A function algebra of the second degree on non-localness

Let $A$ be a function algebra with uniform convergence containing the constants, and let $\mathfrak M_A$ be its maximal ideal space. A continuous function $f$ on $\mathfrak M_A$ is called $f$-local if it coincides, in a neighborhood of each point $m\in\mathfrak M_A$, with some function from the algebra $A$. The algebra $A$ is called local if it contains all $A$-local functions, and nonlocal otherwise. A well-known example of a nonlocal algebra has been constructed by E. Kallin. She also raised the question: is there a smallest local closed subalgebra in $C(\mathfrak M_A)$ containing all the $A$-local functions?
In this work we give a negative answer to this question. The appropriate algebra is realized as a subalgebra in $C(S)$, where $S$ is a compactum in $C^5$, and is generated by acertain family of rational functions.
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