# 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.

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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
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