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Regularity of group algebras. (English. Russian original) Zbl 1191.46045
Sb. Math. 200, No. 8, 1165-1179 (2009); translation from Mat. Sb. 200, No. 8, 63-78 (2009).
A Banach algebra \(A\) is said to be a Wermer algebra if there exists a locally compact Abelian group \(G\) with the following properties:
(i) \(C_{00}(G)\subset A\subset L^1(G)\), where both embeddings are continuous and \(C_{00}(G)\) is dense in \(A\);
(ii) \(A\) is translation invariant and the map \((t,f)\mapsto\tau_t(f)\) from \(G\times A\) into \(A\) is continuous;
(iii) \(\sup_{t\in G}\|\tau_t\|\int_{tK}| f|<\infty\) for each \(f\in A\) and each compact subset \(K\) of \(G\);
(iv) \(\|\tau_t\|\geq 1\) for each \(t\in G\).
Here, \(\tau_t\) stands for the translation operator corresponding to \(t\in G\). The weighted \(L^p\)-algebras are Wermer algebras.
The author proves that a Wermer algebra \(A\) is regular if and only if \(\sum_{n=1}^\infty\frac{\log\|\tau_{na}\|}{n^2}<\infty\).

MSC:
46J10 Banach algebras of continuous functions, function algebras
46J20 Ideals, maximal ideals, boundaries
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
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