A N N A L E S S O C I E T A T I S M A T H E M A T I C A E P O L O N A E S e r i e s I : C O M M E N T A T I O N E S M A T H E M A T I C A E X V I I ( 1 9 7 3 )
R O C Z N I K I P O L S K I E G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O S é r i a I : P R A C E M A T E M A T Y C Z N E X V I I ( 1 9 7 3 )
Bonald G. Mosier and Hidegoro Ж AK an о (Detroit, Mich.) Discrete linear lattices
Let В be a space, i.e. a set. A linear lattice L of functions which map В into the real numbers is called a function lattice on В if for all
<p, ye L, co€ В and real numbers a, we have
(cupfitp)(x) = a<p(x)-{-(5y)(x), (cpv y))(x) = Max{<p(x), ip(x)}.
A function lattice on В is called a vector lattice on В if it contains the characteristic function %r of each single element re B. This concept of a vector lattice is a modest generalization of finite-dimensional vector lattices.
We can easily prove for vector lattices on В that
(1) <p = V if and only if <p(r) = supç>A(r) for all r e B,
A e A A e A
and
(2) <p = A <Paif and °nly if <p(r) = inf <px(r) for all r e B .
A e A A e A
The characterization of function lattices is discussed in [2]. In this paper we consider a characterization of vector lattices.
Let L be a linear lattice. An element x e L is said to be discrete if
® Ф 0 and if for any y e L such that \y\ < \x\ there is a real number a such that у = ax. It is an easy matter to prove :
(3) If x is discrete and а Ф 0, then ax is discrete.
(4) If X€ L is discrete, then \x\ — x or —x .
(5) An x is discrete if and only if x Ф 0 and for any 0 < у < \x\ there is an a> 0 such that y = a\x\.
An element x e L is said to be arcMmedean if Д £\x\ = 0. The linear f>0
lattice L is said to be archimedean if every xe L is archimedean.
Theorem 1. For 0 Ф xe L to be discrete it is necessary and sufficient that (i) x be archimedean and (ii) for y , z e L i f \x\ > |y|, \z\ and у J_ z, then
<tt least one of у , z is 0.
Proof. Let xe L be discrete. If 0 < y < Ç\x\ for all £ > 0, then there is an a > 0 such that y — a\x\ and we conclude a = 0. Therefore x is archimedean by definition. Furthermore if \x\ > \y\, \z\, then у = ax and z — fix for some real numbers a, fi by definition, and so if у J_ z we have either a = 0 or fi — 0.
Now suppose an xe L satisfies (i) and (ii). If 0 < у < \x\, then setting a = inf £ we have a < 1 and y < a \x\ because x is archimedean. If a > 0, then for any 0 < fi < a we have ( y — fi\x\)+ _L{y — fi\x\)~ and 0 < (y-~P\x\)+ < \x\, 0 < ( y —fi\x\)~ < |a?|. Since (у — fi\x\)+ Ф 0, we ob
tain {y — fi\x\)~ — 0 by assumption (ii). This says y ^ f i \x\ for any f i< a . Thus we conclude у ^ a \x\ because x is archimedean. Therefore for any 0 < У < \x\ there is an a > 0 such that у — a\x|.
Th e o r e m 2. I f L is archimedean, then 0 Ф xe L is discrete if and only if x is normaldble and for any y e L there is a real a such that [#•] у — ax.
Proof. Let x be discrete. Then \x\ is discrete and \x\ = x or —x by (4). For any у ^ 0, since 0 < у av\x\< v \x\ (v = 1, 2, ...), we can find by (3) aP {v = 1 , 2, ...) such that 0 < у л v \x\ = av\x\ (v = 1 , 2 , . . . ) . If aP (v = 1 , 2 , . . . ) are unbounded, then there is a v0 such that av > 0
1 00
for and 0 ^ |a?| ^ — y j. 0 since L is archimedean. But x Ф 0
« V v = v o с о
by definition. Thus we conclude that the ap are bounded. Let a„ f a .
CO *" = 1
By Theorem 2.6 of [3], V (у лг \x\) = a' \x\- = ax for some real number a.
v= 1
Therefore by Theorem 6.11 of [3], x is normalable and [x]y — ax. For a general ye L, [x~\y = [x](y+ — y~) = \_x]y+ — \x ’\y~ = ax + fix = (a-\-fi)x for some real numbers a and fi.
