Remarks on perturbations of function algebras
Let A be a Banach algebra. By an \(\epsilon\)-perturbation of A we mean any multiplication \(\times\) defined on the Banach space A such that \(\| f\times g-f\cdot g\| \leq \epsilon \| f\| \| g\|\), for f,g\(\in A\). An algebra A is said to be \(\epsilon\)-stable if for any \(\epsilon\)-perturbation \(\times\) of A the algebras A and (A,\(\times)\) are isomorphic. The author proves that if \(\{A_{\lambda}\}\) is a family of \(\epsilon\)- stable function algebras then \(\oplus_{\lambda}A_{\lambda}\), a direct sum of algebras \(\{A_{\lambda}\}\), is also \(\epsilon '\)-stable, for some \(\epsilon '>0\). The converse implication is proved under an additional assumption that the Choquet boundary of \(A_{\lambda}\) is connected for each \(\lambda\in \Lambda\). It seems to be an open question whether the converse implication holds without this additional assumption.
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