Bands And Ideals Of Banach Lattices

The ideas and bands of Banach lattices (or Riesz spaces) play very important and interesting role in the literature of Banach lattice (Riesz space) and operator theory, especially in discussion of the structure properties of Banach lattices and Riesz spaces and operators on them. There are number of results concerning the ideas and bands of Banach lattices or Riesz spaces, but there still are some problems worth to consideration and study, such as the exact relations between ideas and bands; the projection properties of bands (or ideas) and the structure properties of some concrete spaces, and so on. In this thesis we devote to investigate the Banach lattices on which each idea is a band, the projection properties of ideas/or bands and the structure of ideas and bands of the classical sequence spaces. Some related properties are also discussed. The thesis mainly consists of three parts.In the first part we will first prove that every finite dimensional idea of a Riesz space must be band, and then we characterize Banach lattices on which every (principal) idea being a band. Some related properties are included as well.In the second part we consider the projection properties of ideas and bands. The famous Kantorovich Theorem claimed that every band of a Dedekind complete Banach lattice must be a image of a positive (order) projection. The typical results in this part are:(1) If an idea of a Riesz space is the image of a positive projection, then it must be a band; (2) Each continuous positive linear functional on a Banach lattice admits a positive extension, and as application; (3) Every finite dimensional Riesz sublattice of a Banach lattice must be an image of a positive projection. Some related counter-examples are also provided.In the third part we mainly characterize the bands of classical sequence spaces (including both separable and non-separable cases). And based on the results we deduce the structure properties of ideas of classical sequence spaces. Some examples show that the results are almost best.