Printer Friendly

A Stone Approximation Theorem for TM-partition spaces.

ABSTRACT -- By a generalized Kakutani's representation theorem, a similar result of Schaefer's Stone Approximation Theorem for the so-called TM-partition space is obtained. Key words: M-partition space; TM-partition space; Stone Approximation Theorem; inductive topology.

**********

For terminologies not defined explicitly in this paper, the reader is referred to Luxemburg and Zaann (1971), Schaefer (1974), and Wu (1986).

A collection of positive elements P={[e.sub.i]: i [epsilon] [OMEGA]} in a vector lattice is said to be saturated if for any two elements [e.sub.i], [e.sub.j] in P, [e.sub.i] V [e.sub.j] is also in P (Wu, 1986:4). For any collection D of positive elements in V, the collection of all finite suprema of elements in D is referred to as the saturation of D. If a vector lattice V contains a saturated collection of positive elements P={[e.sub.i]: i [epsilon] [OMEGA]} satisfying the conditions, then (V,P) is termed an M-partition lattice with an M-partition P: 1) if |v| [LAMBDA] [e.sub.i] = 0 for all [e.sub.i] in P, then |v| = 0; 2) for any [e.sub.i] in P and v in V, there exist [e.sub.j] in P and r > 0 such that |v| [LAMBDA]n[e.sub.i] [less than or equal to] r[e.sub.j] for all n in N; and 3) for any [e.sub.i] in P and [epsilon] > 0, there exists [e.sub.j] in P such that for all [e.sub.k] in P, [e.sub.k] [LAMBDA] n[e.sub.i] [less than or equal to] (1 + [epsilon])[e.sub.j] for all n in N. If P is finite, P is termed a finite M-partition. For each [e.sub.i] in P, [p.sub.i](v) = inf{r > 0: there exists [e.sub.j] in P such that |v| [LAMBDA] n[e.sub.i] [less than or equal to] r[e.sub.j] for all n in N} is a seminorm on V. The topology [T.sub.P] inducted by {[p.sub.i]: i [epsilon] [OMEGA]} is referred to as the M-partition topology induced by P. A topological vector lattice (V,[tau]) is termed an M-partition space if (V, P) is an M-partition lattice for some M-partition P such that [tau] is equivalent to the M-partition topology [T.sub.P] induced by P. For convenience, Theorem 7 in Wu (1986:12) is copied as Theorem 1 herein.

Theorem 1. -- A topological vector lattice (E,[tau]) is homeomorphic and lattice isomorphic to a dense subspace V of (C(Y),k) for some locally compact Hausdorff space Y such that V [intersection] [C.sub.[infinity]](Y) is dense in ([C.sub.o](Y), ||*||) if, and only if, (E,[tau]) is an M-partition space. The space Y is unique up to homeomorphism.

Schaefer (1974:169) defined an orthogonal set S of nonzero positive elements in a Banach lattice E to be a topological orthogonal system (t.o.s.) of E if the ideal generated by S is dense in E. Schaefer (1974:177) obtained the so-called Stone Approximation Theorem.

Stone Approximation Theorem. -- Let E be a Banach lattice, and let H be a vector sublattice in which closure contains a t.o.s. S of E. If H separates the points of [V.sub.S](E), then H is dense in E, where [V.sub.S](E) is the strong representation space for E.

In this paper, by Theorem 1, a similar result of this Stone Approximation Theorem will be obtained for the so-called TM-partition spaces.

TM-PARTITION SPACES AND TM-PARITIONS

Let (V,T) be a topological vector lattice containing a saturated collection P={[e.sub.i]: i [epsilon] [OMEGA]} of positive elements generating a dense Archimedean ideal [I.sub.P] in (V,T) and satisfying the following condition(*): for every [e.sub.i] in P and every [epsilon] > 0, there exists an [e.sub.j] in P such that for all [e.sub.k] in P, [e.sub.k] [LAMBDA] n[e.sub.i] [less than or equal to] (1 + [epsilon])[e.sub.j] for all n in N. Then (V,T) will be called a TM-partition space with TM-partition P. It is obvious from the first section that an Archimedean topological orthogonal system of a Banach lattice E is a TM-partition of E.

