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Séminaire Dubreil. Algèbre et théorie des nombres L ÁSSLÓ F UCHS Riesz vector spaces and Riesz algebras Séminaire Dubreil. Algèbre et théorie des nombres, tome 19, no 2 (1965-1966), exp. no 2324, p. 1-9 <http://www.numdam.org/item?id=SD_1965-1966__19_2_A9_0> © Séminaire Dubreil. Algèbre et théorie des nombres (Secrétariat mathématique, Paris), 1965-1966, tous droits réservés. L’accès aux archives de la collection « Séminaire Dubreil. Algèbre et théorie des nombres » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Seminaire DUBREIL-PISOT (Algèbre et Theorie des Nombres) 19e annee, 1965/66, nO 23-24 23-01 20 et 23 mai 1966 RIESZ VECTOR SPACES AND RIESZ ALGEBRAS by Lássló FUCHS 1. Introduction. In 1940, F. RIESZ investigated the bounded linear functionals spaces S, and showed that they form a vector lattice whenever possess the following interpolation property. (A) Riesz interpolation property. -If f , in S such h S such that E Clearly, (choose e. that if S g. h = is f1 (1) a polynomial (2) v (0 , 1) , not f if and 1) in the unit interval only if f(x) ~ 0 for all x E where some put 0 f > is the for so- ordering). coefficients, defined everywhere in (0 , D, pointwise ordering. positive integer k, the k-times differentiable under the pointwise ordering. a real functions in The real analytic functions in (5) The real (6) The real-valued continuous functions in the unit interval (0, 1) , under the (0, the trigonometric functions compact Hausdorff (this ordering will be (7) w9 (0, 1) (this (4) on a there is interpolation property but there exist a number of important A lattice-ordered and have the Riesz interpolation The rational functions with real For and j = 1 , 2 , then g 2 ~, f~ are polynomials pointwise under the (3) 1 , 2 functions are g~ For instance : The real called i = is assumed to S lattice then it has the Riesz a function spaces which property. for gj real function on space) if called the f > 0 in has the (1)-(5) ordering (or f(x) inequalities). ipeaning ordering by strict The function spaces mentioned in pointwise ordering. under the > 0 is pointwise. more generally, for all x ordering by strict ine- qualities. (8) perty The cartesian product of function spaces with the Riesz has again the same property. interpolation p ro- As further example of a property, lation reals) the over ing, iy x + we partially x + x > means > iy These examples whow that it 0 (considered night 0 , as a interpo- vector space in another order- or reals). y are be of y > and 0 x > means (x, 0 ordered vector space with the Riesz complex numbers may mention the where 0 > a interest to investigate some systemati- cally the partially ordered vector spaces with the Riesz interpolation property. Our present aim is to give a brief survey of the results 9 some of the theorems can be formulated more generally for partially ordered groups. For the proofs, we refer to 2. Bas i c By our definitions. vector space a is said to be such that ~ papers V, V in b shall implies the field over mean one ordered if it is partially a we partially c $ b a + + R ordered under for all c V of real numbers. a binary relation ~a ~ ~b and V c E positive real X . The set P of all a E V , satisfying a ~ 0 , is called the positivity domain of Vy and the elements of P are called positive. b is equivalent to P completely determines the partial order of V since a for all b-» aE P . such that a ~ and c ordered. If a, b E V b ~ is (union) be u, is directed if and V a ~ b V , either lattice under its a b,y a v b of the elements a, b lattice. We shall write 1. c. V , there exists c some E V is the only if every element positive elements. difference of two If for all a, b E if, to each pair is called directed V a A or b fi V , then in a partial order, V even if vector a (intersection) might exist which totally then it is called for the g. 1. b. E is V and the V is not lattice-ordered. A directed partially ordered vector space with the Riesz inter- polation property will be called polation property (A) is a equivalent Riesz vector space. Note that the Riesz interto either (B) Decomposition property. - If bi ~ 0 , then there exist al ’ a2 E V (c) b. J such of Refinement property. - If a V , then there exist elements that 0 a b + b such that a = + a~ = c..