Arens Regularity of the Algebra of Operators on a Banach Space

Daws, Matthew

doi:10.1112/s0024609303003072

Cited by 16 publications

(27 citation statements)

References 10 publications

(7 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It was proved by Daws in 2004 [15] that the converse is almost true, in that B(E) is Arens regular whenever E is super-reflexive. Thus the Banach algebras B( p ), defined for 1 ≤ p ≤ ∞, are Arens regular if and only if 1 < p < ∞.…”

Section: Background and Notationmentioning

confidence: 99%

Weighted convolution algebras on subsemigroups of the real line

Dales

Dedania

2009

Dissertationes Math.

View full text Add to dashboard Cite

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.

show abstract

Section: Background and Notationmentioning

confidence: 99%

Weighted convolution algebras on subsemigroups of the real line

Dales

Dedania

2009

Dissertationes Math.

View full text Add to dashboard Cite

show abstract

“…In [3], it was shown that BðEÞ; the Banach algebra of operators on a Banach space, is Arens regular whenever E is super-reflexive. The proof uses an injective homomorphism BðEÞ 00 -BðF Þ (for either Arens product) where F is another reflexive Banach space-one can take F ¼ ðl 2 ðEÞÞ U where ðl 2 ðEÞÞ U is an ultrapower.…”

Section: Introduction and Algebraic Backgroundmentioning

confidence: 99%

Semisimplicity of B(E)″

Daws¹,

Read²

2005

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

We study the semi-simplicity of the second dual of the Banach algebra of operators on a Banach space, BðEÞ 00 ; endowed with either Arens product. It was previously shown that if E is a Hilbert space, then BðEÞ is Arens regular and BðEÞ 00 is semi-simple. We show that for a large class of Banach spaces E; including subspaces of L p spaces not isomorphic to a Hilbert space, BðEÞ 00 is not semi-simple. This is achieved by deriving a new representation of Bðl p Þ 0 ; and then constructing a member of the radical of Bðl p Þ 00 ; for pa2: r

show abstract

“…As noted in [10], if A is a closed subalgebra of B(E) for a super-reflexive E (or A is super-reflexive) then every even dual of A is Arens regular.…”

Section: Arens Regularitymentioning

confidence: 99%

“…Ultrapowers of Banach spaces have been used in [2] to study representations of Banach algebras and representations of groups; see also the similar ideas used in [10] and [29].…”

Section: Proposition 21 For a Banach Algebra A An Ultrapower (A) Umentioning

confidence: 99%

See 1 more Smart Citation

Amenability of ultrapowers of Banach algebras

Daws

2009

Proceedings of the Edinburgh Mathematical Society

Self Cite

View full text Add to dashboard Cite

We study when certain properties of Banach algebras are stable under ultrapower constructions. In particular, we consider when every ultrapower of A is Arens regular, and give some evidence that this is so if and only if A is isomorphic to a closed subalgebra of operators on a super-reflexive Banach space. We show that such ideas are closely related to whether one can sensibly define an ultrapower of a dual Banach algebra. We study how tensor products of ultrapowers behave, and apply this to study the question of when every ultrapower of A is amenable. We provide an abstract characterisation in terms of something like an approximate diagonal, and consider when every ultrapower of a C * -algebra, or a group L 1 -convolution algebra, is amenable.

show abstract

Arens Regularity of the Algebra of Operators on a Banach Space

Abstract: A short proof is given that if E is a super-reflexive Banach space, then B(E), the Banach algebra of operators on E with composition product, is Arens regular. Some remarks are made on necessary conditions on E for B(E) to be Arens regular.

Cited by 16 publications

References 10 publications

Weighted convolution algebras on subsemigroups of the real line

Weighted convolution algebras on subsemigroups of the real line

Semisimplicity of B(E)″

Amenability of ultrapowers of Banach algebras

Contact Info

Product

Resources

About