# Complex homomorphisms on self-conjugate algebras of functions

Complex homomorphisms on self-conjugate algebras of functions Let X be a non-void set and A be a subalgebra of $${\mathbb{C}^{X}}$$ . We call a $${\mathbb{C}}$$ -linear functional $${\varphi}$$ on A a 1-evaluation if $${\varphi(f) \in f(X) }$$ for all $${f\in A}$$ . From the classical Gleason–Kahane–Żelazko theorem, it follows that if X in addition is a compact Hausdorff space then a mapping $${\varphi}$$ of $${C_{\mathbb{C}}(X) }$$ into $${\mathbb{C}}$$ is a 1-evaluation if and only if $${\varphi}$$ is a $${\mathbb{C}}$$ -homomorphism. In this paper, we aim to investigate the extent to which this equivalence between 1-evaluations and $${\mathbb{C}}$$ -homomorphisms can be generalized to a wider class of self-conjugate subalgebras of $${\mathbb{C}^{X}}$$ . In this regards, we prove that a $${\mathbb{C}}$$ -linear functional on a self-conjugate subalgebra A of $${\mathbb{C}^{X}}$$ is a positive $${\mathbb{C}}$$ -homomorphism if and only if $${\varphi}$$ is a $${\overline{1}}$$ -evaluation, that is, $${\varphi(f) \in\overline{f\left(X\right)}}$$ for all $${f\in A}$$ . As consequences of our general study, we prove that 1-evaluations and $${\mathbb{C}}$$ -homomorphisms on $${C_{\mathbb{C}}\left( X\right)}$$ coincide for any topological space X and we get a new characterization of realcompact topological spaces. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Complex homomorphisms on self-conjugate algebras of functions

, Volume 14 (4) – Jun 10, 2010
9 pages

/lp/springer_journal/complex-homomorphisms-on-self-conjugate-algebras-of-functions-OoAM00UnDt
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-010-0067-7
Publisher site
See Article on Publisher Site

### Abstract

Let X be a non-void set and A be a subalgebra of $${\mathbb{C}^{X}}$$ . We call a $${\mathbb{C}}$$ -linear functional $${\varphi}$$ on A a 1-evaluation if $${\varphi(f) \in f(X) }$$ for all $${f\in A}$$ . From the classical Gleason–Kahane–Żelazko theorem, it follows that if X in addition is a compact Hausdorff space then a mapping $${\varphi}$$ of $${C_{\mathbb{C}}(X) }$$ into $${\mathbb{C}}$$ is a 1-evaluation if and only if $${\varphi}$$ is a $${\mathbb{C}}$$ -homomorphism. In this paper, we aim to investigate the extent to which this equivalence between 1-evaluations and $${\mathbb{C}}$$ -homomorphisms can be generalized to a wider class of self-conjugate subalgebras of $${\mathbb{C}^{X}}$$ . In this regards, we prove that a $${\mathbb{C}}$$ -linear functional on a self-conjugate subalgebra A of $${\mathbb{C}^{X}}$$ is a positive $${\mathbb{C}}$$ -homomorphism if and only if $${\varphi}$$ is a $${\overline{1}}$$ -evaluation, that is, $${\varphi(f) \in\overline{f\left(X\right)}}$$ for all $${f\in A}$$ . As consequences of our general study, we prove that 1-evaluations and $${\mathbb{C}}$$ -homomorphisms on $${C_{\mathbb{C}}\left( X\right)}$$ coincide for any topological space X and we get a new characterization of realcompact topological spaces.

### Journal

PositivitySpringer Journals

Published: Jun 10, 2010

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