Premium
Non‐affine functions and realcompact spaces
Author(s) -
Azouzi Youssef,
Benamor Fethi,
Boulabiar Karim
Publication year - 2014
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200327
Subject(s) - mathematics , homomorphism , affine transformation , banach space , pure mathematics , bounded function , identity function , space (punctuation) , discrete mathematics , mathematical analysis , philosophy , linguistics
Let ψ ∈ C ( R ) and X be a topological space. An identity preserving a bounded map h : C ( X ) → R is called a ψ‐ homomorphism if h is additive and h ∘ ψ = ψ ∘ h . We call ψ a realcompact function if, whenever X is a realcompact space, any ψ‐homomorphism h : C ( X ) → R is an evaluation at some point of X . By classical results of Hewitt and Shirota, respectively, the square as well as the absolute value are examples of realcompact functions. This paper extends these results and gives a complete description of realcompact functions. Indeed, it turns out that ψ ∈ C ( R ) is a realcompact function if and only if ψ is non‐affine. This leads to a Banach‐Stone type theorem, namely, the realcompact spaces X and Y are homeomorphic if and only if C ( X ) and C ( Y ) are ψ‐isomorphic for some non‐affine function ψ ∈ C ( R ) .