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An arbitrary intersection of \(L_{p}\)-spaces. (English) Zbl 1246.43001

In this paper for a locally compact group \(G\) and a subset \(J\) of \([1, \infty]\) the space \(IL_J (G) = \{f \in \bigcap_{p\in J} L^p (G); \sup_{p\in J} \| f \| p < \infty\}\) is introduced as a Banach space in \(\bigcap_{p\in J}L^{p}(G)\), equipped with the norm \(\| f \|{}_{{IL}_{J}} = \sup_{p\in J} \| f \| p\). Moreover, \(IL_{J}(G)\) is studied as a Banach algebra under a convolution product, and it is proven that the existence of a bounded approximate identity in \(IL_{J}(G)\) is equivalent to the discreteness of \(G\). Some results on the amenability, weak amenability and approximate amenability of \(IL_{J}(G)\) are given. Finally, under some conditions it is shown that \(IL_{J}(G)\) can be considered as an ideal in its second dual if and only if \(G\) is compact.

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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