Unbounded Non-Commutative Integration

Jurzak, J. P.

doi:10.1007/978-94-009-5231-7

Cited by 2 publications

(4 citation statements)

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“…Contrary to current ideas, these spaces are far from being pathological [7] and are very adapted to the functional analysis (this is due to the order structure associated to the cone of positive functions).…”

Section: Preliminariesmentioning

confidence: 99%

“…, where (v, w) stands for the scalar product of vectors v, w in H. We refer to [7] for a general non-commutative study. …”

Section: Simple Factsmentioning

confidence: 99%

“…with the locally convex topology inductive limit (not strict in general) of the sequence of normed spaces (A Φ j , · Φ j ) j≥0 : following [7], we use the notation for this topology and (A, ) for the locally convex topological vector space so defined. We note that the map (5.1) restricted to A is generally undefined because it sends A outside itself.…”

Section: More Generally For a Linear Spacementioning

confidence: 99%

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Dominated Convergence and Stone-Weierstrass Theorem

Jurzak¹

2005

Journal of Applied Analysis

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Abstract. Let C(X; R) the algebra of continuous real valued functions defined on a locally compact space X. We consider linear subspaces A ⊂ C(X; R) having the following property: there is a sequence (Φj) j∈N of positive functions in A with limx→∞ Φj(x) = +∞ for every j ∈ N, such that A consists of functions f ∈ C(X; R) bounded above for the absolute value by an homothetic of some Φn (n depends on each f ). Dominated convergence of a sequence (gn) n≥1 in A is an estimation of the form |gn(x) − g(x)| ≤ εn|h(x)| for allx ∈ X and all n ∈ N where gn, g, h ∈ A and εn → 0 as n → ∞. We extend the Stone-Weierstrass theorem to subalgebras or lattices B ⊂ A and we show that the dominated convergence for sequences is exactly the convergence of sequences when A is endowed with a locally convex (DF)-space topology.

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Section: Preliminariesmentioning

confidence: 99%

“…, where (v, w) stands for the scalar product of vectors v, w in H. We refer to [7] for a general non-commutative study. …”

Section: Simple Factsmentioning

confidence: 99%

Section: More Generally For a Linear Spacementioning

confidence: 99%

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Dominated Convergence and Stone-Weierstrass Theorem

Jurzak¹

2005

Journal of Applied Analysis

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“…During the past 30 years many works concerning (topological) algebras of unbounded operators and their physical applications appeared in the literature (see e.g. [1,10,12,22] and references therein).…”

Section: Introductionmentioning

confidence: 99%