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## Pointwise approximate identities in Banach function algebras

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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 -