On geometric topological algebras (Q2367621)

The paper under review, one of related works by the author [see also the review above], is influenced by, and subsumed under, a programme of sheaf-theoretic treatment of differential geometry, hence the term ``geometric'' for the algebras in the title. Indeed, they are the ones which can be represented as the global section spaces of certain soft sheaves of topological algebras, each of them constructed over the spectrum of the initial algebra (Gel'fand sheaves) [cf. also \textit{B. Kramm}, J. Funct. Anal. 37, 249-269 (1980; Zbl 0405.46046)]. Typical examples are \(C^ \infty(X)\) on a suitable differentiable manifold \(X\), \(O(K)\) with \(K\) a Stein compact set [compare with \textit{P. de Bartolomeis}, Rend. Accad. Naz. XL, V. Ser. 1-2, 105-144 (1976; Zbl 0429.32011) and/or \textit{J. L. Taylor}, Pac. J. Math. 103, 163-203 (1982; Zbl 0528.13007)], \(C_ c(X)\), where \(X\) is a completely regular Hausdorff space etc. Thus, their class being convenient for localization, is interesting for various cohomological computations. The author provides sufficient conditions under which a given topological algebra is a geometric one and mentions several applications.