A theorem of Gel'fand–Mazur type

Volume 191 / 2009

Hung Le Pham Studia Mathematica 191 (2009), 81-88 MSC: Primary 46H05. DOI: 10.4064/sm191-1-6


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.


  • Hung Le PhamDepartment of Mathematical and Statistical Sciences
    University of Alberta
    Edmonton, Alberta T6G 2G1, Canada

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