Algebraic Real Analysis

An effort to initiate the subject of the title: the basic tool is the study of the abstract closed interval equipped with certain equational structures. The title is wishful thinking; there ought to be a subject that deserves the name “algebraic real analysis.” Herein is a possible beginning. For reasons that can easily be considered abstruse we were led to the belief that the closed interval—not the entire real line—is the basic structure of interest. Before describing those abstruse reasons, a theorem: Let G be a compact group and I the closed interval. (We will not say which closed interval; to do so would define it as a part of the reals, belying the view of the closed interval as the fundamental structure.) Let C(G) be the set of continuous maps from G to I. We wish to view this as an algebraic structure, where the word “algebra” is in the very general sense, something described by operations and equations. In the case at hand, the only operators that will be considered right now are the constants, “top” and “bottom,” denoted > and ⊥, and the binary operation of “midpointing,” denoted x|y. (There are axioms that will define the notion of “closed midpoint algebra” but since the theorem is about specific examples they’re not now needed.) C(G) inherits this algebraic structure in the usual way (f |g, for example, is the map that sends σ ∈ G to (fσ)|(gσ) ∈ I). We use the group structure on G to define an action of G on C(G), thus obtaining a representation of G on the group of automorphisms of the closed midpoint algebra. (Fortunately no knowledge of the axioms is necessary for the definition of automorphism, or even homomorphism.) Let (C(G),I) be the set of closed-midpoint-algebra homomorphisms from C(G) to I. Again we obtain an action of G. 0.1. Theorem. There is a unique G-fixed point in (C(G),I) There is an equivalent way of stating this: Special thanks to Mike Barr and Don von Osdol for editorial assistance and to the Executive Director of the FMS Foundation for making it all possible. Received by the editors 2008-02-09 and, in revised form, 2008-06-25. Transmitted by Michael Barr. Published on 2008-07-02. 2000 Mathematics Subject Classification: 03B45, 03B50, 03B70, 03D15, 03F52, 03F55, 03G20, 03G25, 03G30, 08A99, 18B25, 18B30, 18F20, 26E40, 28E99, 46M99, 34A99.