~0 ~-J b following conditions : of the one + b in V for a al + a2 for (i bi E V , where positive elements =1 ~ 2 ; j= 1 , a. , 2) A a , b E dered (i. W, it contains e. necessarily has a V E x partially ordered vector space and a ~ x ~ b imply x E W . If of W subspace o-ideal is defined A" convex. as o-ideals of THEOREM 1. - The Riesz vector space a V/W W subspace becomes THEOREM 2. I If V/I and A vector space is V homomorphism that isomorphism o-isomorphism. an distributive sub- a positive vector coset a I is a + (w a + w its mod V E is defined W W) . V , then o-ideal of an convex are Riesz vector spaces. A’ sector space be a Riesz vector space and a form ordered vector space if partially o-simple. ordered vector space partially a whenever it contains positive both a of V V . lattice in the lattice of all subspaces of The factor space V or- then it is subspace. If convex if convex trivially ), b a o-ideals other than the trivial ones, it is called no to be directed a is subspace a such that a,y b two elements no is said to be V positivity preserves that is order is called an o-homomorphism. preserving in both directions is said to 3. Algebraic theory. In order to first type In investigate the have to introduce the we a ordered vector space partially a A b whenever other go 1. b. or simplest and,at the same algebraic point of view, time,the most important of Riesz vector spaces : the antilattices. g. 1. b. no Riesz vector spaces from the a n b existence of lattice is a exist~ io is called b , = v a a ~ b b implies b a Ab antilattice. Because of b a v = a Riesz vector space in which a, b have a . A Riesz vector space the existence of e. an or V,y the elements in = b or in a v an A always A, implies (- in which a n b = a b) , the thus an anti- g. 1. b. and 1. u, b. b = - a A - antilattice, only the trivial a exist. It is easy to conclude that if have a g. Examples § 1. Also, an 1. b., say for antilattices all totally a finite number of elements of then ... A a == b , are abundant ;9 see ordered vector spaces antilattice which is lattice-ordered is The following is the structure theorem on a. = b our are a for examples an antilattice some ’ o (~,~~,~~(2~, (4)-(7) in antilattices. On the other hand, totally ordered vector space. Riesz vector spaces. It shows that the antilattices play a similar role in the theory of Riesz vector spaces as the to- in the tally ordered vector spaces theory of vector lattices. (a E I) ces A sian product fl and such that A ~il notation n Here the if, considered as § 0 that means and onto an (..., element , of antilatti- subspace of the carte- a u, b. too. ...) aa , for every 0 only if, family a preserves g. 1. be and 1. cp - V of o-isomorphism y an there exists V THEOREM 3. - To every Riesz vector space a E E ~! is to be A Cc I . More information about the structure of antilattices may be obtained from the following two results , THEOREM 4. - If Thus antilattice is an THEOREM 5. - Assume that extension of embedded for If is A in o-isomorphically an H a a means that ... , C ...~ of A is product ~ B~ cartesian b03B2 , n C > 0 is defined 0 . Then of if, totally and A can be ordered vector only if, bo b03B2 > 0 all 03B2 . A THEOREM is as in theorem 5 and is in addition to the real 6* - If is o-isomorphic to the ordering is numbers, and is an A a we o-smmple, then all the B are then A get : o-simple antilattice such that vector space of real-ralued functions (1 C = 0 9 on some set 0 where given by strict inequalities. Note that by choosing suitable 4. trivially ordered antilattice such that the intersection j) c B o-isomorphic ding maximal trivially ordered vector space by a of all maximal trivially ordered subspaces Here a totally ordered. is an is ordered vector spaceo totally spaces C antilattice and an A/C A, then of subspace is A to elements of A isomorphisms B- ~ R , the functions correspon- will be bounded. Topological questions. The continuous functions with the uniform topology. on a compact Hausdorff space Q are usually furnished This topology the one may be regarded as obtained by ordering the continuous functions by strict inequalities and then taking as a subbase of neighbourhoods at 0 the open intervals (- f , f) with functions f > 0 . In accordance with this remark, follows. we introduce the open-interval topology as V Let topology as 0 . This rise to gives We open-interval topology. the as take the open inter- V , in neighbourhoods at subbase of open which will be referred to V on a 0 u > Riesz vector space. For all a (- u , u) vals a be have : open-interval topology THEOREM 7. - For the of a the fol- V Riesz vector space lowing hold : (i) It is Hausdorff ordered vially (ii) (iii) V is Hausdorff, then