On some algebraic characteristics of the algebra of all continuous functions on a locally connected compactum

In the first part of this paper the algebra $C(X)$ is studied, and in the case of a locally connected compactum $X$ a characteristic of the algebra $C(X)$ is given from the point of view of the plentitude of roots of certain algebraic equations that it contains. In the second part a general method is given for constructing uniform algebras $A$ on suitable compacta $X$ which are different from $C(X)$ but have a number of properties in common with $C(X)$ (normality, algebraic closure, complete closure, etc.). In particular, these methods allow us to give, as a general concept, a new solution to a problem of Gleason concerning peak points.
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