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Abstract M- and Abstract L-Spaces of Polynomials on Banach Lattices

Published online by Cambridge University Press:  13 February 2015

Qingying Bu ,
Gerard Buskes  and
Yongjin Li
Qingying Bu
Affiliation:
Department of Mathematics, University of Mississippi, University, MS 38677, USA (qbu@olemiss.edu; mmbuskes@olemiss.edu)
Gerard Buskes
Affiliation:
Department of Mathematics, University of Mississippi, University, MS 38677, USA (qbu@olemiss.edu; mmbuskes@olemiss.edu)
Yongjin Li
Affiliation:
Department of Mathematics, University of Mississippi, University, MS 38677, USA (qbu@olemiss.edu; mmbuskes@olemiss.edu)
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Abstract

In this paper we use the norm of bounded variation to study multilinear operators and polynomials on Banach lattices. As a result, we obtain when all continuous multilinear operators and polynomials on Banach lattices are regular. We also provide new abstract M- and abstract L-spaces of multilinear operators and polynomials and generalize all the results by Grecu and Ryan, from Banach lattices with an unconditional basis to all Banach lattices.

Keywords

homogeneous polynomialspositive tensor productsAM- and AL-spacesPrimary 46G2546B2846B42
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 3 , October 2015 , pp. 617 - 629
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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References

1.Alencar, R., Multilinear mappings of nuclear and integral type, Proc. Am. Math. Soc. 94 (1985), 33–38.CrossRefGoogle Scholar
2.Aliprantis, C. D. and Burkinshaw, O., Positive operators (Springer, 2006).Google Scholar
3.Benyamini, Y., Lassalle, S. and Llavona, J. G., Homogeneous orthogonally additive polynomials on Banach lattices, Bull. Lond. Math. Soc. 38 (2006), 459–469.CrossRefGoogle Scholar
4.Boulabiar, K., On products in lattice-ordered algebras, J. Austral. Math. Soc. 75 (2003), 23–40.CrossRefGoogle Scholar
5.Bu, Q. and Buskes, G., Polynomials on Banach lattices and positive tensor products, J. Math. Analysis Applic. 388 (2012), 845–862.CrossRefGoogle Scholar
6.Buskes, G. and van, A.Rooij, Bounded variation and tensor products of Banach lattices, Positivity 7 (2003), 47–59.CrossRefGoogle Scholar
7.Cartwright, D. I. and Lotz, H. P., Some characterizations of AM- and AL-spaces, Math. Z. 142 (1975), 97–103.CrossRefGoogle Scholar
8.Diestel, J. and Uhl, J. J., Vector measures, Mathematical Surveys, Volume 15 (American Mathematical Society, Providence, RI, 1977).CrossRefGoogle Scholar
9.Dineen, S., Complex analysis on infinite dimensional spaces (Springer, 1999).CrossRefGoogle Scholar
10.Fremlin, D. H., Tensor products of Archimedean vector lattices, Am. J. Math. 94 (1972), 778–798.CrossRefGoogle Scholar
11.Fremlin, D. H., Tensor products of Banach lattices, Math. Annalen 211 (1974), 87–106.CrossRefGoogle Scholar
12.Grecu, B. and Ryan, R. A., Polynomials on Banach spaces with unconditional bases, Proc. Am. Math. Soc. 133 (2005), 1083–1091.Google Scholar
13.Loane, J., Polynomials on Riesz spaces, PhD dissertation, National University of Ireland, Galway (2007).Google Scholar
14.Meyer-Nieberg, P., Banach lattices (Springer, 1991).CrossRefGoogle Scholar
15.Palazuelos, C., Peralta, A. M. and Villanueva, I., Orthogonally additive polynomials on C *-algebras, Q. J. Math. 59 (2008), 363–374.Google Scholar
16.Perez, D. and Villanueva, I., Orthogonally additive polynomials on spaces of continuous functions, J. Math. Analysis Applic. 306 (2005), 97–105.Google Scholar
17.Schaefer, H. H., Aspects of Banach lattices, in Studies in functional analysis (ed. Bartle, R. G.), Studies in Mathematics, Volume 21, pp. 158–221 (Mathematical Association of America, Washington, DC, 1980).Google Scholar
18.Schep, A. R., Factorization of positive multilinear maps, Illinois J. Math. 28 (1984), 579–591.Google Scholar