# d-Independence and d-bases

d-Independence and d-bases Positivity 7: 95–97, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands. Y.A. ABRAMOVICH and A.K. KITOVER Community College of Philadelphia, Department of Mathematics, 1700 Spring Garden Street, Philadelphia, PA 19130, USA The notion of a d -basis is an extremely important and convenient tool for in- vestigation of properties of disjointness preserving operators (see, for instance, [1, 2, 4–6]). Nevertheless, some very basic and natural questions about d -bases remain open and this, in our opinion, might be on of the major obstacles in obtain- ing a complete understanding of the structure of disjointness preserving operators between vector lattices. To formulate these questions we need some deﬁnitions. All undeﬁned terms or symbols can be found in [1]. DEFINITION 1. A collection {x } of elements in a vector lattice X is said to γ γ ∈ be d -independent if for any band B in X, for any ﬁnite subset {γ ,... ,γ } of , l n and for any non-zero scalars c ,... ,c the condition c x ⊥B implies that 1 n i γ i=1 i x ⊥B for i = 1,... ,n. DEFINITION 2. A d -independent system {x } is called a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# d-Independence and d-bases

Positivity, Volume 7 (2) – Oct 17, 2004
3 pages

/lp/springer_journal/d-independence-and-d-bases-o2lqtW0lKR
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1025876218015
Publisher site
See Article on Publisher Site

### Abstract

Positivity 7: 95–97, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands. Y.A. ABRAMOVICH and A.K. KITOVER Community College of Philadelphia, Department of Mathematics, 1700 Spring Garden Street, Philadelphia, PA 19130, USA The notion of a d -basis is an extremely important and convenient tool for in- vestigation of properties of disjointness preserving operators (see, for instance, [1, 2, 4–6]). Nevertheless, some very basic and natural questions about d -bases remain open and this, in our opinion, might be on of the major obstacles in obtain- ing a complete understanding of the structure of disjointness preserving operators between vector lattices. To formulate these questions we need some deﬁnitions. All undeﬁned terms or symbols can be found in [1]. DEFINITION 1. A collection {x } of elements in a vector lattice X is said to γ γ ∈ be d -independent if for any band B in X, for any ﬁnite subset {γ ,... ,γ } of , l n and for any non-zero scalars c ,... ,c the condition c x ⊥B implies that 1 n i γ i=1 i x ⊥B for i = 1,... ,n. DEFINITION 2. A d -independent system {x } is called a

### Journal

PositivitySpringer Journals

Published: Oct 17, 2004

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