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\(\overline{1}\)-evaluating linear functionals on function spaces. (English) Zbl 1434.46018

Summary: Let \(C(X)\) denotes the Riesz algebra of all real valued continuous functions on a Tychonoff space \(X\), and let \(L\) be a vector subspace of \(C(X)\). A linear functional \(H:L\rightarrow \mathbb{R}\) is said to be \(\overline{1}\)-evaluating if and only if \(H(f) \in \overline{f(X)}\) for all \(f\in L\). In this work, we investigate \(\overline{1}\)-evaluating linear functionals on certain vector subspaces of \(C(X)\). Our main result in this direction asserts that, when \(L\) is a Riesz subalgebra of \(C(X)\) that contains idempotents, any linear functional \(H:L\rightarrow\mathbb{R}\) is \(\overline{1}\)-evaluating if and only if it acts like a Riesz homomorphism on boundedly clean vector subspaces of \(L\).

MSC:

46E25 Rings and algebras of continuous, differentiable or analytic functions
54C40 Algebraic properties of function spaces in general topology
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References:

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