After collecting a number of results on interval and almost interval preserving lin-ear maps and vector lattice homomorphisms, we show that direct systems in variouscategories of normed vector... Show moreAfter collecting a number of results on interval and almost interval preserving lin-ear maps and vector lattice homomorphisms, we show that direct systems in variouscategories of normed vector lattices and Banach lattices have direct limits, and thatthese coincide with direct limits of the systems in naturally associated other cate-gories. For those categories where the general constructions do not work to establishthe existence of general direct limits, we describe the basic structure of those directlimits that do exist. A direct system in the category of Banach lattices and contractivealmost interval preserving vector lattice homomorphisms has a direct limit. Whenthe Banach lattices in the system all have order continuous norms, then so does theBanach lattice in a direct limit. This is used to show that a Banach function space overa locally compact Hausdorff space has an order continuous norm when the topologieson all compact subsets are metrisable and (the images of) the continuous compactlysupported functions are dense. Show less
Let G be a Polish locally compact group acting on a Polish space X" role="presentation">X with a G-invariant probability measure μ" role="presentation">μ. We factorize the integral with respect to... Show moreLet G be a Polish locally compact group acting on a Polish space X" role="presentation">X with a G-invariant probability measure μ" role="presentation">μ. We factorize the integral with respect to μ" role="presentation">μ in terms of the integrals with respect to the ergodic measures on X, and show that Lp(X,μ)" role="presentation">Lp(X,μ) (1≤p<∞" role="presentation">1≤p<∞) is G-equivariantly isometrically lattice isomorphic to an Lp" role="presentation">Lp-direct integral of the spaces Lp(X,λ)" role="presentation">Lp(X,λ), where λ" role="presentation">λ ranges over the ergodic measures on X. This yields a disintegration of the canonical representation of G as isometric lattice automorphisms of Lp(X,μ)" role="presentation">Lp(X,μ) as an Lp" role="presentation">Lp-direct integral of order indecomposable representations. If (X′,μ′)" role="presentation">(X′,μ′) is a probability space, and, for some 1≤q<∞" role="presentation">1≤q<∞, G acts in a strongly continuous manner on Lq(X′,μ′)" role="presentation">Lq(X′,μ′) as isometric lattice automorphisms that leave the constants fixed, then G acts on Lp(X′,μ′)" role="presentation">Lp(X′,μ′) in a similar fashion for all 1≤p<∞" role="presentation">1≤p<∞. Moreover, there exists an alternative model in which these representations originate from a continuous action of G on a compact Hausdorff space. If (X′,μ′)" role="presentation">(X′,μ′) is separable, the representation of G on Lp(X′,μ′)" role="presentation">Lp(X′,μ′) can then be disintegrated into order indecomposable representations. The notions of Lp" role="presentation">Lp-direct integrals of Banach spaces and representations that are developed extend those in the literature. Show less
Let A be a Banach algebra with a bounded left approximate identity {eλ}λ∈Λ" role="presentation">{eλ}λ∈Λ, let π" role="presentation">π be a continuous representation of A on a Banach space X, and let SShow moreLet A be a Banach algebra with a bounded left approximate identity {eλ}λ∈Λ" role="presentation">{eλ}λ∈Λ, let π" role="presentation">π be a continuous representation of A on a Banach space X, and let S be a non-empty subset of X such that limλπ(eλ)s=s" role="presentation">limλπ(eλ)s=s uniformly on S. If S is bounded, or if {eλ}λ∈Λ" role="presentation">{eλ}λ∈Λ is commutative, then we show that there exist a∈A" role="presentation">a∈A and maps xn:S→X" role="presentation">xn:S→X for n≥1" role="presentation">n≥1 such that s=π(an)xn(s)" role="presentation">s=π(an)xn(s) for all n≥1" role="presentation">n≥1 and s∈S" role="presentation">s∈S. The properties of a∈A" role="presentation">a∈A and the maps xn" role="presentation">xn, as produced by the constructive proof, are studied in some detail. The results generalize previous simultaneous factorization theorems as well as Allan and Sinclair’s power factorization theorem. In an ordered context, we also consider the existence of a positive factorization for a subset of the positive cone of an ordered Banach space that is a positive module over an ordered Banach algebra with a positive bounded left approximate identity. Such factorizations are not always possible. In certain cases, including those for positive modules over ordered Banach algebras of bounded functions, such positive factorizations exist, but the general picture is still unclear. Furthermore, simultaneous pointwise power factorizations for sets of bounded maps with values in a Banach module (such as sets of bounded convergent nets) are obtained. A worked example for the left regular representation of C0(R)" role="presentation">C0(R) and unbounded S is included. Show less
