The notions of approximate amenability and weak amenability in Banach algebras are formally stronger than that of approximate weak amenability. We demonstrate an example confirming that approximate weak amenability is indeed actually weaker than either approximate or weak amenability themselves. As a consequence, we examine the (failure of) approximate amenability for -sums of finite-dimensional normed algebras.
We study the structure of Lipschitz algebras under the notions of approximate biflatness and Johnson pseudo-contractibility. We show that for a compact metric space , the Lipschitz algebras and are approximately biflat if and only if is finite, provided that . We give a necessary and sufficient condition that a vector-valued Lipschitz algebras is Johnson pseudo-contractible. We also show that some triangular Banach algebras are not approximately biflat.
We discuss the convergence of approximate identities in Musielak-Orlicz spaces extending the results given by Cruz-Uribe and Fiorenza (2007) and the authors F.-Y. Maeda, Y. Mizuta and T. Ohno (2010). As in these papers, we treat the case where the approximate identity is of potential type and the case where the approximate identity is defined by a function of compact support. We also give a Young type inequality for convolution with respect to the norm in Musielak-Orlicz spaces.
In this paper, we shall study contractive and pointwise contractive Banach function algebras, in which each maximal modular ideal has a contractive or pointwise contractive approximate identity, respectively, and we shall seek to characterize these algebras. We shall give many examples, including uniform algebras, that distinguish between contractive and pointwise contractive Banach function algebras. We shall describe a contractive Banach function algebra which is not equivalent to a uniform algebra....
We continue our study of derivations, multipliers, weak amenability and Arens regularity of Segal algebras on locally compact groups. We also answer two questions on Arens regularity of the Lebesgue-Fourier algebra left open in our earlier work.
The aim of the paper is twofold. First we give a survey of some recent results concerning the asymptotic behavior of the entropy and approximation numbers of compact Sobolev embeddings. Second we prove new estimates of approximation numbers of embeddings of weighted Besov spaces in the so called limiting case.