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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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