Proposition 2

The ideal [I.sub.P] in V generated by a collection P of positive elements in V is an M-partition lattice with the M-partition P if, and only if, P satisfies the condition (*).

Proof. -- It is enough to show that ([I.sub.P],P) satisfies conditions 1) and 2) for an M-partition lattice. Let v be an element in [I.sub.P], pick [k.sub.1], [k.sub.2],..., [k.sub.m] in N and [e.sub.1], [e.sub.2],..., [e.sub.m] in P such that |v| [less than or equal to] [k.sub.1][e.sub.1] + [k.sub.2][e.sub.2] + ... + [k.sub.m][e.sub.m]. Let k = max{m[k.sub.t]: t = 1,2,..., m} and [e.sub.j] = [e.sub.1] V [e.sub.2] V ... V [e.sub.m]. Then for every [e.sub.i] in P, |v| [LAMBDA] n[e.sub.i] [less than or equal to] |v| [less than or equal to] [k.sub.1][e.sub.1] + [k.sub.2][e.sub.2] + ... + [k.sub.m][e.sub.m] [less than or equal to] k[e.sub.j] for all n in N. If u is an element in [I.sub.P] such that |u| [LAMBDA] [e.sub.i] = 0 for all [e.sub.i] in P. Then for any [n.sub.1], [n.sub.2],..., [n.sub.k] in N, |u| [LAMBDA] ([n.sub.1][e.sub.1] + [n.sub.2][e.sub.2] + ... + [n.sub.k][e.sub.k]) [less than or equal to] |u| [LAMBDA] [n.sub.1][e.sub.1] + ... + |u| [LAMBDA] [n.sub.k][e.sub.k] [less than or equal to] [n.sub.1] (|u| [LAMBDA] [e.sub.1]) + [n.sub.2](|u| [LAMBDA] [e.sub.2]) + ... + [n.sub.k] (|u| [LAMBDA] [e.sub.k]) = 0. Inasmuch as [I.sub.P] is the ideal generated by P, thus |u| [LAMBDA] v = 0 for all v in [I.sub.P]. Let K be the Dedekind completion of [I.sub.P], by Theorem 24.2 in Luxemburg and Zaann (1971:131), [I.sub.p.sup.[perpendicular to]] [direct sum] ([I.sub.p.sup.[perpendicular to]])[.sup.[perpendicular to]] = K and [I.sub.p.sup.[perpendicular to]] [intersection] ([I.sub.p.sup.[perpendicular to]])[.sup.[perpendicular to]] = {0}. Because [I.sub.P] [subset] ([I.sub.p.sup.[perpendicular to]])[.sup.[perpendicular to]] and |u| [epsilon] [I.sub.p.sup.[perpendicular to]], thus |u| = 0. For the converse, it is obvious from the definition for an M-partition lattice.

Proposition 3

An M-partition space (V,[T.sub.P]) with the M-partition P is a TM-partition space with the TM-partition P.

Proof. -- This is clear from Proposition 3 in Wu (1986:5) that the ideal [I.sub.P] in an M-partition lattice (V,P) generated by P is dense in (V,[T.sub.P]).

Corollary 4

Let V be a vector lattice having an order unit u. Then V with the order unit topology is a TM-partition space with TM-partition {u}.

Proof. -- This is obvious from Proposition 5 in Wu (1986:5) that an order unit space is an M-partition space with a finite M-partition.

TM-REPRESENTATION SPACES

Let (V,T) be a TM-partition space with a TM-partition P and let [I.sub.P] be the ideal in V generated by P. Proposition 2 implies that ([I.sub.P],P) is an M-partition lattice with the M-partition P. Let [T.sub.P] be the M-partition topology induced by P, then ([I.sub.P],[T.sub.P]) is an M-partition space. By Theorem 1, there is a homeomorphism and lattice isomorphism H from ([I.sub.P],[T.sub.P]) onto a dense subspace of (C(Y),k) for some locally compact Hausdorff space Y. Y is unique up to homeomorphism. This Y will be called a TM-representation space of (V,T,P).

