Minimal latticesubspaces
Author:
Ioannis A. Polyrakis
Journal:
Trans. Amer. Math. Soc. 351 (1999), 41834203
MSC (1991):
Primary 46B42, 52A21, 15A48, 53A04
Published electronically:
April 20, 1999
MathSciNet review:
1621706
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this paper the existence of minimal latticesubspaces of a vector lattice containing a subset of is studied (a latticesubspace of is a subspace of which is a vector lattice in the induced ordering). It is proved that if there exists a Lebesgue linear topology on and is closed (especially if is a Banach lattice with order continuous norm), then minimal latticesubspaces with closed positive cone exist (Theorem 2.5). In the sequel it is supposed that is a finite subset of , where is a compact, Hausdorff topological space, the functions are linearly independent and the existence of finitedimensional minimal latticesubspaces is studied. To this end we define the function where . If is the range of and the convex hull of the closure of , it is proved:  (i)
 There exists an dimensional minimal latticesubspace containing if and only if is a polytope of with vertices (Theorem 3.20).
 (ii)
 The sublattice generated by is an dimensional subspace if and only if the set contains exactly points (Theorem 3.7).
This study defines an algorithm which determines whether a finitedimensional minimal latticesubspace (sublattice) exists and also determines these subspaces.
 1.
Edward
Tutaj, On Schauder bases which are unconditional like, Bull.
Polish Acad. Sci. Math. 33 (1985), no. 34,
137–146 (English, with Russian summary). MR 805027
(87b:46010)
 2.
C. D. Aliprantis, D. Brown, I. Polyrakis, and J. Werner, Portfolio dominance and optimality in infinite security markets, J. Math. Economics 30 (1998), 347366. CMP 99:04
 3.
C. D. Aliprantis, D. Brown, and J. Werner, Minimumcost portfolio insurance, J. Economic Dynamics and Control (to appear).
 4.
Charalambos
D. Aliprantis and Owen
Burkinshaw, Positive operators, Pure and Applied Mathematics,
vol. 119, Academic Press Inc., Orlando, FL, 1985. MR 809372
(87h:47086)
 5.
J.
A. Kalman, Continuity and convexity of projections and barycentric
coordinates in convex polyhedra, Pacific J. Math. 11
(1961), 1017–1022. MR 0133732
(24 #A3557)
 6.
Peter
MeyerNieberg, Banach lattices, Universitext, SpringerVerlag,
Berlin, 1991. MR
1128093 (93f:46025)
 7.
Shizuo
Miyajima, Structure of Banach quasisublattices, Hokkaido Math.
J. 12 (1983), no. 1, 83–91. MR 689259
(84g:46033)
 8.
Susanna
Papadopoulou, On the geometry of stable compact convex sets,
Math. Ann. 229 (1977), no. 3, 193–200. MR 0450938
(56 #9228)
 9.
A. Peresini, Ordered topological vector spaces, Harper & Row, New York, 1967.
 10.
Ioannis
A. Polyrakis, Schauder bases in locally solid lattice Banach
spaces, Math. Proc. Cambridge Philos. Soc. 101
(1987), no. 1, 91–105. MR 877703
(89b:46020), http://dx.doi.org/10.1017/S0305004100066433
 11.
Ioannis
A. Polyrakis, Latticesubspaces of 𝐶[0,1] and positive
bases, J. Math. Anal. Appl. 184 (1994), no. 1,
1–18. MR
1275938 (95g:46040), http://dx.doi.org/10.1006/jmaa.1994.1178
 12.
Ioannis
A. Polyrakis, Finitedimensional latticesubspaces
of 𝐶(Ω) and curves of 𝑅ⁿ, Trans. Amer. Math. Soc. 348 (1996), no. 7, 2793–2810. MR 1355300
(96k:46031), http://dx.doi.org/10.1090/S000299479601639X
 1.
 Y. A. Abramovich, C. D. Aliprantis, and I. A. Polyrakis, Latticesubspaces and positive projections, Proc. Roy. Irish Acad. 94 A (1994), no. 2, 237253. MR 87b:46010
 2.
 C. D. Aliprantis, D. Brown, I. Polyrakis, and J. Werner, Portfolio dominance and optimality in infinite security markets, J. Math. Economics 30 (1998), 347366. CMP 99:04
 3.
 C. D. Aliprantis, D. Brown, and J. Werner, Minimumcost portfolio insurance, J. Economic Dynamics and Control (to appear).
 4.
 C. D. Aliprantis and O. Burkinshaw, Positive operators, Academic Press, New York & London, 1985. MR 87h:47086
 5.
 J. A. Kalman, Continuity and convexity of projections and barycentric coordinates in convex polyhedra, Pacific J. of Math. (1961), 10171022. MR 24:A3557
 6.
 P. MeyerNieberg, Banach lattices, SpringerVerlag, 1991. MR 93f:46025
 7.
 S. Miyajima, Structure of Banach quasisublattices, Hokkaido Math. J. 12 (1983), 8391. MR 84g:46033
 8.
 S. Papadopoulou, On the geometry of stable compact convex sets, Math. Annalen 229 (1977), 193200. MR 56:9228
 9.
 A. Peresini, Ordered topological vector spaces, Harper & Row, New York, 1967.
 10.
 I. A. Polyrakis, Schauder bases in locally solid lattice Banach spaces, Math. Proc. Cambridge Philos. Soc. 101 (1987), 91105. MR 89b:46020
 11.
 , Latticesubspaces of and positive bases, J. Math. Anal. Appl. 184 (1994), 118. MR 95g:46040
 12.
 , Finitedimentional latticesubspaces of and curves of , Trans. American Math. Soc. 384 (1996), 27932810. MR 96k:46031
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (1991):
46B42,
52A21,
15A48,
53A04
Retrieve articles in all journals
with MSC (1991):
46B42,
52A21,
15A48,
53A04
Additional Information
Ioannis A. Polyrakis
Affiliation:
Department of Mathematics National Technical University of Athens Zographou 157 80, Athens, Greece
Email:
ypoly@math.ntua.gr
DOI:
http://dx.doi.org/10.1090/S0002994799023843
PII:
S 00029947(99)023843
Received by editor(s):
March 16, 1997
Published electronically:
April 20, 1999
Additional Notes:
This research was supported by the 1995 PENED program of the Ministry of Industry, Energy and Technology of Greece
Article copyright:
© Copyright 1999 American Mathematical Society
