Nonassociative normed algebras: geometric aspects

Angel Rodríguez Palacios

Banach Center Publications (1994)

  • Volume: 30, Issue: 1, page 299-311
  • ISSN: 0137-6934

Abstract

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Introduction. The aim of this paper is to review some relevant results concerning the geometry of nonassociative normed algebras, without assuming in the first instance that such algebras satisfy any familiar identity, like associativity, commutativity, or Jordan axiom. In the opinion of the author, the most impressive fact in this direction is that most of the celebrated natural geometric conditions that can be required for associative normed algebras, when imposed on a general nonassociative normed algebra, imply that the algebra is actually "nearly associative". We shall explain this idea by selecting four favourite topics, namely: • Nonassociative Vidav-Palmer theorem, • Nonassociative Gelfand-Naimark theorem, • Nonassociative smooth normed algebras, and • One-sided division absolute valued algebras. Although there are classical nice forerunners in this circle of ideas, as for example the Albert-Urbanik-Wright determination of (nonassociative) absolute valued algebras with a unit ([2], [3], [42], and [41]), a systematic treatment of questions of this type has been made only recently, more precisely since 1980 [34].

How to cite

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Rodríguez Palacios, Angel. "Nonassociative normed algebras: geometric aspects." Banach Center Publications 30.1 (1994): 299-311. <http://eudml.org/doc/262555>.

@article{RodríguezPalacios1994,
abstract = { Introduction. The aim of this paper is to review some relevant results concerning the geometry of nonassociative normed algebras, without assuming in the first instance that such algebras satisfy any familiar identity, like associativity, commutativity, or Jordan axiom. In the opinion of the author, the most impressive fact in this direction is that most of the celebrated natural geometric conditions that can be required for associative normed algebras, when imposed on a general nonassociative normed algebra, imply that the algebra is actually "nearly associative". We shall explain this idea by selecting four favourite topics, namely: • Nonassociative Vidav-Palmer theorem, • Nonassociative Gelfand-Naimark theorem, • Nonassociative smooth normed algebras, and • One-sided division absolute valued algebras. Although there are classical nice forerunners in this circle of ideas, as for example the Albert-Urbanik-Wright determination of (nonassociative) absolute valued algebras with a unit ([2], [3], [42], and [41]), a systematic treatment of questions of this type has been made only recently, more precisely since 1980 [34]. },
author = {Rodríguez Palacios, Angel},
journal = {Banach Center Publications},
keywords = {nonassociative Vidav-Palmer theorem; nonassociative Gelfand-Naimark theorem; nonassociative smooth normed algebras; one-sided division absolute valued algebras; geometry of nonassociative normed algebras; noncommutative -algebra; Gelfand-Naimark axiom; Gelfand-Mazur theorem},
language = {eng},
number = {1},
pages = {299-311},
title = {Nonassociative normed algebras: geometric aspects},
url = {http://eudml.org/doc/262555},
volume = {30},
year = {1994},
}

TY - JOUR
AU - Rodríguez Palacios, Angel
TI - Nonassociative normed algebras: geometric aspects
JO - Banach Center Publications
PY - 1994
VL - 30
IS - 1
SP - 299
EP - 311
AB - Introduction. The aim of this paper is to review some relevant results concerning the geometry of nonassociative normed algebras, without assuming in the first instance that such algebras satisfy any familiar identity, like associativity, commutativity, or Jordan axiom. In the opinion of the author, the most impressive fact in this direction is that most of the celebrated natural geometric conditions that can be required for associative normed algebras, when imposed on a general nonassociative normed algebra, imply that the algebra is actually "nearly associative". We shall explain this idea by selecting four favourite topics, namely: • Nonassociative Vidav-Palmer theorem, • Nonassociative Gelfand-Naimark theorem, • Nonassociative smooth normed algebras, and • One-sided division absolute valued algebras. Although there are classical nice forerunners in this circle of ideas, as for example the Albert-Urbanik-Wright determination of (nonassociative) absolute valued algebras with a unit ([2], [3], [42], and [41]), a systematic treatment of questions of this type has been made only recently, more precisely since 1980 [34].
LA - eng
KW - nonassociative Vidav-Palmer theorem; nonassociative Gelfand-Naimark theorem; nonassociative smooth normed algebras; one-sided division absolute valued algebras; geometry of nonassociative normed algebras; noncommutative -algebra; Gelfand-Naimark axiom; Gelfand-Mazur theorem
UR - http://eudml.org/doc/262555
ER -

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