V is C subspaces If it is of is non-discrete in the V the intersection if, and only if, of all maximal tri- n C 0 ; topological group in this topology ; only if, it is and open-interval topology if, antilattice ; an (iv) If is not non-discrete then every dense V subspace V of is an antilat- tice. Notice that the hypothesis of theorem 5 is thus equivalent to simple topolo- a gical condition. Let be A gy. We know antilattice that is an by (ii) that A Hausdorff space in the a is then a topological topological completion A* is again a topological group ..By the structure of its gical groups, space over Cauchy then If Air R . If is constructed which of means is, in particular, the antilattice inequalities, (0, 1) , continuous functions in This is, however, in § 1. not true in approximation antilattice, general, a , b E A and if x E A satisfies u > 0 in x if the A A* A as a , b then following for A c (0, 1) by condition is satisfied + u . such that It is easy to in order that A* be c see a case. (1) example our approximation property, A under the vector space of all vector lattice in this a , c e a ordered vector space. is shown for instance has the x a just call we positive elements, polynomials in is is A , there exists proximation property is necessary THEOREM 8. - Let A* then thus We shall say that the antilattice of all of vector a if Cauchy nets, and topolo- on is, in addition, positive whenever it contains a net consisting readily checks that A* is likewise a partially the ordering by strict an by may ask for so we standard results net one A A* group, and open-interval topolo- o or is A Given a , b , and that the ap- vector lattice. be an antilattice which is Hausdorff in its open interval " topology. Then : (i) If 0), then (ii) is metrizable A it has e. is countable system of a A* topological completion its A* (i. is The canonical map of is not necessarily A A~’ into is of o-isomorphism. an of lations, but the image A~~ in A Riesz vector space ; a if, vector lattice if,and only a has the A course contains more approximation property. continuous a A embedding The neighbourhoods at ~ A~’ isomorphism, but it preserves order positive elements in re- general, namely, the positivity domain of the image of A in A~~ is the least closed set P containing the image of the positivity domain P of A . For instance, if A is the vector space of polynomials in (0 , 1) under the ordering by strict inequalities, then in A* a polynomial f will be positive if it can be approximated by polynomials positive in A, i. e. in the image of A in A~~ the ordering will be pointwise. Let be L a L vector lattice. We say that can be the antilat- approximated by A, if : tice (a) A is a A* completion The L ( A partially ordered vector positivity domain P by its clusure P ~. by replacing its A space obtained from (b) subspace of vector of is A a denotes the vector lattice containing L as vector a sublattice. : THEOREM mating L approximating there exists It is 5. an we whi ch our groups, but also to say, we also vector lattice. There exists L , and every dense subspace Conversely, if are B has R x A this is its a proper dense topological Ay which vector spaces in the A that the antilattice subspace then A = L . B. THEOREM 10. - An antilattice it is are not only topological is mapping is continuous topology if,and only if, an open-interval topology. That ~a (where R carries its open-interval topology). It is easy to prove into is approxi- to pological vector spaces. (~. y a) 2014~ of A of is dense and which satisfies B attention to antilattices assume B A antilattice y A cases an an antilattice approximating L , is in which A in which problem open turn = a antilattice an Antilattices Next A such that L be L 9. - Let A is a usual : topological vector o-simple. topology ; actually, space in its open-interval For the sake of brevity, shall call we antilattice an A that is a topological vector space topological. Let us fix u) U = (- u , for an arbitrary (which element some 0 the unit ball of a E A , we ~...