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We study property ($T$) and the fixed-point property for actions on$L$^{$p$}and other Banach spaces. We show that property ($T$) holds when$L$^{2}is replaced by$L$^{$p$}(and even a subspace/quotient of$L$^{$p$}), and that in fact it is independent of 1≤$p$<∞. We show that the fixed-point property for$L$^{$p$}follows from property ($T$) when 1<$p$< 2+$ε$. For simple Lie groups and their lattices, we prove that the fixed-point property for$L$^{$p$}holds for any 1<$p$<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces.

A new proof is given for a theorem by V. G. Maz'ya. It gives a necessary and sufficient condition on the open set Ω in$R$^{$N$}for the functions in$W$_{0}^{$m,p$}(Ω) to have the ordinary norm equivalent to the norm obtained when including only the highest order derivatives in the definition. The proof is based on a kind of polynomial capacities, Maz'ya capacities.

After recalling briefly the main properties of the amalgamated duplication of a ring$R$along an ideal$I$, denoted by $R\bowtie I$ , see M.$D’Anna$and M.$Fontana$, to appear in$J. Algebra Appl.$, we restrict our attention to the study of the properties of $R\bowtie I$ , when$I$is a multiplicative canonical ideal of$R$, see W. J.$Heinzer$, J. A.$Huckaba$and I. J.$Papick$,$Comm. Algebra$. In particular, we study when every regular fractional ideal of $R\bowtie I$ is divisorial.

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