A remark on the homomorphism on 𝐶(𝑋)

Ercan, Zafer; Önal, Süleyman

doi:10.1090/s0002-9939-05-07930-x

Cited by 14 publications

(8 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Later, Shirota [52] showed that X is realcompact if and only if to each algebra homomorphism ϕ from C (X) onto the real field R there corresponds a point x of X such that ϕ (f ) = f (x) for all f ∈ C (X). This remarkable necessary and sufficient condition for a Tychonoff space to be realcompact was obtained very recently by Ercan and Önal [26] via an elementary approach. These observations will be used next to obtain an alternative characterization of realcompact spaces which is a little more fit for our study.…”

Section: Concrete Situationsmentioning

confidence: 74%

Recent Trends on Order Bounded Disjointness Preserving Operators

Boulabiar¹

2008

Irish Math. Soc. Bull.

View full text Add to dashboard Cite

Section: Concrete Situationsmentioning

confidence: 74%

Recent Trends on Order Bounded Disjointness Preserving Operators

Boulabiar¹

2008

Irish Math. Soc. Bull.

View full text Add to dashboard Cite

“…It should be pointed out that the idea of using projections in such a context is essentially due do to Aron‐Fricke in and Ercan‐Onal in .…”

Section: Realcompact Functionsmentioning

confidence: 99%

Non-affine functions and realcompact spaces

2013

View full text Add to dashboard Cite

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).

show abstract

“…Proof of Theorem 1.1. In the proof we mainly follow the arguments of [3], so to be as self contained as possible, we repeat some parts of the proof therein. As indicated in [1], the implications (i) =⇒ (ii) and (ii) =⇒ (iii) are obvious.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%