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Book review

Author(s) Huijsmans, C.B. (ed.)
Luxemburg, W.A.J. (ed.)
Title Positive Operators and Semigroups on Banach Lattices
Publisher Kluwer Academic Publishers
Year of publication 1992
Reviewed by S. Cobzas

From June 18 through June 22, 1990 on the Caribbean island of Curacao (Netherlands Antilles) a small workshop was held devoted to the theory of positive operators and their semigroups. The purpose of the workshop, conducted by C.B. Huijsmans, W.A.J. Luxemburg and B. de Pagter, was to present to a group of interested mathematicians from the Caribbean and Latin America an up-to-date account of the main results of the theory of positive operators on Banach lattices. The workshop was attended by mathematicians from Florida U.S.A., Guyana, Panama, Surinam and Venezuela.

The workshop was followed by a conference from June 25 to June 29, devoted primarily to recent advances in this area. The workshop and the conference took place under the auspices of the Caribbean Mathematical Foundation (CMF) under the directorship of Dr. J. Martinez. The main purpose of the conference, organized by C.B. Huijsmans and W.A.J. Luxemburg was to bring together a group of prominent specialists from the U.S.A. and Western Europe to present their recent results and to discuss their research interests.

The volume contains the worked-out version of the papers presented at the conference and of some related contribution. The papers included in the volume have a survey character, present new results or are of both character. There are included fourteen papers headed as follows: Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, Positive Operators on Krein Spaces, 1-22; A. Abramovich and W. Filter, A Remark on the Representation of Vector Lattices as Spaces of Continuous Real-Valued Functions, 23-26; W. Arendt and J. Voigt, Domination of Uniformly Continuous Semigroups, 27-32; S.J. Bernau, Sums and Extensions of Vector Lattices Homomorphisms, 33-46; B. Eberhardt and G. Greiner, Baillon's Theorem and Maximal Regularity, 47 - 54; A.W. Hager and J. Martinez, Fraction-Dense Algebras and Spaces, 55-66; C.B. Huijsmans and W.A.J. Luxemburg, An Alternative Proof of Radon-Nikodym Theorem for Lattice Homomorphism, 67-72; C.B. Huijsmans and B. de Pagter, Some Remarks on Disjointness Preserving Operators, 73-78; L. Maligranda, Weakly Compact Operators and Interpolation, 79 -90; B. de Pagter, A Wiener-Young Type Theorem for Dual Semigroups, 101-110; A.R. Schep, Krivine's Theorem and Indices of a Banach Lattice, 111-122; A. W. Wickstead, Representations of Archimedean Riesz Spaces by Continuous Functions, 123-134; X-D. Zhang, Some Aspects of the Spectral Theory of Positive Operators, 134-142.

A Problem Section, 143-152, containing sixteen open problems is also included.

The volume is a valuable contribution to these very active domains of research - theory of algebraic structures and lattices, Banach lattices and positive operators.

The book appears in excellent typographical conditions and undoubtely, will get a large audience between mathematicians as well as physicists and other scientists needing to apply (and to make) order in their research .