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MAXIMAL D-IDEALS in a RIESZ SPACE

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MAXIMAL D-IDEALS in a RIESZ SPACE Can. J. Math., Vol. XXXV, No. 6, 1983, pp. 1010-1029 MAXIMAL d-IDEALS IN A RIESZ SPACE CHARLES B. HUIJSMANS AND BEN DE PAGTER 1. Introduction. We recall that the ideal / in an Archimedean Riesz space L is called a d-ideal whenever it follows from/ e / that {/} c /. Several authors (see [4], [5], [6], [12], [13], [15] and [18] ) have considered the class of all d-ideals in L, but the set^ of all maximal d-ideals in L has not been studied in detail in the literature. In [12] and [13] the present authors paid some attention to certain aspects of the theory of maximal d-ideals, however neglecting the fact that^, equipped with its hull-kernel topology, is a structure space of the underlying Riesz space L. The main purpose of the present paper is to investigate the topological properties of^ and to compare^ to other structure spaces of L, such as the space Jt of minimal prime ideals and the space Qe of all ^-maximal ideals in L (where e > 0 is a weak order unit). This study of^ makes it for instance possible to place some recent results of F. K. Dashiell, A. W. Hager and M. Henriksen [7], concerning the order completion of C(X) in the general setting of Riesz spaces. After establishing some preliminary results in Section 2, it is proved in Section 3 that for any uniformly complete Riesz space L with weak order unit the spaced is a compact quasi-F-space. This result makes it possible to use^ for the description of the order completion of L, which is done in Section 4. Furthermore we consider what conditions have to be imposed on L in order that^ is an F-space, basically disconnected or extremally disconnected respectively. In Section 5 it is investigated what effect e certain homeomorphisms between^, Jt9 and £ have on L. Finally in Section 6, we interpret some of the results in the case that L is the Riesz space C(X). It turns out that ^ is precisely the minimal quasi-F-cover K(X) of X, as introduced in [7]. We can apply therefore the results of Sections 4 and 5 to the space K(X). Moreover, it is shown that the representation of the order completion of L which we have obtained in Section 4 is in the case L = C(X) precisely the realization of the order completion as obtained in [7]. Received March 19, 1982 and in revised form January 21, 1983. Work by the second author on this paper was supported by a NATO-Science Fellowship from the Netherlands Organization for the Advancement of Pure Research (Z.W.O.). 1010 Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 07:38:29, subject to the Cambridge Core terms of use. MAXIMAL D-IDEALS 1011 2. Sème preliminary results. All Riesz spaces considered in this paper are supposed to be Archimedean. For the terminology and unproved properties of Riesz spaces we refer to [16]. Throughout this paper relatively uniformly complete (closed) is called uniformly complete (closed). Let L be an Archimedean Riesz space. Definition 2.1 (see [12], Section 4). The ideal / in L is called a d-ideal if it follows from/ e / that {f}dd c /. It is immediate from the definition that the ideal / is a d-ideal if and only if/G/,gGL and {f}dd = {g}dd imply gel. All er-ideals in L (and hence all bands) are d-ideals. Moreover, every minimal prime ideal is a d-ideal. For any non-empty subset D of L, the d-ideal E(D) generated by D is the intersection of all d-ideals containing D. The proof of the next lemma is routine. LEMMA 2.2. E(D) = {/ e L3d{, dn e D such that /e {|ddV...V|d,?