Representation Theorems for Riesz Spaces

Abstract

First of all, we shall briefly recall several fundamental notions from the theory of Riesz spaces, that is, from the theory of vector lattices. A Riesz space, denote it by L, is a real vector space which is at the same time a lattice such that the vector space structure and the lattice structure are compatible (i.e., if f, g ∈ L satisfy f ≦g, then f+h ≦ g+h for every h ∈L and af ≦ ag for every real number a≧0). The subset L + = (f:f ∈ L, f ≧ 0) is called the positive cone of L and the elements in the positive cone are called positive elements. We shall use the familiar lattice notations f v g and f ∧ g for the supremum and infimum of f and g. The notations $$\matrix{ {{f^ + } = f{\rm{v0,}}} & {{f^ - } = \left( { - f} \right){\rm{v0,}}} & {\left| f \right| = f{\rm{v}}\left( { - f} \right)} \cr } $$ for any f ∈L are also familiar, and it is a simple theorem that f +, f − and |f| are positive elements in L with f = f +— f − and |f| =f ++f −. We say that f is infinitely small with respect to g if f ≠ 0 and n |f|≦ |g| for n = l, 2, …, and f is called an infinitely small element if f is infinitely small with respect to some g. A Riesz space containing no infinitely small elements is called Archimedean. In the present exposition we shall restrict ourselves to Archimedean Riesz spaces, and we list some examples (i) The set C (X) of all real continuous functions on a topological space X, with the usual definitions for the vector space operations and the partial order, is an Archimedean Riesz space. (ii) We generalize the preceding example by admitting also extended real-valued continuous functions on X (i.e., continuous mappings from X into the topological space R∞ of all extended real numbers), with the restriction however that if f is a function of this kind, then the open set (x|f (x)|<∞) should be dense in X. The system of all these functions is denoted by C∞(X). In general, C∞(X) is no vector space, simply because if f and g in C∞(X) are given and we define already h(x) = f (x)+g (x) for all x for which f(x) and g(x) are finite, then it is not always possible to define h(x) in the remaining points x in such a manner that h becomes continuous in the extended sense on X. However, if X is extremally disconnected (i.e., the closure of every open set is open), then C∞(X) is a vector space, and so C∞(X) is then also a Riesz space. In the other cases, although C∞(X) as a whole is not always a vector space, certain subsets of C∞(X) may be Riesz spaces. (iii) If μ is a measure in the point set X and L p (X, μ), for some number p satisfying 0 <p < ∞, is the family of all p-th power μ-integrable real functions on X (or rather equivalence classes of these functions in the well-known manner), then L p (X, μ) is a Riesz space. Of course, the partial order has to be defined appropriately, i.e., f ≦g means that f (x) ≦ g (x) holds for μ-almost every x (iv) If H is a (complex) Hubert space of dimension at least two and ℋ is the real vector space of all bounded Hermitian operators on H, partially ordered in the usual manner, then ℋ is a partially ordered vector space but no Riesz space. Certain appropriate subspaces of ℋ, however, are Riesz spaces, and these subspaces can be used very neatly to give a simple proof of the spectral theorem for Hermitian and normal operators.