Conversely if x Ф 0 is normalable and for any y e L there is an ay such that [ж]у = ayx, then for 0 < у < jo?| we have by Theorems 6 . 8 and 6.9 of [3] that у — [x]y. Hence there is an ay > 0 such that у = ay \x\
by our assumption. Therefore by (6), x is discrete.
A linear lattice L is said to be discrete if for any 0 Ф x e L there exists a discrete de L such that \x\ > \d\. This is a weaker definition than is used in [1]. In [1 ] the additional condition that L be universally continuous is imposed and the characterization of such spaces is discussed.
Th e o r e m 3. A linear lattice L is isomorphic to a vector lattice on some space if and only if L is discrete and archimedean.
Proof. Let F be a vector lattice of functions on a space E. It is clear that vector lattices are archimedean. We can easily prove that every characteristic function %r {reR) is a discrete element of V. For any 0 Ф Ф <pe V we can find an re E such that <p{r) Ф 0. Then \cp\ > \p{r)\%r and
\<p(r)\Xr is discrete by definition.
Discrete linear lattices 181
On the other hand, let L be discrete and archimedean. By the Maximal Theorem in [2] there exists a maximal set В of positive, mutually ortho
gonal discrete elements of L. For each x e L we define a function <px on В as follows:
<px{d) — У for [d]x = yd and d e l ) .
Set V = {epx: xe L}. It is not hard to show that F is a vector lattice on I), and we have
Фхуу Ф х ^ Ф у ! Фах+Ру ~ a(Px~^~ @*Ру
and <pd is the characteristic function of d for each de B.
Setting T{x) = <px (xe L) we obtain a linear operator T from L onto V by definition. If x Ф 0, then there is a positive discrete d' < |a?| such that 0 < [d]d' < [d]\x\ for all d e l ). Thus if T(x) = 0 and x Ф 0, then [d]|&| = 0 for all de J) and В и {d'} is a set of positive, mutually ortho
gonal discrete elements which properly contains the set B. This contradicts the maximality of B, so T(x) = 0 implies x = 0. Hence T is one- to-one.
Finally, by definition we have x > y if and only if x v y = x. But x v y = x if and only if (pxvy = <px v <py = <px, since T is one-to-one. But
<px v<py — (px is true if and only if <px > <py, so T is an isomorphism from L onto the vector lattice V, proving the theorem.
Taking into account the result of Theorem 1, in proving Theorem 3 we also proved
(6) In a vector lattice V on В, 0 Ф <pe V is discrete if and only if <p = a%d for some de R.
By definition obviously we have
(7) For a vector lattice V on R, if <pe V, then [#r]<p = (p{r)%r for all Ге R.
By (6), (7), and Theorem 3, we obtain
(8) For an arehimedian discrete linear lattice L and for any 0 < x e L, let В be a maximal set of mutually orthogonal, positive, discrete elements of L such that d ^ x for every de B. Then x = V [d]x.
dt D
An xe L is said to be simple if we can find a finite number of discrete П
elements xve L such that x = ^ xv.
v — 1
By Theorem 1 and (4), any simple x ^ 0 can be written as a linear combination of mutually orthogonal, positive, discrete elements.
П
(9) I f x e L is simple, say f?xv = x, where the xv are diswete and mu-
v — l
tually orthogonal (v = 1 ,2 , ..., w), then x is archimedean, and if L
П П
is archimedean, then x is normalable and [x] = V [#„] = ^ [a?„].
V = 1 V = 1
Proof. Finite sums of archimedean elements are archimedean, so x is archimedean. When L is archimedean, discrete elements are normalable by Theorem 2. Finite sums of normalable elements are normalable, so x is normalable. The left equality comes from (8) and Theorems 5.31 and 6.7 of [3]. The right equality follows from .Theorem 5.23 of [3].
If every element of L is simple, then we say I is simply discrete.
It is clear that L is simply discrete if every positive element of L is simply discrete.
