Boolean-Valued Introduction to the Theorv of Vector Lattices

The theory of vector lattices appeared in early thirties of this century and is connected with the names of L. V. Kantorovich, F. Riesz, and H. Freudenthal. The study of vector spaces equipped with an order relation compatible with a given norm structure was evidently motivated by the general circumstances that brought to life functional analysis in those years. Here the general inclination to abstraction and uniform approach to studying functions, operations on functions, and equations related to them should be noted. A remarkable circumstance was that the comparison of the elements could be added to the properties of functional objects under consideration. At the same time, the general concept of a Banach space ignored a specific aspect of the functional spaces-the existence of a natural order structure in them, which makes these spaces vector-lattices. Along with the theory of ordered spaces, the theory of Banach algebras was being developed almost at the same time. Although at the beginning these two theories advanced in parallel, soon their paths parted. Banach algebras were found to be effective in function theory, in the spectral theory of operators, and in other related flelds. The theory of vector lattices was developing more slowly and its achievements related to the characterization of various types of ordered spaces and to the description of operators acting in them was rather unpretentious and specialized. In the middle of the seventies the renewed interest in the theory of vector lattices led to its fast development which was related to the general explosive developments in functional analysis; there were also some specific reasons, the main one being the use of ordered vector space in the mathematical approach to social phenomena, economics in particular. The scientific work and the unique personality of L. V. Kantorovich also played important role in the development of the theory of ordered spaces and in relating this theory to economics and optimization. Another, though less evident, reason for the interest in vector lattices was their rather unexpected role in the theory of nonstandard-Boolean-valued-models of set theory. Constructed by D. Scott, R. Solovay, and P. Vopenka in connection with the well-known results by P. G. Cohen about the continuum hypothesis, these models proved to be inseparably linked with the theory of vector lattices. Indeed, it was discovered that the elements of such lattices serve as images of real numbers in a suitably selected Boolean model. This fact not only gives a precise meaning to the initial idea that abstract ordered spaces are derived from real numbers, but also provides a new possibility to infer common