On a characterization of the maximal ideal spaces of algebraically closed commutative C*-algebras

摘要

Let C(X) be the algebra of all complex-valued continuous functions on a compact Hausdorff space X. We say that C(X) is algebraically closed if each monic polynomial equation over C(X) has a continuous solution. We give a necessary and sufficient condition for C(X) to be algebraically closed for a locally connected compact Hausdorff space X. In this case, it is proved that C( X) is algebraically closed if each element of C(X) is the square of another. We also give a characterization of a first-countable compact Hausdorff space X such that C(X) is algebraically closed. [References: 10]