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Vladikavkaz Mathematical Journal 2009, Vol. 11, No 2, pp. 43-45
UDC 517.98
WHEN ARE THE NONSTANDARD HULLS OF NORMED LATTICES DISCRETE OR CONTINUOUS?
Dedicated to Safak Alpay on the occasion of his sixtieth birthday
V. G. Troitsky
This note is a nonstandard analysis version of the paper «When are ultrapowers of normed lattices discrete
or continuous?» by W. Wnuk and B. Wiatrowski.
Mathematics Subject Classification (2000): 46S20, 46B42.
Key words: normed lattice, discrete element, nonstandard hull.
In Functional Analysis, the ultrapower and the nonstandard analysis approaches are equivalent: results obtained by one of these two methods can usually be translated into the other. In this short note, we present nonstandard analysis versions of the main results of [5], where they were originally presented in the ultrapower language. We believe that in this new form the ideas of the proofs are more transparent.
Suppose that E is a Archimedean vector lattice. Recall that an element 0 < e G E is said to be discrete if 0 ^ x ^ e implies that x is a scalar multiple of e or, equivalently, the interval [0, e] doesn't contain two non-zero disjoint vectors (see [3, Theorem 26.4]). We say that E is continuous if it contains no discrete elements and discrete if every non-zero positive vector dominates a discrete element or, equivalently, E has a complete disjoint system consisting of discrete elements (see [1, p. 40]).
If E is a normed space. We will write *E for the nonstandard extension of E and E for the nonstandard hull of E. We refer the reader to [2, 6] for terminology and details on nonstandard hulls of normed spaces and normed lattices. We will use the following standard fact (see, e. g., [4, Remark 4]).
Lemma 1. Suppose that E is a normed lattice and a,x,b G *E such that a ^ b and a ^ X ^ b. Then there exists y G *E such that y ~ x and a ^ y ^ b.
The following is a variant of Theorem 2.2 of [5]:
Theorem 2. Let E be a normed lattice. Then the following are equivalent.
(i) E is continuous;
(ii) 3e > 0 Vx G E+ 3a, b G [0,x] a ± b and ||a|| A ||b|| ^ e||x||.
< (i) ^ (ii) Suppose that E fails (ii). Let e be a positive infinitesimal. Then there exists a vector x G *E+ such that for all a, b G *[0, x] with a ± b we have ||a|| A ||b|| < e||x||. Without loss of generality, ||x|| = 1. Let a, b G [0, x] and a ± b. By Lemma 1, we may assume that
© 2009 Troitsky V. G.
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Troitsky V. G.
a, b £ *[0, x]. Furthermore, a ±6 implies that a A b w 0. Let u = a — a A b and v = b — a A b, then u, v £ *[0, x] and u ± v, so that ||u|| A ||v|| < e. It follows that either ||u|| or ||v|| is infinitesimal. Say, u ~ 0. Then a = u + a A b is infinitesimal as well, so that a = 0. Thus, x is discrete in E
(ii) ^ (i) Suppose that (ii) holds for some (standard) e > 0. Let x £ E+, show that x is not discrete. Without loss of generality, x £ *E+ and ||x|| = 1. By (ii), we can find a, b £ *[0, x] such that a ± b and ||a|| A ||b|| ^ e. It follows that neither a nor b is infinitesimal, so that a, b are two non-zero disjoint elements of [0, x]. Hence, x is not discrete. >
Recall that a normed lattice satisfies the Fatou property if 0 ^ xa | x implies ||xa|| ^ ||x||, and the a-Fatou property if 0 ^ xn | x implies ||xn|| ^ ||x||, see, e. g., [1]. We will use the following simple lemma.
Lemma 3. Suppose that E is a normed lattice with the Fatou property and S C E+ such that x = sup S exists. Then for every e > 0 there is a finite subset y of S such that || sup Y|| ^ (1 — e)||x||. The same is true for countable families if E satisfies the a-Fatou property.
< Let A be the collection of all finite subsets of S, ordered by inclusion. Clearly, supaeA sup a = x. Let xa = sup a, then (xa)aeA is an increasing net and 0 ^ xa | x. It follows from the Fatou property that | xa 11 ^ 11 x 11, so that there exists y £ A with ||x71| ^ (1 — e)||x||.
Now suppose that E satisfies a-Fatou property and x = V¿=1 x%. Let Zk = V(=1 xi, then xk ^ Z( ^ x, so that x = \/¿=1 Z(. Now a-Fatou property guarantees that ||z(|| ^ ||x||, so that (1 — e)||x|| ^ ||zm|| = ||x1 V ■ ■ ■ V xm|| for some m. >
The following is a variant of Theorem 3.1 of [5].