Th e o r e m 4 . A linear lattice L is simply discrete if it is continuous
00 oo 00
and if xv I 0 implies the existence of nv f + oo and 0 < av f + oo such that
oo v — 1 v — 1 v — 1
a„xnv \ . I f a linear lattice L is simply discrete, then it is continuous and
v = 1
oo oo oo 00
xv I 0 implies the existence of nv f + oo and 0 < av f + oo such that avxnv j 0.
V = 1 V = 1 V = 1 V = 1
Proof. If L is simply discrete, then it is archimedean and discrete, and by Theorem 3 it is isomorphic to a vector lattice V on D, a maximal subset of mutually orthogonal, positive, discrete elements of L. Further-
П more, V is simply discrete; i.e., every <peV can be written <p = ]£ %%a ?
Л*— 1 **
where d ^ D and a^ is a real number (p — 1 , 2, . . . , n).
w 7 ь г*
Suppose О and <px = £ <Pi(dM)xaM- Then q>v = = ( % IXdJjV*
^ = 1 ^ = 1
by (8). Thus <pv = by (7). Setting <рр(йм) | we have by (2)
OO n / i = l v — 1
that <pv j a^Xd € V since e 7 (^u = 1 , 2 , . . : , w). Hence V is con
v e x M= i M 11
tinuous. oo 00
Suppose % I 0. Then we have % (d j j 0 (p = 1 , 2 , . . . , n) by (2). Hence
oo v = l 00 v = l 00
there are nv f + 00and a* f -f 00 such that a^Adf) j 0 (p — 1 , 2 , ..., n),
V — 1 00 v = 1 v = 0
and we have av% j 0 by (2).
V = 1
On the other hand, assume that L is a continuous linear lattice and
OO 00 00 00
that for any xv j 0 there are nv f 00 and 0 < av f -f- 00 such that avxnv j .
V — 1 V — 1 v = l v = l
Consider a sequence y ^ L (p = 1 ,2, . . . ) of mutually orthogonal,
Discrete linear lattices 183
positive elements of L such that £ Уц ts convergent. Setting x — ^ y^,
/л— l f l ~ 1
V OO 0 0 00
we have [x —^ y^ j 0, and by assumption there are nv f + со and 0 < av f + oo
1 V = 1 V — 1 V = * l
n v CO V
such that {a^x — £ o-vy^) | . This says that avx — аРУм^ агх (v = 1 , 2 , ...).
( 1 = 1 V = 1 ( 1 = 1 n v
Consequently for q > nv we have а„[уе]а? = [уе](а„ж — £ а„у^ < ах [уе]х.
С О /1 = 1
Since х = ]}Уц, we have [уе] х = ув and hence avyQ < агув, which is to
( l — l o o
say 0 ^ (а„ — ax)ye > 0. Since av f + oo, there exists v0 such that a„o> ax,
v — 1
and thus there is a v0 such that q ^ v0 implies y Q = 0.
We have shown that for a sequenceyv*L (v = 1 , 2 , . . . ) of orthogon-
o o
al, positive elements, if yv is convergent, then there is a v0 such that у, = 0 for v > v0. V =1
Now suppose x > 0 is not discrete. Then by Theorem 1, there are mutually orthogonal y x, zx > 0 such that x > y x + zx. If neither y x nor zx is discrete we can find mutually orthogonal y 2, z2 > 0 such that zx ^ у2 + г2•
Therefore if there is no positive discrete element < x, we may proceed in this manner and find a mutually orthogonal sequence у^ (/4 = 1 ,2 , . . . )
OO OO
such that Ур> 0 (у = 1 , 2 , . . . ) and x > V уй = ^ У и- This is a con- (i=i (i=i
tradiction. Thus I is a discrete linear lattice.
Furthermore, for any x > 0, if В is the maximal set of mutually orthogonal, positive discrete elements such that d < x for every de B, then ^ «iQj and âi^âiin by the above B/Sscrtioiij is finitCy say
deD n n
В = {dxi d2, . . . , dn}. Thus by (8) we have x = [d^x = £ avdv,
V = 1 V = 1
and x is simple. Therefore L is simply discrete.
For a linear lattice L , consider the following conditions.
OO 00 o o
(a) L *xv I 0 implies the existence of nv f + 0 0 and 0 < av f + oo such
00 V = 1 » = 1 V = 1
that avxnv I .
