S ep 2 00 0 Infinite - dimensional Grassmann - Banach algebras
نویسنده
چکیده
A short review on infinite-dimensional Grassmann-Banach algebras (IDGBA) is presented. Starting with the simplest IDGBA over K = R with l 1-norm (suggested by A. Rogers), we define a more general IDGBA over complete normed field K with l 1-norm and set of generators of arbitrary power. Any l 1-type IDGBA may be obtained by action of Grassmann-Banach functor of projective type on certain l 1-space. In non-Archimedean case there exists another possibility for constructing of IDGBA using the Grassmann-Banach functor of injective type. Infinite-dimensional Grassmann-Banach algebras (IDGBA) and their modifications are key objects for infinite-dimensional versions of superanalysis (see [1]-[5] and references therein). They are generalizations of finite-dimensional Grassmann algebras to infinite-dimensional Banach case (for infinite-dimensional topological Grassmann algebras see also [6]). Any IDGBA is an associative Banach algebra with unit over some complete normed field K [7], whose linear space G is a Banach space with the norm ||.|| satisfying ||a · b|| ≤ ||a||||b|| for all a, b ∈ G and ||e|| = 1, where e is the unit. (For applications in superanalysis K should be non-discrete,
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