An Algebraic Characterization for Rings of Continuous Functions on Compact Topological Spaces
by
V. K. Zakharov
Moscow State University
Coauthors: A.A. Seredinski (Moscow State University)

In 1975, J.-P. Delfosse announced the remarkable result [1].

Theorem. A commutative ring A with a unity 1 is isomorphic to a ring C(K) of all continuous real-valued functions on a compact topological space K iff it satisfies the following properties: 1) for every a, b in A there exists c in A such that a² + b² = c²; 2) for every a in A there exist b in A and c in A such that a=b² - c² and bc=0; 3) for every a there exists (1+a²)^-1; 4) if for given a there exists a sequence (b_n | n in N) such that n(a² + b²_n) = 1 then a=0; 5) for every a there are b in A and n in N such that a² + b² = n1; 6) if (a_n) is a sequence for which there is a sequence (m_k in N | k in N) such that k((a_m - a_n)² + b²)=1 for all m, n >= m_k and for corresponding b = b(k, m, n), then there exists a for which there exists a sequence (n_k | k in N ) such that k((a - a_n)² + c²) = 1 for all n >= n_k and for corresponding c = c(k, n) .

We have found no proof of this result. Here we give another characterization for C(K), where assertions 4)-6) are substituted by the following ones: 4') if for given a there exists a sequence b_n such that n²(a² + b²_n) = 1, then a = 0; 5') for every a there are b in A and n in N such that a² + b² = n²1; 6') if a_n is a sequence for which there exists a sequence (m_k in N) such that k²((a_m - a_n)² + b²) = 1 for all m, n >= m_k and for corresponding b = b(k, m, n) then there exists a for which there is a sequence n_k such that k²((a - a_n)² + c²) = 1 for all n >= n_k and for corresponding c = c(k, n).

Theorem. A commutative ring A with a unity 1 is isomorphic to a ring C(K) of all continuous real-valued functions on a compact topological space K iff it satisfies properties 1) - 3), 4'), 5'), and 6').

We discuss the problem of characterization for a ring C(K, C) of all continuous complex-valued functions on a compact topological space K .

References

1. J.-P. Delfosse, Caracterizations d'anneaux de fonctions continues, Ann. Soc. Sci. Bruxelles, ser.1 89(1975), 364-368.

Date received: May 18, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeu-10.