Intermediate rings of complex-valued continuous functions

2020/01/27 by Acharyya, Amrita, Acharyya, Sudip Kumar, Bag, Sagarmoy +1
#FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.2001.09659

Abstract

Let Σ(X,ℂ) denote the collection of all the rings between C^*(X,ℂ) and C(X,ℂ). We show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal ideals/z-ideals/z^∘-ideals in the rings P(X,ℂ) in Σ(X,ℂ) and in their real-valued counterparts P(X,ℂ)∩ C(X). It is shown that the structure space of any such P(X,ℂ) is βX. We show that for any maximal ideal M in C(X,ℂ), C(X,ℂ)/M is an algebraically closed field. We give a necessary and sufficient condition for the ideal C_P(X,ℂ) of C(X,ℂ) to be a prime ideal, and we examine a few special cases thereafter.

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