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A discontinuous function does not operate on the real part of a function algebra. (English) Zbl 0577.46054

Let A be a function algebra on a compact space X and let h be a real valued function defined on an interval I. We say that h operates by composition on Re A\(=\{Ref:\) \(f\in A\}\) iff \(h\circ u\in Re A\) whenever \(u\in Re A\) has the range in I. It was an old conjecture that if h operates by composition on Re A and h is not affine, then \(A=C(X)\). S. J. Sidney [Pac. J. Math. 80, 265-272 (1979; Zbl 0377.46042)] and O. Hatori [Proc. Am. Math. Soc. 83, 565-568 (1981; Zbl 0493.46045)] proved that the answer is positive for any continuous function h. In this note we prove that the answer is also positive for a non-continuous function h. In this case we get even more:
Theorem. A non-continuous function h operates by composition on the real part of a function algebra A iff A is finite dimensional.

MSC:

46J10 Banach algebras of continuous functions, function algebras
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