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The $σ$ -property of a Riesz space (real vector lattice) $B$ is: For each sequence ${b_{n}}$ of positive elements of $B$ , there is a sequence ${λ_{n}}$ of positive reals, and $b \in B$ , with $λ_{n} b_{n} \leq b$ for each $n$ . This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “ $σ$ ” obtains for a Riesz space of continuous real-valued functions $C (X)$ . A basic result is: For discrete $X$ , $C (X)$ has $σ$ iff the cardinal $| X | < 𝔟$ , Rothberger’s bounding number. Consequences and...

Theorems of the alternative for cones and Lyapunov regularity of matrices

Bryan Cain, Daniel Hershkowitz, Hans Schneider (1997)

Czechoslovak Mathematical Journal

Standard facts about separating linear functionals will be used to determine how two cones $C$ and $D$ and their duals $C^{*}$ and $D^{*}$ may overlap. When $T V \to W$ is linear and $K \subset V$ and $D \subset W$ are cones, these results will be applied to $C = T (K)$ and $D$ , giving a unified treatment of several theorems of the alternate which explain when $C$ contains an interior point of $D$ . The case when $V = W$ is the space $H$ of $n \times n$ Hermitian matrices, $D$ is the $n \times n$ positive semidefinite matrices, and $T (X) = A X + X^{*} A$ yields new and known results about the existence of block diagonal...

Theorems of the Stone-Weierstrass type for lattices of uniformly continuous functions.

Л.Н. Мохова (1981)

Sibirskij matematiceskij zurnal

Topological properties and matrix transformations of certain ordered generalized sequence spaces.

Gupta, Manjul, Kaushal, Kalika (1995)

International Journal of Mathematics and Mathematical Sciences

Torsion classes and subdirect products of Carathéodory vector lattices

Ján Jakubík (2006)

Mathematica Slovaca

Trace inequalities for spaces in spectral duality

O. Tikhonov (1993)

Studia Mathematica

Let (A,e) and (V,K) be an order-unit space and a base-norm space in spectral duality, as in noncommutative spectral theory of Alfsen and Shultz. Let t be a norm lower semicontinuous trace on A, and let φ be a nonnegative convex function on ℝ. It is shown that the mapping a → t(φ(a)) is convex on A. Moreover, the mapping is shown to be nondecreasing if so is φ. Some other similar statements concerning traces and real-valued functions are also obtained.