Characterizing algebras of smooth functions on manifolds.pdf

arXiv:math/9404228v1 [] 1 Apr 1994 To appear in: Comm. Math. Univ. Carolinae CHARACTERIZING ALGEBRAS OF SMOOTH FUNCTIONS ON MANIFOLDS Peter W. Michor Ji ˇr ′? Van ˇzura Erwin Schr¨odinger International Institute of Mathematical Physics, Wien, Austria J. Vanˇzura: Mathematical Institute of the AV ˇCR, department Brno, Ziˇzkova 22, CZ 616 62 Brno, Czech Republic April 8, 1994 allC ∞-algebras we characterize those which are algebras of smooth functions on smooth separable Hausdor? manifolds. ∞--algebra is mutative ringAwith unit together with a ring homomorphismR→A. Then every mapp:R n→R mwhich is given by anm-tuple of real polynomials (p 1, . . . , p m) can be interpreted as a mapping A(p) :A n→A min such a way that projections, composition, and identity are preserved, by just evaluating each polynomialp ion ann-tuple (a 1, . . . , a n)∈A n. AC ∞-algebraAis a real algebra in which we can moreover interpret all smooth mappingsf:R n→R m. There is a corresponding mapA(f) :A n→A m, and again projections, composition, and the identity mapping are preserved. More precisely, aC ∞-algebraAis a product preserving functor fromthe category C ∞to the category of sets, whereC ∞has as objects all spacesR n,n≥0, and all smooth mappings between them as arrows. Morphisms betweenC ∞-algebras are then natural transformations: they correspond to those