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Matrix Compact Sets and Operator Approximation Properties

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Corran Webster

Functional Analysis · Submitted 20 Apr 1998

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper we introduce ``compact matrix sets'' which correspond to the two different operator approximation properties, and show that a space has the operator approximation property if and only if the ``operator compact'' operators are contained in the closure of the finite rank operators.

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arXiv IDmath/9804093 (version 1)
Pith ID2RIOIBSP, Pith's permanent record of this paper and its review
CitationCorran Webster. "Matrix Compact Sets and Operator Approximation Properties." arXiv:math/9804093 (1998). https://pith.science/paper/2RIOIBSP
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,607 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.78 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

The relationship between the operator approximation property and the strong operator approximation property has deep significance in the theory of operator algebras. The original definitions of Effros and Ruan, unlike the classical analogues, make no mention of compact operators or compact sets. In this paper we introduce ``compact matrix sets'' which correspond to the two different operator approximation properties, and show that a space has the operator approximation property if and only if the ``operator compact'' operators are contained in the closure of the finite rank operators. We also investigate when the two types of compactness agree, and introduce a natural condition which guarantees that they do.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The relationship between the operator approximation property and the strong operator approximation property has deep significance in the theory of operator algebras. background 1.00 0.04 0.03 asserted 1.00
s1 The original definitions of Effros and Ruan, unlike the classical analogues, make no mention of compact operators or compact sets. background 0.79 0.06 0.02 asserted 1.00
s2 In this paper we introduce ``compact matrix sets'' which correspond to the two different operator approximation properties, and show that a space has the operator approximation property if and only if the ``operator compact'' operators are contained in the closure of the finite rank operators. contribution 0.89 0.96 0.84 asserted 1.00
s3 We also investigate when the two types of compactness agree, and introduce a natural condition which guarantees that they do. contribution 0.89 0.94 0.07 asserted 0.92

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

The relationship between the operator approximation property and the strong operator approximation property has deep significance in the theory of operator algebras. The original definitions of Effros and Ruan, unlike the classical analogues, make no mention of compact operators or compact sets. In this paper we introduce ``compact matrix sets'' which correspond to the two different operator approximation properties, and show that a space has the operator approximation property if and only if the ``operator compact'' operators are contained in the closure of the finite rank operators. We also investigate when the two types of compactness agree, and introduce a natural condition which guarantees that they do.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it

Cites it as background 1 paper
  • The notion of complete compactness we use is an operator space version of compactness which was introduced by Saar [ref] and subsequently studied by Oikhberg [ref] and Webster [ref] .

    Compactness properties of operator multipliers · Introduction · 0.98

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