Positivity

C(X) denotes the ring of continuous real-valued functions on a Tychonoff space X and P a prime ideal of C(X). We summarize a lot of what is known about the reside class domains C(X)/P and add many new results about this subject with an emphasis on determining when the ordered C(X)/P is a valuation domain (i.e., when given two nonzero elements, one of them must divide the other). The interaction between the space X and the prime ideal P is of great importance in this study. We summarize first what is known when P is a maximal ideal, and then what happens when C(X)/P is a valuation domain for every prime ideal P (in which case X is called an SV-space and C(X) an SV-ring). Two new generalizations are introduced and studied. The first is that of an almost SV-spaces in which each maximal ideal contains a minimal prime ideal P such that C(X)/P is a valuation domain. In the second, we assume that each real maximal ideal that fails to be minimal contains a nonmaximal prime ideal P such that C(X)/P is a valuation domain. Some of our results depend on whether or not βω ω contains a P-point. Some concluding remarks include unsolved problems.

This article is aimed at a general reader familiar with the basics of functional analysis. It begins with a quick summary of the most basic ‘facts of life’ of positivity for Hilbert space operators, or for algebras of operators on a Hilbert space. It being impossible to adequately survey the fundamental role of positivity in the field of operator algebras, since this is so extensive and ubiquitous, in the present article we review selectively some of the general principles in the subject, and give some examples of how positivity plays a central role in the field, even in settings where positivity is not at first in evidence. The topics become more progressively more specialized towards our own current interests, ending with some very recent work of ours and of others.

We wrote a survey [18] on lattice ordered algebras five years ago. Why do we return to f-algebras once more? We hasten to say that there is only little overlap between the current paper and that previous survey.We have three purposes for the present paper. In our previous survey we remarked that one aspect that we did not discuss, while of some historical importance to the topic, is the theory of averaging operators. That theory has its roots in the nineteenth century and predates the rise of vector lattices. Positivity is a crucial tool in averaging, and positivity has been a fertile ground for the study of averaging-like operators. The fruits of positivity in averaging have recently (see [24]) started to appear in probability theory (to which averaging operators are close kin) and statistics. In the first section of our paper, we survey the literature for our selection of old theorems on averaging operators, at the same time providing some new perspectives and results as well.

This is a survey on bilinear maps on products of vector lattices

We will deal exclusively with the integration of scalar (i.e., ℝ or ℂ)-valued functions with respect to vector measures. The general theory can be found in [36, 37, 32], [44, Ch. I II] and [67, 124], for example. For applications beyond these texts we refer to [38, 66, 80, 102, 117] and the references therein, and the survey articles [33, 68]. Each of these references emphasizes its own preferences, as will be the case with this article. Our aim is to present some theoretical developments over the past 15 years or so (see §1) and to highlight some recent applications. Due to space limitation we restrict the applications to two topics. Namely, the extension of certain operators to their optimal domain (see §2) and aspects of spectral integration (see §3). The interaction between order and positivity with properties of the integration map of a vector measure (which is defined on a function space) will become apparent and plays a central role.

A frame is a complete lattice in which finite meets distribute over arbitrary joins.

Frames have only recently made a formal entry into the development of lattice-ordered groups. On the other hand, the work of Paul Conrad and some of his students of the sixties and seventies, analyzing a lattice-ordered group through its lattice of convex ℓ-subgroups, is frame theory in disguise. In more recent work, pure frame theory has found application to problems in ℓ-groups, producing, in several cases, theorems which had not been possible with more traditional techniques. And now this turning of the tables has been taken a step further: proving theorems from the theory of ℓ-groups in frame-theoretic settings, without invoking the Axiom of Choice or other axioms which imply the existence of points in spectra.

This article aims to inform and convince the reader: inform, in broad terms, and convince that the phenomena discussed in the preceding paragraph constitute an honorable research activity. This is a survey article of modest length: selectivity is a must - with the choices of illustrations being left, for good or ill, to the taste and prejudices of the author.

The exposition is in three parts, following the three (chronological) aspects of the role of frame theory in the development of ℓ-groups. First up is the famous theorem of Conrad on finite-valued ℓ-groups. This is followed by an account of dimension theory, particularly as it applies to the ℓ-dimension of rings of continuous functions. Finally, there is an account of the recent and ongoing work on the epicompletion in a category of regular frames, and related issues concerning archimedean frames.

In this paper we survey some aspects of the theory of non-commutative Banach function spaces, that is, spaces of measurable operators associated with a semi- finite von Neumann algebra. These spaces are also known as non-commutative symmetric spaces. The theory of such spaces emerged as a common generalization of the theory of classical (“commutative”) rearrangement invariant Banach function spaces (in the sense of W.A.J. Luxemburg and A.C. Zaanen) and of the theory of symmetrically normed ideals of bounded linear operators in Hilbert space (in the sense of I.C. Gohberg and M.G. Krein). These two cases may be considered as the two extremes of the theory: in the first case the underlying von Neumann algebra is the commutative algebra L _∞ on some measure space (with integration as trace); in the second case the underlying von Neumann algebra is B (ℌ), the algebra of all bounded linear operators on a Hilbert space ℌ (with standard trace). Important special cases of these non-commutative spaces are the non-commutative L _p-spaces, which correspond in the commutative case with the usual L _p-spaces on a measure space, and in the setting of symmetrically normed operator ideals they correspond to the Schatten p-classes \( \mathfrak{S}_p \) .

Throughout this paper we denote by L ^p the Banach lattice of p-integrable functions on a σ-finite measure space (X, B, μ), where 1 ≤ p ≤ ∞. We will consider those aspects of the theory of positive linear operators, which are in some way special due to the fact the operators are acting on L ^p-spaces. For general information about positive operators on Banach lattices we refer to the texts [1]. [20], and [36]. Our focus on L ^p-spaces does not mean that in special cases some of the results can not be extended to a larger class of Banach lattices of measurable function such as Orlicz spaces or re-arrangement invariant Banach function spaces. However in many cases the results in these extensions are not as precise or as complete as in the case of L ^p-spaces. We will discuss results related to the boundedness of positive linear operators on L ^p-spaces. The most important result is the so-called Schur criterion for boundedness. This criterion is the most frequently used tool to show that a concrete positive linear operator is bounded from L ^p to L ^q. Then we will show how this result relates to the change of density result of Weis [33]. Next the equality case of Schur’s criterion is shown to be closely related to the question whether a given positive linear operator attains its norm. We discuss in detail the properties of norm attaining operators on L ^p-spaces and discuss as an example the weighted composition operators on L ^p-spaces. Then we return to the Schur criterion and show how it can be applied to the factorization theorems of Maurey and Nikišin. Most results mentioned in this paper have appeared before in print, but sometimes only implicitly and scattered over several papers. Also a number of the proofs presented here are new.

If X and Y are Banach lattices then there are several spaces of linear operators between them that may be studied. \( \mathcal{L}^r \) (X, Y) is the space of all norm bounded operators from X into Y. There is no reason to expect there to be any connection between the order structure of X and Y and that of \( \mathcal{L} \) (X, Y). \( \mathcal{L}^r \) (X, Y) is the space of regular operators, i.e., the linear span of the positive operators. This at least has the merit that when it is ordered by the cone of positive operators then that cone is generating. \( \mathcal{L}^b \) (X, Y) is the space of order bounded operators, which are those that map order bounded sets in X to order bounded sets in Y. We always have \( \mathcal{L}^r (X,Y) \subseteq \mathcal{L}^b (X,Y) \subseteq \mathcal{L}(X,Y) \) and both inclusions may be proper.