A generalization of space of maps (Q1326735)

Let \(A\) be a Fréchet *-algebra and let \(M\) be a smooth manifold, with \(C^ \infty (M)\) the Fréchet *-algebra of complex valued smooth functions. Then let \(M(A)\) be the space of all continuous *-algebra homomorphisms \(\varphi: C^ \infty (M)\to A\) which have compact support in the obvious sense (adapted from the theory of distributions). \(M(A)\) is equipped with the topology of pointwise convergence. It is proved that \(M\mapsto M(A)\) is a covariant functor from the category of smooth finite dimensional manifolds into the category of topological spaces, which even preserves products in a certain weak sense (the natural equivalence \((M_ 1 \times M_ 2) (A)\to M_ 1 (A)\times M_ 2(A)\) is continuous and bijective, but the inverse is in general only sequentially continuous). If \(N\) is a smooth compact manifold then \(M (C^ \infty (N))= C^ \infty (N,M)\), and similar results are also hinted at for Sobolev completions, and this explains the title. Finally it is shown that \(M(A)\) is a smooth manifold itself, modelled on Fréchet spaces. Reviewer's remarks: The ideas in this paper are well known in the field called `synthetic differential geometry' as used in the book by \textit{A. Kock} [`Synthetic differential geometry' (1981; Zbl 0466.51008)], the view nearest to this paper is in the book by \textit{I. Moerdijk} and \textit{G. E. Reyes} [`Models for smooth infinitesimal analysis', Springer-Verlag (1991; Zbl 0715.18001)]. A careful study of the manifold structure on \(M(A)\) for finite dimensional \(A\) can be found in the following papers: \textit{D. J. Eck} [Product preserving functors on smooth manifolds, J. Pure Appl. Algebra 42, 133-140 (1986; Zbl 0615.57019)], \textit{G. Kainz} and the reviewer [Czech. Math. J. 37(112), 584-607 (1987; Zbl 0654.58001)] and \textit{O. O. Luciano} [Nagoya Math. J. 109, 69-89 (1988; Zbl 0661.58007)]. The difficulties with product come from the choice of topology on \(M(A)\); the system of bounded sets inherited from \(L (C^ \infty (M), A)\) is the better choice.