Semisimplicity of B(E)″

Daws, Matthew; Read, C. J.

doi:10.1016/j.jfa.2004.04.014

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…For a discussion of this result, see [12,Chapter 6]. It is interesting that, for 1 < p < ∞, the second dual algebra (B( p ) , 2) is semisimple if and only if p = 2 [17].…”

Section: Background and Notationmentioning

confidence: 99%

Weighted convolution algebras on subsemigroups of the real line

Dales

Dedania

2009

Dissertationes Math.

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In this memoir, we shall consider weighted convolution algebras on discrete groups and semigroups, concentrating on the group (Q, +) of rational numbers, the semigroup (Q +• , +) of strictly positive rational numbers, and analogous semigroups in the real line R. In particular, we shall discuss when these algebras are Arens regular, when they are strongly Arens irregular, and when they are neither, giving a variety of examples. We introduce the notion of 'weakly diagonally bounded' weights, weakening the known concept of 'diagonally bounded' weights, and thus obtaining more examples. We shall also construct an example of a weighted convolution algebra on N that is neither Arens regular nor strongly Arens irregular, and an example of a weight ω on Q +• such that lim infs→0+ ω(s) = 0. 2000 Mathematics Subject Classification: Primary 46H20; Secondary 43A20.

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“…For a discussion of this result, see [12,Chapter 6]. It is interesting that, for 1 < p < ∞, the second dual algebra (B( p ) , 2) is semisimple if and only if p = 2 [17].…”

Section: Background and Notationmentioning

confidence: 99%

Weighted convolution algebras on subsemigroups of the real line

Dales

Dedania

2009

Dissertationes Math.

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“…In [16] it is shown that semisimple B(ℓ p ) fails to have a semisimple second dual if p = 2. This can also happen for operator algebras: (1) If A * * is semiprime (resp.…”

Section: Example: Adjoining a Root To An Algebramentioning

confidence: 99%

Ideals and hereditary subalgebras in operator algebras

2012

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This paper may be viewed as having two aims. First, we continue our study of algebras of operators on a Hilbert space which have a contractive approximate identity, this time from a more Banach algebraic point of view. Namely, we mainly investigate topics concerned with the ideal structure, and hereditary subalgebras (HSA's), which are in some sense generalization of ideals. Second, we study properties of operator algebras which are hereditary subalgebras in their bidual, or equivalently which are 'weakly compact'. We also give several examples answering natural questions that arise in such an investigation.

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“…The study has not been restricted to those Banach algebras coming from abstract harmonic analysis. One particularly striking result is a theorem of Daws and Read [6] which states that, for 1 < p < ∞, the algebra B(ℓ p ) ′′ is semisimple if and only if p = 2.…”

Section: Introductionmentioning

confidence: 99%

The radical of the bidual of a Beurling algebra

White

2018

The Quarterly Journal of Mathematics

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We prove that the bidual of a Beurling algebra on Z, considered as a Banach algebra with the first Arens product, can never be semisimple. We then show that rad (ℓ 1 (⊕ ∞ i=1 Z) ′′ ) contains nilpotent elements of every index. Each of these results settles a question of Dales and Lau. Finally we show that there exists a weight ω on Z such that the bidual of ℓ 1 (Z, ω) contains a radical element which is not nilpotent.

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Semisimplicity of B(E)″

Cited by 3 publications

References 9 publications

Weighted convolution algebras on subsemigroups of the real line

Weighted convolution algebras on subsemigroups of the real line

Ideals and hereditary subalgebras in operator algebras

The radical of the bidual of a Beurling algebra

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