Ground algebras for superanalysis
It is generally agreed now that any reasonable version of superanalysis should include an algebra serving as a supply of odd (anticommuting) constants (so to say, `superconstants'); the presence of such an algebra is inevitable when dealing with supersymmetry and many other matters of primary importance from the point of view of intended applications [see, e.g., \textit{A. Rogers}, Commun. Math. Phys. 105, 375-384 (1986; Zbl 0602.58003)]. Such an algebra should be a `Grassmannian' one, and there are numerous attempts to make this latter term precise. The most well- known among algebras of super-constants exploited so far are: the usual finite-dimensional Grassmann algebras; the Banach-Grassmann algebras [cf. \textit{A. Jadczyk} and \textit{K. Pilch}, ibid. 78, 373-390 (1981; Zbl 0464.58006)], the DeWitt-Fréchet algebra [cf. \textit{B. S. DeWitt}, Supermanifolds (1984; Zbl 0551.53002)], a nuclear LB-algebra introduced by \textit{Sh. Nagamachi} and \textit{Y. Kobayashi} [Lett. Math. Phys. 14, 15- 23 (1987; Zbl 0638.46054)] etc. The present paper is an attempt to systematize all those examples and propose more. The notion `supernumber algebra' is given a precise meaning: so is called a Hausdorff topological associative unital graded commutative algebra decomposing into the direct sum of the scalar field and the closed ideal topologically generated by the odd part. In particular, many new examples of Banach-Grassmann algebras are constructed, in addition to the only one previously known since the work of \textit{A. Rogers} [J. Math. Phys. 21, 1352-1365 (1980; Zbl 0447.58003)].
Hausdorff topological associative unital graded commutative
Hausdorff topological associative unital graded commutative algebra
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