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Amenability of Banach algebras of compact operators

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Niels Gronbaek, Barry E. Johnson, George A. Willis

Functional Analysis · Submitted 8 Sep 1992

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

The authors' main claim We give a symmetrized approximation property of X which is proved to be such a condition.

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arXiv IDmath/9209211 (version 1)
Pith ID2KVWFUXC, Pith's permanent record of this paper and its review
CitationBarry E. Johnson, George A. Willis, Niels Gronbaek. "Amenability of Banach algebras of compact operators." arXiv:math/9209211 (1992). https://pith.science/paper/2KVWFUXC
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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.65 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.11 · mentions code or data availability 0.01.

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

In this paper we study conditions on a Banach space X that ensure that the Banach algebra K(X) of compact operators is amenable. We give a symmetrized approximation property of X which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. For direct sums this question is closely related to factorization of linear operators. In the final section we discuss some open questions, in particular, the converse problem of what properties of X are implied by the amenability of K(X).

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this paper we study conditions on a Banach space X that ensure that the Banach algebra K(X) of compact operators is amenable. contribution 0.80 0.76 0.08 asserted 0.54
s1 We give a symmetrized approximation property of X which is proved to be such a condition. contribution 0.65 0.95 0.43 asserted 1.00
s2 This property is satisfied by a wide range of Banach spaces including all the classical spaces. result 0.82 0.72 0.14 asserted 1.00
s3 We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. contribution 1.00 0.89 0.05 unclear 0.51
s4 Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. result 0.96 0.85 0.14 asserted 0.77
s5 For direct sums this question is closely related to factorization of linear operators. interpretation 0.70 0.50 0.04 asserted 0.95
s6 In the final section we discuss some open questions, in particular, the converse problem of what properties of X are implied by the amenability of K(X). future work 0.95 0.59 0.04 unclear 0.74

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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

In this paper we study conditions on a Banach space X that ensure that the Banach algebra K(X) of compact operators is amenable. We give a symmetrized approximation property of X which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. For direct sums this question is closely related to factorization of linear operators. In the final section we discuss some open questions, in particular, the converse problem of what properties of X are implied by the amenability of K(X).

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