By Proposition 8 in Wu (1986:7), for each i in [OMEGA], the closure cl([E.sub.i]) of [E.sub.i]={y [epsilon] Y: H([e.sub.i]) (y) > 0} is compact. Let [I.sub.i] be the ideal in V generated by [e.sub.i]. Then H([I.sub.i]) is contained in the set [C.sub.*](cl([E.sub.i]))={f [epsilon] C(Y): |f|[.sup.-1]((0,[infinity])) [subset] [E.sub.i]}. For convenience, let [H.sub.i] = [H.sub.|Ii], the restriction of H on [I.sub.i], [T.sub.i] = [T.sub.|Ii], the relative topology on [I.sub.i], and ||*|| denote the supremum norm.

Lemma 5

The mapping [H.sub.i.sup.-1]: (H([I.sub.i]), ||*||) [right arrow] ([I.sub.i],[T.sub.i]) is continuous.

Proof. -- It is enough to show that [H.sub.i.sup.-1] is continuous at zero. Let [O.sub.i] be a balanced, absorbing, solid open neighborhood of zero in ([I.sub.i],[T.sub.i]). Then there is a balanced, absorbing, solid open neighborhood O of zero in ([I.sub.P],[T.sub.|I.sub.P]) such that O [intersection] [I.sub.i] = [O.sub.i]. By the Corollary of Proposition 8 in Wu (1986:7), there exists [e.sub.j] in P such that H([e.sub.j]) (x) > 1/2 for all x in cl([E.sub.i]). Because O is absorbing, there exists [r.sub.j] > 0 such that [r.sub.j][e.sub.j] is in O. Inasmuch as [r.sub.j][e.sub.j] [LAMBDA] m[e.sub.i] [less than or equal to] [r.sub.j][e.sub.j] for all m in N and O is solid, [r.sub.j][e.sub.j] [LAMBDA] m[e.sub.i] is in O for all m in N. This implies that [r.sub.j][e.sub.j] [LAMBDA] m[e.sub.i] is in O [intersection] [I.sub.i] = [O.sub.i] for all m in N. It is claimed that {v [epsilon] H([I.sub.i]): ||v|| < [r.sub.j]/2} is contained in [H.sub.i]([O.sub.i]). Let u be an element in H([I.sub.i]) such that ||u|| < [r.sub.j]/2. Then |u| (x) < [r.sub.j]/2 for all x in [E.sub.i] and u(x) = 0 for all x in Y - [E.sub.i]. Because H([e.sub.j]) (x) > 1/2 for all x in cl([E.sub.i]), |u| (x) < [r.sub.j]/2 < [r.sub.j]H[e.sub.j]) (x) for all x in Y. Hence |u| [less than or equal to] [r.sub.j]H([e.sub.j]). Because u is in H([I.sub.i]), there exists k in N such that |u| [less than or equal to] kH([e.sub.i]). Therefore, |u| [less than or equal to] [r.sub.j]H([e.sub.j]) [LAMBDA] kH([e.sub.i]) = H([r.sub.j][e.sub.j] [LAMBDA] k[e.sub.i]). Inasmuch as [r.sub.j][e.sub.j] [LAMBDA] k[e.sub.i] is in [O.sub.i] and [O.sub.i] is solid, this implies that |u| is in [H.sub.i]([O.sub.i]). Thus, [H.sub.i.sup.-1] ({v [epsilon] H([I.sub.i]): ||v|| < [r.sub.j]/2}) [subset] [O.sub.i]. Therefore, [H.sub.i.sup.-1] is continuous.

A STONE APPROXIMATION THEOREM FOR TM-PARTITION SPACES

Let H be the homeomorphism and lattice isomorphism from ([I.sub.P],[T.sub.P]) onto the dense subspace of (C(Y),k) defined in the previous section. For each i in [OMEGA], let [g.sub.i]: H([I.sub.i]) [right arrow] H([I.sub.P]) be the inclusion mapping and ||*|| the supremum norm on H([I.sub.i]). Let T* be the finest locally convex topology such that [g.sub.i]: (H([I.sub.i]), ||*||) [right arrow] (H([I.sub.P]), T*) is continuous for all i in [OMEGA]; that is, T* is the inductive topology on H([I.sub.P]) with respect to the family {(H([I.sub.i]), ||*||, [g.sub.i]): i [epsilon] [OMEGA]} (Schaefer, 1971:5). A zero-neighborhood base for T* is given by the family {U} of all radical, convex, circled subsets of H([I.sub.P]) such that for each in in [OMEGA], [g.sub.i.sup.-1] (U) is a zero-neighborhood in (H([I.sub.i]), ||*||).