~ , Recall that topological antilattice This is justified in view in functional), topological antilattice abstract of the fact that, if set and the dual space of an A,y and call a A . is the well-known Ninkowski THEOREM 11. - A norm u > A is then we obtain : normed vector space under the A is an abstract Lebesgue Lebesgue space is a a space. normed vector lattice which is a real Banach space such that The following result gives a fairly good characterisation of topological antilat- tices. THEOREM 12. - For every topological antilattice A,y there exists a topological ~-isomorphism ~ of A,y with a subspace of all real-valued continuous functions on a compact Hausdorff space 0 , where the functions are ordered by strict inequalities. Moreover, 03A6 Here Q can so is the space of all maximal nished with the Gelfand The be chosen following theorem THEOREM 13. - property, then If it is A as to be an trivially isometry ordered as well. wubspaces of A, fur- topology. of Stone-Weierstrass is type may be mentioned. topological antilattice satisfying the approximation o-isomorphic and isometric to a dense subspace of the space a of all real continuous functions compact Hausdorff space Q , ordered by strict inequalities, and the topological completion A* of A is the vector lattice 6. C(Q) on a of all continuous functions on 03A9 under the pointwise ordering. Riesz algebras. An algebra partially over the real numbers is called ordered vector space with the partially ordered algebra if is a property that its possitivity domain is a closed under algebra tion that The last multiplication. inequalities may be is property from the left and from the multiplied to the condi- equivalent right by posi- tive elements. ordered partially A tor space is Riesz vector space. Our a A partially algebra. X then Here identity some have meant we ideal. As a o-ideal of an an o-ideal of an x E Riesz are is said to be x A algebra z X or = 0 . X a obtains that one 1 underlying as an antilattice- an x A and convex vec- algebras. o-ideal in an embedding such that if an satisf ies corollary to theorem 14, o-isomorphic with 7. by (1)-(9) in § be embedded then either 0 z > E X , 3 There is e . x , y > 0 with x, y e Y sope x E with Y can if its algebra antilattice, an X algebra Riesz a examples algebra, that is ordered THEOREM 14. - Every Riesz algebra is called X algebra y = if, directed Riesz a 0 e. for g., algebra antilattice-algebra is antilattice-algebra with identity. an Antilattice-al ebras. If introduce the we multiplication need not be continuous in this criterion for the Call the u > each continuity partially 0 in X , 9 ordered of We are m-hounded if every X v > some ring going to give 0 in X a E a an X satisfies : to such that an Note that if the additive group of is topology. then algebras, ring multiplication. algebra there is in antilattice antilattice-algebra X that is a Hausdorff topology multiplication is continuous if, and only if, THEOREM 15. - In interval open-interval topology antilattice-algebra is space in the open- X is m-bounded. o-simple, then it m-bounded. For Let not m-bounded us turn antilattice-algebras finally to in the o-simple. is a which 8 holds. topological algebras, i. e. open-interval topology, but at the same time toare spaces. THEOREM 16. - An antilattice topology analogue of theorem antilattice-algebras only topological rings pological vector the algebra topological algebra if, X and which is Hausdorff in its open-interval only if its underlying vertor space is 23-09 The main result is the following THEOREM 17. - Let is X be an Hausdorff space in the a then it is space, representation theorem. antilattice open-interval topology. If o-isomorphic to an Q where the functions In this case, X is a normed an isometry as Note that but our If X we is and o-simple are algebra, ordered and the assume as on a a it vector compact by strict inequalities. o-isomorphism may be assumed to well. have not a priori supposed that X was associative or commutative, result indicates that it must be both associative and commutative. is as in theorem Banach algebra which is a then its topological completion is Riesz algebra too. If, in addition, X 17, proximation property, then its completion on Q are ordered where the functions If X X e algebra of continuous functions Hausdorff space be algebra with identity is the real it is easy to see polynomials in that Q will be is the a commutative enjoys the ap- algebra of all continuous functions pointwise. (0 , 1) , ordered homeomorphic to by strict inequalities, then (0 , 1) . p REFERENCES [1] [2] FUCHS [3] FUCHS (Lássló). - On partially ordered vector spaces with the Riesz interpolation property, Publ. Math. Debrecen, t. 12, 1965, p. 335-343. FUCHS (Lássló). - Riesz rings, Math. Annalen, t. 166, 1966, p. 24-33. [4] FUCHS (Lássló). (Lássló). - Riesz groups, Ann. Sc. Norm. Sup. Pisa, t. 19, 1965, p. 1-34. Approximation of lattice-ordered groups, Ann. Univ. Scient. Budapest, t. 8, 1965, p. 187-203.