denote such an element by a 1 2 or p a. The principal ideal generated by e is the set A e = fa 2 A : 9 0 s.t. jaj eg : 7 It is immediate to see that A e is a subalgebra of A with unit e. If A is an Arens algebra, then A e = A. If there exists a linear and strictly positive functional ' : A e! R, then we can de ne d : A A! [0; 1) by d (a; b) = ' (jb aj ^ e) 8a; b 2 A: As in the case of an Arens algebra, d is an invariant metric. As already noted, an Arens algebra, in particular one of L 1 type, is an Archimedean f-algebra with unit. In this paper, other than algebras of L 1 type, we focus also on another particular class of f-algebras: De nition 6 Let A be an Archimedean f-algebra with unit e. We say that A is an f- algebra of L 0 type if and only if A e is an Arens algebra of L 1 type and A is Dedekind complete and d complete. By [6, Theorems 2.28 and 4.7], if A is an f-algebra of L 0 type, d is generated by the Riesz pseudonorm c 7! ' (jcj ^ e), then it is easy to prove that the topology generated by d is linear, locally solid, and Fatou. Moreover, it can be shown that A is universally complete and such that: 1. If a n # 0 and b 0, then a n b # 0 and a n b! d 0; 2. If b 0 and a d n! a, then ba d n! ba. 3 The vector space structure of In this section, we will rst show that a pre-ilbert A-module has a natural structure of vector space. Next, we will show that the A valued inner product h ; i shares most of the properties of standard real valued inner products. In particular, under mild assumptions on A, we will show that it also induces a real valued inner product on, thus making into a pre-ilbert space. We use the outer product to de ne a scalar product: e : R! (; x) 7! (e) x : We next show that e makes the abelian group into a real vector space. Proposition 2 Let A be an Archimedean f-algebra with unit e and an A-module. (; +; e ) is a real vector space. Proof. By assumption, is an abelian group. For each ; 2 R and each x; y 2, we have that 8 (1) e (x + y) = e (x + y) = (e) x + (e) y = e x + e y; (2) ( + ) e x = (( + ) e) x = (e + e) x = (e) x + (e) x = e x + e x; (3) e ( e x) = (e) ((e) x) = ((e) (e)) x = (() e) x = () e x; (4) 1 e x = (1e) x = e x = x. From now on, we will often write x in place of e x. Corollary 1 Let A be an Archimedean f-algebra with unit e and an A-module. If f :! A is an A-linear operator, then f is linear. If A is an Arens algebra, given a nite probability measure m on the Borel -algebra of K we can also de ne h ; i m :! R by Z hx; yi m = hhx; yi ; 'i dm (') 8x; y 2 : K For each ' 2 K, we also de ne and study the functionals h ; i ' :! R de ned by hx; yi ' = hhx; yi ; 'i 8x; y 2 : Note that h ; i ' = h ; i ' for all ' 2 K where ' is the Dirac measure at '. We next show that h ; i m is a symmetric bilinear form which is positive semide nite on. Proposition 3 Let A be an Arens algebra and a pre-ilbert A-module. The following statements are true: 1. h ; i m is a positive semide nite symmetric bilinear form; 2. hx; xi m = 0 implies x = 0, provided m separates points; 3. hx; yi 2 m hx; xi m hy; yi m for all x; y 2 ; 4. hx; a yi m = ha x; yi m for all a 2 A and for all x; y 2 ; 5. hx; yi m = ' (hx; yi ) for all x; y 2, provided m separates points. Proof. We here prove points 2. and 5. and leave the remaining easy ones to the reader. Assume that m separates points. De ne ' as in (1). By Proposition 1, it follows that Z ' (a) = ha; 'i dm (') 8a 2 A (2) is a strictly positive and linear functional. 9 K 5. By de nition of h ; i m and (2), observe that for each x; y 2 Z hx; yi m = hhx; yi ; 'i dm (') = ' (hx; yi ) : K 2. By assumption, hx; xi 0 for all x 2. By point 5., if hx; xi m = 0, then ' (hx; xi ) = 0. Since ' is strictly positive, we have that hx; xi = 0, proving that x = 0. Corollary 2 Let A be an Arens algebra and a pre-ilbert A-module. If A admits a measure m that separates points, then (; +; e ; h ; i m ) is a pre-ilbert space. Proposition 4 Let A be an Arens algebra and a pre-ilbert A-module. The following statements are true: 1. hx; yi 2 hx; xi hy; yi for all x; y 2 ; 2. jhx; yi j hx; xi 1 2 hy; yi 1 2 for all x; y 2 ; 3. hx; yi 2 A khx; xi k A khy; yi k A for all x; y 2 ; 4. khx; yi k A khx; xi k 1 2 A khy; yi k 1 2 A for all x; y 2. Proof. By Proposition 2 and Corollary 1 and since A is, in particular, an Archimedean f-algebra with unit, point 1. is an easy consequence of [23, Theorem 3.4]. Since A is an Arens algebra, each positive element admits a unique square root and point 2. also follows. Since A is an Arens algebra and k k A is also a lattice norm, we have that for each x; y 2 khx; yi k 2 A = hx; yi 2 A khx; xi hy; yi k A khx; xi k A khy; yi k A ; proving points 3. and 4. Remark 2 If A is a Dedekind complete Archimedean f-algebra with unit e, then points 1. and 2. are still true and their proofs remain the same. 3.1 Topological structure Since a pre-ilbert A-module is also a vector space, we can try to endow with a topology induced by either a norm or an invariant metric. In fact, given the structure of A and, we have several di erent competing norms and topologies. The next subsections are devoted to the study of these norms and metric and their relations. Before starting, note that if A is an Arens algebra or a Dedekind complete Archimedean 10 f-algebra with unit, then h ; i de nes a vector-valued norm, 6 N :! A +, via the formula N (x) = hx; xi 1 2
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