Lemma 6

Let W be a vector sublattice of H([I.sub.P]). If W [intersection] H([I.sub.i]) is dense in (H([I.sub.i]), ||*||) for each i in [OMEGA], then W is dense in (H([I.sub.P]), T*).

Proof. -- Let O be an open subset of H([I.sub.P]) in (H([I.sub.P]), T*). Then for each i in [OMEGA], [g.sub.i.sup.-1] (O) is an open set in (H([I.sub.i]), ||*||). Because W [intersection] H([I.sub.i]) is dense in (H([I.sub.i]), ||*||), it follows that (W [intersection] H([I.sub.i])) [intersection] [g.sub.i.sup.-1] (O) [not equal to] [phi]. This implies that W [intersection] [g.sub.i.sup.-1] (O) [not equal to] [phi]; i.e., W [intersection] O [not equal to] [phi]. Thus W is dense in (H([I.sub.P]), T*).

Lemma 7

T* is finer than H([T.sub.|I.sub.P]).

Proof. -- By Lemma 5, for each i in [OMEGA], [H.sub.i.sup.-1]: (H([I.sub.i]),||*||) [right arrow] ([I.sub.i],[T.sub.i]) is a continuous function. Because H is a lattice isomorphism from [I.sub.P] onto H([I.sub.P]), this implies that the inclusion mapping [g.sub.i]: (H([I.sub.i]),||*||) [right arrow] (H([I.sub.P]), H([T.sub.|I.sub.P])) is continuous for all i in [OMEGA]. Inasmuch as T* is the finest topology such that [g.sub.i] is continuous for each i in [OMEGA], hence T* is finer than H([T.sub.|I.sub.P]).

Theorem 8

Let W be a vector sublattice of a TM-partition space (V,T). Then W is dense in (V,T) if, and only if, 1) the closure cl(W) of W in (V,T) contains a TM-partition P, and 2) cl(W) [intersection] [I.sub.P] separates points of TM-representation space Y of (V,T,P).

