Continuous vector bundles over topological algebras

On Topological Algebra Bundles

“…So, one considers a triplet £ = (E, IT, X), where -n: E -» X is a given map of a set E into a topological space X, the latter map being further specified, up to equivalence, by "algebraic" atlases. The preceding is patterned after certain current results of A. Mallios [11,12], as well as some ideas in the above quoted work of B. R. Gelbaum. More particularly, in Section 1 we deal with standard facts of the general theory of topological algebra bundles getting thus results analogous to those of the classical theory of vector bundles [7] (see Theorem 1.3, Remark 1.1). More specifically, we give a characterization of a topological algebra bundle through a system of transition functions on X (Theorem 1.2), a technique which is systematically applied throughout the rest of the paper.…”

“…We also define the zero section O(JC) := y a (x, 0) and moreover, if the algebras M a have identities l a , we still define 1(JC):= <p a (x, l a ), x G C/ a , these sections being also independent of the choice of U a ; furthermore, for all y e F(|), one gets So, let £ be a topological algebra and ^U(K, E) the algebra of continuous £-valued functions on a compact space K endowed with the topology of uniform convergence in K. Then, by definition of the topology u, % U (K, E) is a topological algebra (see [12,Lemma 2.1]). Moreover, the algebra of continuous £-valued functions on a topological space X endowed with the topology of compact convergence in X (denoted by ^( A ' , £)) is also a topological algebra, as this follows by…”

“…So, let £ be a topological algebra and ^U(K, E) the algebra of continuous £-valued functions on a compact space K endowed with the topology of uniform convergence in K. Then, by definition of the topology u, % U (K, E) is a topological algebra (see [12,Lemma 2.1]). Moreover, the algebra of continuous £-valued functions on a topological space X endowed with the topology of compact convergence in X (denoted by ^( A ' , £)) is also a topological algebra, as this follows by…”

Homotopy classification of module bundles via Grassmannians

“…On the other hand, A. Mallios [6] gave a cohomological classification of R-bundles of finite rank n, via GL(n, R)-cocycles, with R being a unital locally m-convex algebra.…”