V = 1
00 CO
(b) L * x v I 0 implies the existence of 0< a vf + ° ° such that the
v = l V = 1
sequence avxv (v = 1 ,2 , . . . ) is bounded.
Condition (a) is stronger than condition (b). Condition (b) is obviously equivalent to each of the following conditions.
OO
(c) For any convergent series JT1 xv, where 0 ^.xve L for all
v = 1
OO V
v = 1 , 2 , ... we can find 0 f +oo such that the sequence амхм
(v = 1 ,2 , . . . ) is bounded. ,(‘ = 1
(d) Every order-convergent sequence is uniformly convergent; i.e.,
00
if lim xv = x, then there is TceL and sv j 0 such that \xv — x \ ^ e vh
v —>oo v—1
(v = 1 ,2 , . . . ).
When L is archimedean, for a sequence 0 < (v = 1 , 2 , . . . )
00
if there is 0 < a„ f +oo such that the sequence avxv {y = 1,2, . . . )
1»=1 00
is bounded, then there is 0 < /?„ f + oo such that lim (3vxv = 0.
» > = 1 v —> o o
Concerning the proper spaces of discrete linear lattices we have
Th e o r e m 5 . I f L is a continuous linear lattice and E is the proper space of L, then
(i) L is discrete if and only if the set of all isolated points is dense in E, and
(ii) L is simply discrete if and only if E consists of only isolated points.
Proof, (i) Let x e L be discrete. By Theorem 2, we conclude that Z7[£C] consists of a single point p e E and hence p is an isolated point of E.
If L is discrete, for any 0 Ф ye L there is a discrete xe L such that Ufyj 2 TJ[X}. Hence the set of isolated points is dense in E.
Conversely, if {p} is open and closed, then there is an xe L such that U[x) = {p} by Theorem 8 . 6 of [3]. Then by Theorem 2 we conclude that a; is a discrete element.
Hence if the isolated points of E are dense, then given О Ф ye L there is an isolated point p e U[yy By (4) on page 31 of [1], [y] > [x].
Finally then by Theorem 2, \y\ = [y]|y| > [#]|y| = a\x\. We know that а Ф 0 because [\x~\y\ = [a?][y] = [x] Ф 0, so by (3), a\x\ is discrete. We have shown that L is discrete.
П (ii) If L is simply discrete, then for О Ф xe L we have [ж] = V Ifyh
v = l
n
where xt is discrete (v = 1, 2 , .. . , n) by (9). Therefore U[x] = (U^[ag)~
П V = 1
= U U[z] by Theorem 8 . 8 of [3]. Because J7la!] is an isolated point,
v = 1
this proves that every open set consists of isolated points. Thus E consists of isolated points.
Conversely, if E consists only of isolated points, then for any О Ф x e L,
П
since U[xj is compact we have U[x] = U U[x ], where the xv are discrete
I> = 1 n
and mutually orthogonal (v = 1 , 2 , . . . , n) . Hence [а?] = V [®r] by
n v= l n
Theorem 8 . 8 of [3], and therefore [a?]a? = x = \f [x^x = ]?[xv]x
n 11 = 1 . v = 1
= ^ a,x. We have proved that L is simply discrete.
Discrete linear lattices 185
Befering to Theorem 13.4 in [3] we can easily prove
Th e o r e m 6 . A discrete linear lattice L is superuniversally continuous П
if and only if L is continuous and x e L implies x = ]?xv, where xv is
discrete or 0 (v — 1 , 2, ...). ” ==1
Because of the results of Theorem 4, Theorem 6 can also be wrtiten as
Th e o r e m 7. A continuous vector lattice V on В is superuniversally continuous if and only if V is a continuous linear lattice manifold of {<p: <p(r)
= 0 except for at most countable reB}.
References
[1 ] ' I. H a lp e r in and H. N a k a n o , Discrete semi-ordered linear spaces, Canadian J. Math. 3 (1951), p. 293-298.
[2] H. N a k a n o , M odern spectral theroy, Maruzen Co. L td., Tokyo 1950.
[3] — L in ea r lattices, W ayne State U niv. Press, D etroit 1960.