Theorem 4. Let E be a discrete normed lattice, and D the set of all discrete elements of norm one in E. If E satisfies the Fatou property (or the a-Fatou property if D is countable) then the discrete elements of E are exactly the positive scalar multiples of the elements of {e | e £ *D}.
< It suffices to show that given x £ *E with ||x|| = 1, then x is discrete in E if and only if x = e for some e £ *D. Suppose that x = e for some e £ *D. Take any a £ *E such that 0 ^ a ^ x. By Lemma 1, we may assume that 0 ^ a ^ e. It follows that a is a scalar multiple of e, hence a is a scalar multiple of x.
Conversely, suppose that x is discrete in E. Note that the set D is a complete disjoint system in E. By [1, Theorem 1.75], we have x = sup{Pex | e £ *D}. For every e £ *D, the vector Pex is a scalar multiple of e, and 0 ^ Pex ^ x, hence 0 ^ Pex ^ x. Therefore, if Pex is not infinitesimal for some e £ *D then x is a scalar multiple of Pex, hence of e.
Suppose now that Pex is infinitesimal for every e £ *D. It follows from x = sup{Pex | e £ *D} and Lemma 3 that there exist n £ *N and e1,...,en £ *D such that ||z|| ^ |, where z = ||Peix V ■ ■ ■ V Penx||. Choose k ^ n in *N so that ||Peix V ■ ■ ■ V Pek-i x|| < 4, while ||Peix V ■ ■ ■ V Pekx|| ^ 4. put u = Peix V ■ ■ ■ V Pekx = Peix +-----+ Pekx. Then
4 < ||u|| < 11Peix V ■ ■ ■ V Pek-ix|| + ||Pefcx|| < 4,
hence ||u|| ~ 4. Put v = z — u, then u ± v, 0 ^ u, v ^ z, and ||u||, ||v|| ^ 4. Therefore, u and e are non-zero and disjoint elements of [0, x]; a contradiction. >
Corollary 5. Suppose that E is an AM-space with a strong unit, and H is a discrete regular sublattice of E. Then H is discrete.
< Let D be a complete disjoint system of discrete elements of norm one in H. Suppose that x £ H+. We will show that x majorizes a discrete vector. Without loss of generality,
When are hulls discrete or continuous?
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x £ *H+ with ||x|| = 1. Then x = sup{Pex | e £ } by [1, Theorem 1.75]. Since E is an AM-space, we can apply Lemma 3 with e ~ 0 and find n £ *N and ei,...,en £ such that ||Peix V ■ ■ ■ V Penx|| ^ (1 — e)||x|| ~ 1. Again, since E is an AM-space, we have ||Peix V ■ ■ ■ V Penx|| = ||Peix|| V ■ ■ ■ V ||Penx||, so that ||Pekx|| w 1 for some k ^ n. Then Pekx is non-zero. It is discrete by Theorem 4 because Pek x is a multiple of . Finally, notice that pekx ^ x. >
References
1. Aliprantis C. D., Burkinshaw O. Locally solid Riesz spaces with applications to economics (Math. Surveys and Monographs, Vol. 105). 2 ed.—Providence (R.I.): Amer. Math. Soc., 2003.—344 p.
2. Emel'yanov E. Yu. Infinitesimals in vector lattices // Nonstandard analysis and vector lattices (Math. Appl., Vol. 525).—Dordrecht: Kluwer, 2000.—P. 161-230.
3. Luxemburg W. A. J., Zaanen A. C. Riesz spaces.—Amsterdam-London: North-Holland Publishing Co, 1971.—Vol. 1.
4. Troitsky V. G. Measures on non-compactness of operators on Banach lattices // Positivity.—2004.— Vol. 8, № 2.—P. 165-178.
5. Wnuk W., Wiatrowski B. When are ultrapowers of normed lattices discrete or continuous? // Positivity IV—theory and applications—Dresden: Tech. Univ. Dresden, 2006.—P. 173-182.
6. Wolff M. P. H. An introduction to nonstandard functional analysis // Nonstandard analysis (Edinburgh, 1996).—Dordrecht: Kluwer Acad. Publ., 1997.—P. 121-151.
Received May 19, 2009. Troitsky Vladimir G.
Department of Mathematical and Statistical Sciences, University of Alberta, Associate Professor Edmonton, AB, T6G2G1. Canada E-mail: [email protected]
