A theorem of Gel'fand-Mazur type

Hung Le Pham

A theorem of Gel'fand-Mazur type

Hung Le Pham

Studia Mathematica (2009)

Volume: 191, Issue: 1, page 81-88
ISSN: 0039-3223

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Abstract

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Denote by any set of cardinality continuum. It is proved that a Banach algebra A with the property that for every collection

a_{α} : α \in \subset A

there exist α ≠ β ∈ such that

a_{α} \in a_{β} A^{}

is isomorphic to

⨁_{i = 1}^{r} (ℂ [X] / X^{d_{i}} ℂ [X]) \oplus E

, where

d ₁, . . ., d_{r} \in ℕ

, and E is either

X ℂ [X] / X^{d ₀} ℂ [X]

for some d₀ ∈ ℕ or a 1-dimensional

⨁_{i = 1}^{r} ℂ [X] / X^{d_{i}} ℂ [X]

-bimodule with trivial right module action. In particular, ℂ is the unique non-zero prime Banach algebra satisfying the above condition.

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Hung Le Pham. "A theorem of Gel'fand-Mazur type." Studia Mathematica 191.1 (2009): 81-88. <http://eudml.org/doc/284875>.

@article{HungLePham2009,
abstract = {Denote by any set of cardinality continuum. It is proved that a Banach algebra A with the property that for every collection $\{a_\{α\}: α ∈ \} ⊂ A$ there exist α ≠ β ∈ such that $a_\{α\} ∈ a_\{β\}A^\{#\}$ is isomorphic to $⨁_\{i=1\}^\{r\} (ℂ[X]/X^\{d_\{i\}\}ℂ[X]) ⊕ E$, where $d₁,...,d_\{r\} ∈ ℕ$, and E is either $Xℂ[X]/X^\{d₀\}ℂ[X]$ for some d₀ ∈ ℕ or a 1-dimensional $⨁_\{i=1\}^\{r\} ℂ[X]/X^\{d_\{i\}\}ℂ[X]$-bimodule with trivial right module action. In particular, ℂ is the unique non-zero prime Banach algebra satisfying the above condition.},
author = {Hung Le Pham},
journal = {Studia Mathematica},
keywords = {Banach algebra; prime algebra; radical algebra; Fréchet algebra; principal ideal; Gelfand-Mazur type theorem},
language = {eng},
number = {1},
pages = {81-88},
title = {A theorem of Gel'fand-Mazur type},
url = {http://eudml.org/doc/284875},
volume = {191},
year = {2009},
}

TY - JOUR
AU - Hung Le Pham
TI - A theorem of Gel'fand-Mazur type
JO - Studia Mathematica
PY - 2009
VL - 191
IS - 1
SP - 81
EP - 88
AB - Denote by any set of cardinality continuum. It is proved that a Banach algebra A with the property that for every collection ${a_{α}: α ∈ } ⊂ A$ there exist α ≠ β ∈ such that $a_{α} ∈ a_{β}A^{#}$ is isomorphic to $⨁_{i=1}^{r} (ℂ[X]/X^{d_{i}}ℂ[X]) ⊕ E$, where $d₁,...,d_{r} ∈ ℕ$, and E is either $Xℂ[X]/X^{d₀}ℂ[X]$ for some d₀ ∈ ℕ or a 1-dimensional $⨁_{i=1}^{r} ℂ[X]/X^{d_{i}}ℂ[X]$-bimodule with trivial right module action. In particular, ℂ is the unique non-zero prime Banach algebra satisfying the above condition.
LA - eng
KW - Banach algebra; prime algebra; radical algebra; Fréchet algebra; principal ideal; Gelfand-Mazur type theorem
UR - http://eudml.org/doc/284875
ER -

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