The Stone-Weierstrass property in quotient algebras, and sets of spectral resolution (Q1320956)

We investigate the Stone-Weierstrass property in quotient Fourier algebras. Combining the results by \textit{P. Malliavin} [C. R. Acad. Sci., Paris, Sér. A 248, 1756-1759 (1959; Zbl 0188.204); 2155-2157 (1959; Zbl 0188.204), Inst. Haut. Études Sci., Publ. Math. 2, 61-68 (1959; Zbl 0101.094)], \textit{Y. Katznelson} and \textit{W. Rudin} [Pac. J. Math. 11, 253-265 (1961; Zbl 0102.328)], we can observe that if an LCA group \(G\) is a set of spectral resolution, then the Fourier algebra \(A(G)\) has the Stone-Weierstrass property, and that the converse does not follow. In this paper, we obtain the same conclusion for quotient Fourier algebras. That is, if \(E\) is a set of spectral resolution in a finite-dimensional metrizable LCA group, then \(A(E)\) has the Stone-Weierstrass property (Theorem 3.16). Although the converse of Theorem 3.16 does not follow, the following theorem gives a partial converse of Theorem 3.16: If a closed set \(E\) of spectral synthesis in the maximal ideal space of a regular semi-simple commutative Banach algebra \(A\) contains a compact set of non-synthesis, then the quotient algebra \(A/I(E)\) contains a strongly separating subspace (not necessarily a subalgebra) that is not dense in \(A/I(E)\). We define two classes of new sets: Stone-Weierstrass sets and Idempotent sets. A closed set \(E\) in \(G\) is said to be a Stone-Weierstrass set if \(A(E)\) has the Stone-Weierstrass property. A closed set \(E\) in \(G\) is said to be an Idempotent set if \(A(E)\) is spanned by its set of idempotents. These sets share many common properties with sets of spectral synthesis and sets of spectral resolution. We also discuss the union problem of these new classes of sets, and the transfer method by Varopoulos applied to non-Stone-Weierstrass sets. As a byproduct, we obtain a new upper bound for the Helson constant of a union of the type \(H \cup \{H+x\}\). If the Helson constant of a Helson set \(H\) is \(\alpha\), then the Helson constant for \(H \cup \{H+x\}\) is less than or equal to \(2 \alpha\).