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The order σ -complete vector lattice of AM-compact operators

Belmesnaoui Aqzzouz, Redouane Nouira (2009)

Czechoslovak Mathematical Journal

We establish necessary and sufficient conditions under which the linear span of positive AM-compact operators (in the sense of Fremlin) from a Banach lattice E into a Banach lattice F is an order σ -complete vector lattice.

The Orthogonal Projection and the Riesz Representation Theorem

Keiko Narita, Noboru Endou, Yasunari Shidama (2015)

Formalized Mathematics

In this article, the orthogonal projection and the Riesz representation theorem are mainly formalized. In the first section, we defined the norm of elements on real Hilbert spaces, and defined Mizar functor RUSp2RNSp, real normed spaces as real Hilbert spaces. By this definition, we regarded sequences of real Hilbert spaces as sequences of real normed spaces, and proved some properties of real Hilbert spaces. Furthermore, we defined the continuity and the Lipschitz the continuity of functionals...

The path space of a higher-rank graph

Samuel B. G. Webster (2011)

Studia Mathematica

We construct a locally compact Hausdorff topology on the path space of a finitely aligned k-graph Λ. We identify the boundary-path space ∂Λ as the spectrum of a commutative C*-subalgebra D Λ of C*(Λ). Then, using a construction similar to that of Farthing, we construct a finitely aligned k-graph Λ̃ with no sources in which Λ is embedded, and show that ∂Λ is homeomorphic to a subset of ∂Λ̃. We show that when Λ is row-finite, we can identify C*(Λ) with a full corner of C*(Λ̃), and deduce that D Λ is isomorphic...

The Point of Continuity Property: Descriptive Complexity and Ordinal Index

Bossard, Benoit, López, Ginés (1998)

Serdica Mathematical Journal

∗ Supported by D.G.I.C.Y.T. Project No. PB93-1142Let X be a separable Banach space without the Point of Continuity Property. When the set of closed subsets of its closed unit ball is equipped with the standard Effros-Borel structure, the set of those which have the Point of Continuity Property is non-Borel. We also prove that, for any separable Banach space X, the oscillation rank of the identity on X (an ordinal index which quantifies the Point of Continuity Property) is determined by the subspaces...

The positive cone of a Banach lattice. Coincidence of topologies and metrizability

Zbigniew Lipecki (2023)

Commentationes Mathematicae Universitatis Carolinae

Let X be a Banach lattice, and denote by X + its positive cone. The weak topology on X + is metrizable if and only if it coincides with the strong topology if and only if X is Banach-lattice isomorphic to l 1 ( Γ ) for a set Γ . The weak * topology on X + * is metrizable if and only if X is Banach-lattice isomorphic to a C ( K ) -space, where K is a metrizable compact space.

The Poulsen simplex

Joram Lindenstrauss, Gunnar Olsen, Y. Sternfeld (1978)

Annales de l'institut Fourier

It is proved that there is a unique metrizable simplex S whose extreme points are dense. This simplex is homogeneous in the sense that for every 2 affinely homeomorphic faces F 1 and F 2 there is an automorphism of S which maps F 1 onto F 2 . Every metrizable simplex is affinely homeomorphic to a face of S . The set of extreme points of S is homeomorphic to the Hilbert space ℓ 2 . The matrices which represent A ( S ) are characterized.

The Powers sum of spatial CPD-semigroups and CP-semigroups

Michael Skeide (2010)

Banach Center Publications

We define spatial CPD-semigroups and construct their Powers sum. We construct the Powers sum for general spatial CP-semigroups. In both cases, we show that the product system of that Powers sum is the product of the spatial product systems of its factors. We show that on the domain of intersection, pointwise bounded CPD-semigroups on the one side and Schur CP-semigroups on the other, the constructions coincide. This summarizes all known results about Powers sums and generalizes them considerably....

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