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Pointwise approximate identities in Banach function algebras

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Pointwise approximate identities in Banach function algebras. / Dales, G.; Ulger, Ali.

In: Dissertationes Mathematicae (Rozprawy Matematyczne), 10.08.2020.

Research output: Contribution to journalJournal article

Harvard

Dales, G & Ulger, A 2020, 'Pointwise approximate identities in Banach function algebras', Dissertationes Mathematicae (Rozprawy Matematyczne).

APA

Dales, G., & Ulger, A. (Accepted/In press). Pointwise approximate identities in Banach function algebras. Dissertationes Mathematicae (Rozprawy Matematyczne).

Vancouver

Dales G, Ulger A. Pointwise approximate identities in Banach function algebras. Dissertationes Mathematicae (Rozprawy Matematyczne). 2020 Aug 10.

Author

Dales, G. ; Ulger, Ali. / Pointwise approximate identities in Banach function algebras. In: Dissertationes Mathematicae (Rozprawy Matematyczne). 2020.

Bibtex

@article{51084cc82c7949b99ee3e266822cd53e,
title = "Pointwise approximate identities in Banach function algebras",
abstract = "In this memoir, we shall study Banach function algebras that have bounded pointwise approximate identities, and especially those that have contractive pointwise approximate identities. ABanach function algebra A is (pointwise) contractive if A and every non-zero, maximal modular ideal in A have contractive (pointwise) approximate identities. Let A be a Banach function algebra with character space Phi_A. We shall show that the existence of a contractive pointwise approximate identity in A depends closely on whether ||varphi|| =1 for each varphi in Phi_A$. The linear span of Phi_A in the dual space A' is denoted by L(A), and this is used to define the BSE norm on A; the algebra A has a BSE norm if this norm is equivalent to the given norm. We shall then introduce and study in some detail the quotient Banach function algebra {mathcal Q}(A)= A''/L(A)^\perp; we shall give various examples, especially uniform algebras and those involving algebras that are standard in abstract harmonic analysis, including Segal algebras with respect to the group algebra of a locally compact group. We shall characterize the Banach function algebrasfor which overline{L(A)}= \ell^{1}(\Phi_A), and then classify contractive and pointwise contractive algebras in the class of unital Banach function algebras that have a BSE norm; they are uniform algebras with specific properties. We shall also give examples of such algebras that do not have a BSE norm. Finally we shall discuss when some classical Banach function algebras of harmonic analysis have non-trivial reflexive closed ideals, and make some remarks on weakly compact homo\-morphisms between Banach function algebras",
author = "G. Dales and Ali Ulger",
year = "2020",
month = aug,
day = "10",
language = "English",
journal = "Dissertationes Mathematicae (Rozprawy Matematyczne)",
issn = "0012-3862",
publisher = "Institute of Mathematics, Polish Academy of Sciences",

}

RIS

TY - JOUR

T1 - Pointwise approximate identities in Banach function algebras

AU - Dales, G.

AU - Ulger, Ali

PY - 2020/8/10

Y1 - 2020/8/10

N2 - In this memoir, we shall study Banach function algebras that have bounded pointwise approximate identities, and especially those that have contractive pointwise approximate identities. ABanach function algebra A is (pointwise) contractive if A and every non-zero, maximal modular ideal in A have contractive (pointwise) approximate identities. Let A be a Banach function algebra with character space Phi_A. We shall show that the existence of a contractive pointwise approximate identity in A depends closely on whether ||varphi|| =1 for each varphi in Phi_A$. The linear span of Phi_A in the dual space A' is denoted by L(A), and this is used to define the BSE norm on A; the algebra A has a BSE norm if this norm is equivalent to the given norm. We shall then introduce and study in some detail the quotient Banach function algebra {mathcal Q}(A)= A''/L(A)^\perp; we shall give various examples, especially uniform algebras and those involving algebras that are standard in abstract harmonic analysis, including Segal algebras with respect to the group algebra of a locally compact group. We shall characterize the Banach function algebrasfor which overline{L(A)}= \ell^{1}(\Phi_A), and then classify contractive and pointwise contractive algebras in the class of unital Banach function algebras that have a BSE norm; they are uniform algebras with specific properties. We shall also give examples of such algebras that do not have a BSE norm. Finally we shall discuss when some classical Banach function algebras of harmonic analysis have non-trivial reflexive closed ideals, and make some remarks on weakly compact homo\-morphisms between Banach function algebras

AB - In this memoir, we shall study Banach function algebras that have bounded pointwise approximate identities, and especially those that have contractive pointwise approximate identities. ABanach function algebra A is (pointwise) contractive if A and every non-zero, maximal modular ideal in A have contractive (pointwise) approximate identities. Let A be a Banach function algebra with character space Phi_A. We shall show that the existence of a contractive pointwise approximate identity in A depends closely on whether ||varphi|| =1 for each varphi in Phi_A$. The linear span of Phi_A in the dual space A' is denoted by L(A), and this is used to define the BSE norm on A; the algebra A has a BSE norm if this norm is equivalent to the given norm. We shall then introduce and study in some detail the quotient Banach function algebra {mathcal Q}(A)= A''/L(A)^\perp; we shall give various examples, especially uniform algebras and those involving algebras that are standard in abstract harmonic analysis, including Segal algebras with respect to the group algebra of a locally compact group. We shall characterize the Banach function algebrasfor which overline{L(A)}= \ell^{1}(\Phi_A), and then classify contractive and pointwise contractive algebras in the class of unital Banach function algebras that have a BSE norm; they are uniform algebras with specific properties. We shall also give examples of such algebras that do not have a BSE norm. Finally we shall discuss when some classical Banach function algebras of harmonic analysis have non-trivial reflexive closed ideals, and make some remarks on weakly compact homo\-morphisms between Banach function algebras

M3 - Journal article

JO - Dissertationes Mathematicae (Rozprawy Matematyczne)

JF - Dissertationes Mathematicae (Rozprawy Matematyczne)

SN - 0012-3862

ER -