/|}^}. In particular, every principal d-ideal is of the form {/} . The proper d-ideal J in L is called a maximal d-ideal if L is the only d-ideal in which J is properly contained. We denote the set of all maximal d-ideals by^. Every maximal d-ideal is a (proper) prime ideal (see [12], Section 6). If L has in addition a weak order unit e > 0, then, by a Zorn argument, every proper d-ideal is contained in some maximal d-ideal. We équipa with its hull-kernel topology by choosing as a base for the open sets all subsets of the form {J}u = {J G Jd\u & J} with 0 ^ u G L ( [16], Section 36). The closed sets in this topology are precisely the hulls of subsets of L, i.e., sets of the form h(D) = [J e fd\D c /}. Clearly, we can replace D in this formula by the d-ideal E(D) generated by D. From the next proposition it follows thâtjy is a structure space of L. PROPOSITION 2.3. Let L be an Archimedean Riesz space with weak order unit e > 0. (i)c^ ^ a compact Hausdorff space. (ii) n {JJ e^} = {0}. Proof, (i) This is similar to the proof of [16], Theorem 36.4. (ii) For any 0 < u e L there exists n G N such that (nu — e)+ > 0. Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 07:38:29, subject to the Cambridge Core terms of use. 1012 C.B. HUIJSMANS AND B. DE PAGTER Since (nu - e)+ £ I = {(nu - e)~}dd, the ideal / is, as a proper d-ideal, contained in some maximal d-ideal J. We claim that u & J. Indeed, u ^ J would imply that (nu — e)+ e J. Combined with (nu — e)~ <E J this would result in e e /, a contradiction. For future purposes we collect finally some simple properties of maximal d-ideals. The proof is omitted. PROPOSITION 2.4. In any Archimedean Riesz space L with a weak order unit e > 0 the following statements hold. dd (i) {u} = n {J <=fd:u G J} for allO ^ u e L. (ii) {J}u c {^}v if and only if {u}dd c {v}^(0 ^ M, v G L). In [13], Remark 7.6 (iii) it is observed that in any Archimedean Riesz space with a weak order unit every a-ideal is an intersection of maximal d-ideals, so Proposition 2.4 (i) is a special case of the former result. 3. The quasi-F-space ^. The main purpose of this section is to prove that for any uniformly complete Archimedean Riesz space L with a weak order unit the spaced is a quasi-F-space. We recall that a completely regular Hausdorff space X is said to be a quasi-F-space whenever every bounded continuous function on a dense cozero-set has a continuous extension to the whole of X, i.e., every dense cozero-set in X is Q-embedded. Quasi-F-spaces were orginally defined in [7], Definition 3.6. The following lemma gives a simple characterization of quasi-F-spaces. LEMMA 3.1 ( [18], Lemma 11.9). Let X be a completely regular Hausdorff space. The following are equivalent. (i) X is a quasi-F-space. (ii) For any two zero-sets Z\ and Z2 in X with disjoint interiors, the sets int Z\ and int Z^ are completely separated by a continuous function. The next theorem gives more information about the topological structure of^. THEOREM 3.2. Let L be an Archimedean uniformly complete Riesz space with weak order unit e > 0. Then the space J^ is a quasi-F-space. Proof. We use the criterion of Lemma 3.1, so suppose that Z\ and Z2 are zero-sets in^ such that Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 07:38:29, subject to the Cambridge Core terms of use. MAXIMAL D-IDEALS 1013 (int Zj) n (int Z2) = 0. Following [22], Lemma 6.2, there exist 0 ^ i/,7 G L, 0 ^ vw e L (n = 1, 2, . ) such that OO CO Zj = n {/}„, Z. = n L/}v. Since Z! and Z2 are closed, Zj = h(I\) and Z2 = h(I2) for appropriate d-ideals ^ and I2 in L. From h(I\) c {/}Mn it follows that £(/j, ww) = L (n = 1, 2, . ). There exists, therefore, an element 0 ^ wn <E J^ such /J that e <E {w„ V w,?}' (see Lemma 2.2), i.e., un V wn is a weak order unit. Without loss of generality we may assume 0 < wn = e (n = \, 2, . .). Analogously, there exist elements zn E 72, 0 = zn ~ e such that v„ V zn are weak order units in L («=1,2,...). Now define CO CO W P = 2 2- HV <? = 2 2-»z„ /I = 1 /7 = 1 (^-uniformly convergent series). Clearly p e I\ and g G I2 . We assert that/? V g is a weak order unit in L. To this end, take 0 ^ y e L such that j A (/? V </) = 0. It follows that y A w„ = y A zn = 0 (w = 1,2,...). dd Since un V H>„ is a weak order unit we have y e. {W„ V wn} - Hence, y A {*(«BVH'H)} TA >% so J> A (k Un) TA- V tW which implies>> e {w,7} . By Proposition 2.4 (ii), {/}v c {J}Ufi. Similarly, it is shown that {J}y c {</}v,;. It follows easily that {J}v c int (Z0 n int (Z2), so {/}r = 0, i.e., y = 0 (Proposition 2.3 (ii) ). Therefore/? V g is a weak order unit.
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