Proof. -- If W is dense in (V,T), then cl(W) = V. It is obvious that cl(W) contains a TM-partition P and cl(W) [intersection] [I.sub.P] = [I.sub.P]. From Wu (1986:6), it is clear that each x in the TM-representation space Y of (V,T,P) is a real continuous lattice homomorphism on [I.sub.P]. If x and y are two different points in Y, then (x - y) is a nonzero real continuous lattice homomorphism on the M-partition space ([I.sub.P],[T.sub.P]), by the Corollary of Proposition 3 in Wu (1986:5), there is an [e.sub.i] in P such that (x - y) ([e.sub.i]) > 0. Because (x - y) ([e.sub.i]) = x([e.sub.i]) - y([e.sub.i]) = [e.sub.i](x) - [e.sub.i](y). This implies that [e.sub.i](x) [not equal to] [e.sub.i](y); that is, cl(W) [intersection] [I.sub.P] separates points of Y. For the converse, it will first be shown that H(Cl(W) [intersection] [I.sub.i]) is dense in (H([I.sub.i]),||*||). Because H([I.sub.i]) is contained in [C.sub.*](cl([E.sub.i])), it is sufficient to show that H(cl(W) [intersection] [I.sub.i]) is dense in ([C.sub.*](cl([E.sub.i]),||*||). For all x in [E.sub.i], because sup{H([e.sub.j]) (x): j [epsilon] [OMEGA]} = 1, for every [epsilon] > 0, there exists an [e.sub.x] in P such that 1 [greater than or equal to] 1 H([e.sub.x]) (x) [greater than or equal to] 1 - [epsilon]. Pick r > 0 such that rH([e.sub.i]) (x) > H([e.sub.x]) (x). Then H([e.sub.x]) [LAMBDA] rH([e.sub.i]) is in H(cl(W) [intersection] [I.sub.i]) with 1 - [epsilon] < H([e.sub.x]) (x) = H([e.sub.x]) [LAMBDA] rH([e.sub.i]) (x) [less than or equal to] 1 and 0 [less than or equal to] H([e.sub.x]) (y) [LAMBDA] rH([e.sub.i]) (y) [less than or equal to] 1 for all y in [E.sub.i]; i.e., 1 - [epsilon] < ||H([e.sub.x]) [LAMBDA] rH([e.sub.i]) || [less than or equal to] 1. Thus, 1 - (H([e.sub.x]) (x) [LAMBDA] rH([e.sub.i]) (x)) / ||H([e.sub.x]) [LAMBDA] rH([e.sub.i])|| < 1 - (1 - [epsilon]) = [epsilon]. Next, it will be shown that H(cl(W) [intersection] [I.sub.i]) separates points of [E.sub.i]. Because cl(W) [intersection] [I.sub.P] separates points of Y (that is, H(cl(W) [intersection] [I.sub.P]) separates points of Y), for every x, y in [E.sub.i], there exists a v in cl(W) [intersection] [I.sub.P] such that H(v) (x) [not equal to] H(v) (y). This implies that either H(v)[.sup.+](x) [not equal to] H(v)[.sup.+](y) or H(v)[.sup.-](x) [not equal to] H(v)[.sup.-](y). Without loss of generality, assume that H(v)[.sup.+](x) [not equal to] H(v)[.sup.+](y). Let r > 0 such that rH(v)[.sup.+](x) < H([e.sub.i]) (x) and rH(v)[.sup.+](y) < H([e.sub.i])(y). Then rH(v)[.sup.+] [LAMBDA] H([e.sub.i]) is in H(cl(W) [intersection] [I.sub.i]) and rH(v)[.sup.+](x) [LAMBDA] H([e.sub.i])(x) = rH(v)[.sup.+] (x) [not equal to] rH(v)[.sup.+](y) = rH(v)[.sup.+](y) [LAMBDA] H([e.sub.i])(y). Thus, H(cl(W) [intersection] [I.sub.i]) separates points of [E.sub.i]. By the Corollary of Theorem 2 in Wu (1986:3), H(cl(W) [intersection] [I.sub.i]) is dense in ([C.sub.*](cl([E.sub.i])), ||*||), this implies that H(cl(W) [intersection] [I.sub.i]) is dense in (H([I.sub.i]),||*||). Therefore, for each i in [OMEGA], H(cl(W)) [intersection] H([I.sub.i]) = H(cl(W) [intersection] [I.sub.i]) is dense in (H([I.sub.i]),||*||). Lemma 6 implies that H(cl(W) [intersection] [I.sub.P]) is dense in (H([I.sub.P]), T*). Because T* is finer than H([T.sub.|I.sub.P]), thus cl(W) [intersection] [I.sub.P] is dense in ([I.sub.P], [T.sub.|I.sub.P]). By the denseness of [I.sub.P] in (V,T), this implies that cl(W) is dense in (V,T) and, therefore, W is dense in (V,T).

LITERATURE CITED

Luxemburg, W. A. J., and A. C. Zaann. 1971. Riesz space. North-Holand, Amsterdam, 1:1-514.

Schaefer, H. H. 1971. Topological vector spaces, Springer-Verlag, New York, 294 pp.

______. 1974. Banach lattices and positive operations, Springer-Verlag, New York, 376 pp.

Wu, H. J. 1986. A vector lattice representation theorem and a characterization of locally compact Hausdorff spaces, J. Funct. Anal., 65:1-14.

HUEYTZEN J. WU

Department of Mathematics, Texas A & I University, Kingsville, Texas 78363
COPYRIGHT 1990 Texas Academy of Science
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 1990 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Wu, Hueytzen J.
Publication:The Texas Journal of Science
Date:May 1, 1990
Words:3274
Previous Article:Soil differences between native brush and cultivated fields in the lower Rio Grande Valley of Texas.
Next Article:The eastward recession of the piney woods of northeastern Texas, 1815 to 1989.
Topics:

Terms of use | Privacy policy | Copyright © 2019 Farlex, Inc. | Feedback | For webmasters