A theorem of Gel'fand–Mazur type

Hung Le Pham ¹

¹ Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta T6G 2G1, Canada

Studia Mathematica, Tome 191 (2009) no. 1, pp. 81-88

Résumé

Denote by ${\mathfrak c}$ any set of cardinality continuum. It is proved that a Banach algebra $A$ with the property that for every collection $\{a_\alpha :\alpha\in{\mathfrak c}\}\subset A$ there exist $\alpha\neq \beta\in{\mathfrak c}$ such that $a_\alpha\in a_\beta A^\#$ is isomorphic to \[ \bigoplus_{i=1}^r ({\mathbb C}[X]/X^{d_i}{\mathbb C}[X]) \oplus E, \] where $d_1,\ldots, d_r\in\mathbb N$, and $E$ is either $X{\mathbb C}[X]/X^{d_0}{\mathbb C}[X]$ for some $d_0\in\mathbb N$ or a $1$-dimensional $\bigoplus_{i=1}^r {\mathbb C}[X]/X^{d_i}{\mathbb C}[X]$-bimodule with trivial right module action. In particular, ${\mathbb C}$ is the unique non-zero prime Banach algebra satisfying the above condition.

Détail

DOI : 10.4064/sm191-1-6

Mots-clés : denote mathfrak set cardinality continuum proved banach algebra property every collection alpha alpha mathfrak subset there exist alpha neq beta mathfrak alpha beta isomorphic bigoplus mathbb mathbb oplus where ldots mathbb either mathbb mathbb mathbb dimensional bigoplus mathbb mathbb bimodule trivial right module action particular mathbb unique non zero prime banach algebra satisfying above condition