The order topology for a von Neumann algebra

Emmanuel Chetcuti; Jan Hamhalter; Hans Weber

Displaying similar documents to “The order topology for a von Neumann algebra”

A hit-and-miss topology for $2^{X}$ , Cₙ(X) and Fₙ(X)

Benjamín Espinoza, Verónica Martínez-de-la-Vega, Jorge M. Martínez-Montejano (2009)

Colloquium Mathematicae

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A hit-and-miss topology ( $τ_{H M}$ ) is defined for the hyperspaces $2^{X}$ , Cₙ(X) and Fₙ(X) of a continuum X. We study the relationship between $τ_{H M}$ and the Vietoris topology and we find conditions on X for which these topologies are equivalent.

Further new generalized topologies via mixed constructions due to Császár

Erdal Ekici (2015)

Mathematica Bohemica

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The theory of generalized topologies was introduced by Á. Császár (2002). In the literature, some authors have introduced and studied generalized topologies and some generalized topologies via generalized topological spaces due to Á. Császár. Also, the notions of mixed constructions based on two generalized topologies were introduced and investigated by Á. Császár (2009). The main aim of this paper is to introduce and study further new generalized topologies called $μ_{12}^{C}$ via mixed constructions...

The Spaces of Closed Convex Sets in Euclidean Spaces with the Fell Topology

Katsuro Sakai, Zhongqiang Yang (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $C o n v_{F} (ℝ ⁿ)$ be the space of all non-empty closed convex sets in Euclidean space ℝ ⁿ endowed with the Fell topology. We prove that $C o n v_{F} (ℝ ⁿ) \approx ℝ ⁿ \times Q$ for every n > 1 whereas $C o n v_{F} (ℝ) \approx ℝ \times$ .

A compact Hausdorff topology that is a T₁-complement of itself

Dmitri Shakhmatov, Michael Tkachenko (2002)

Fundamenta Mathematicae

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Topologies τ₁ and τ₂ on a set X are called T₁-complementary if τ₁ ∩ τ₂ = X∖F: F ⊆ X is finite ∪ ∅ and τ₁∪τ₂ is a subbase for the discrete topology on X. Topological spaces $(X, τ_{X})$ and $(Y, τ_{Y})$ are called T₁-complementary provided that there exists a bijection f: X → Y such that $τ_{X}$ and $f^{- 1} (U) : U \in τ_{Y}$ are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size $2^{}$ which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact...

Locally $G_{δ}$ -spaces

Zdeněk Frolík (1962)

Czechoslovak Mathematical Journal

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Sequential closures of $σ$ -subalgebras for a vector measure

Werner J. Ricker (1996)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a locally convex space, $m : Σ \to X$ be a vector measure defined on a $σ$ -algebra $Σ$ , and $L^{1} (m)$ be the associated (locally convex) space of $m$ -integrable functions. Let $Σ (m)$ denote ${χ_{_{E}}; E \in Σ}$ , equipped with the relative topology from $L^{1} (m)$ . For a subalgebra $𝒜 \subseteq Σ$ , let $𝒜_{σ}$ denote the generated $σ$ -algebra and ${\bar{𝒜}}_{s}$ denote the closure of $χ (𝒜) = {χ_{_{E}}; E \in 𝒜}$ in $L^{1} (m)$ . Sets of the form ${\bar{𝒜}}_{s}$ arise in criteria determining separability of $L^{1} (m)$ ; see [6]. We consider some natural questions concerning ${\bar{𝒜}}_{s}$ and, in particular, its relation to $χ (𝒜_{σ})$ . It is shown that...

Inverse topology in MV-algebras

Fereshteh Forouzesh, Farhad Sajadian, Mahta Bedrood (2019)

Mathematica Bohemica

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We introduce the inverse topology on the set of all minimal prime ideals of an MV-algebra $A$ and show that the set of all minimal prime ideals of $A$ , namely $Min (A)$ , with the inverse topology is a compact space, Hausdorff, $T_{0}$ -space and $T_{1}$ -space. Furthermore, we prove that the spectral topology on $Min (A)$ is a zero-dimensional Hausdorff topology and show that the spectral topology on $Min (A)$ is finer than the inverse topology on $Min (A)$ . Finally, by open sets of the inverse topology, we define and study a congruence...

The AR-Property of the spaces of closed convex sets

Katsuro Sakai, Masato Yaguchi (2006)

Colloquium Mathematicae

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Let $C o n v_{H} (X)$ , $C o n v_{A W} (X)$ and $C o n v_{W} (X)$ be the spaces of all non-empty closed convex sets in a normed linear space X admitting the Hausdorff metric topology, the Attouch-Wets topology and the Wijsman topology, respectively. We show that every component of $C o n v_{H} (X)$ and the space $C o n v_{A W} (X)$ are AR. In case X is separable, $C o n v_{W} (X)$ is locally path-connected.

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

S. Gabriyelyan, J. Kąkol, G. Plebanek (2016)

Studia Mathematica

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Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of $C_{k} (X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{ℝ}$ -space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space $C_{k} (X)$ is Ascoli iff $C_{k} (X)$ is a $k_{ℝ}$ -space iff X is locally compact. Moreover, $C_{k} (X)$ endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability...

Some properties of the topology of $α$ -sets

D. Andrijević (1984)

Matematički Vesnik

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Subharmonicity in von Neumann algebras

Thomas Ransford, Michel Valley (2005)

Studia Mathematica

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Let ℳ be a von Neumann algebra with unit $1_{ℳ}$ . Let τ be a faithful, normal, semifinite trace on ℳ. Given x ∈ ℳ, denote by $μ_{t} {(x)}_{t \geq 0}$ the generalized s-numbers of x, defined by $μ_{t} (x)$ = inf||xe||: e is a projection in ℳ i with $τ (1_{ℳ} - e)$ ≤ t (t ≥ 0). We prove that, if D is a complex domain and f:D → ℳ is a holomorphic function, then, for each t ≥ 0, $λ \mapsto \int_{0}^{t} l o g μ_{s} (f (λ)) d s$ is a subharmonic function on D. This generalizes earlier subharmonicity results of White and Aupetit on the singular values of matrices.