We present a unified approach to the processes of inversion and duality for quasilinear and $1$-quasilinear maps; in particular, for centralizers and differentials generated by interpolation methods.

We study the stability of the differential process of Rochberg and Weiss associated with an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces, we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families of up to three Köthe spaces distributed in arcs on the unit circle while there is no (bounded) stability for families of four or more Köthe spaces. In the context of arbitrary pairs of Banach spaces, we present some local stability results and some global isometric stability results.

In this article, via the classical complex interpolation method and some interpolation methods traced to Gagliardo, the authors obtain an interpolation theorem for Morrey spaces on quasimetric measure spaces, which generalizes some known results on ${{\mathbb{R}}^{n}}$.

We show that the multiplier algebra of the Fourier algebra on a locally compact group $G$ can be isometrically represented on a direct sum on non-commutative ${{L}^{p}}$ spaces associated with the right von Neumann algebra of $G$. The resulting image is the idealiser of the image of the Fourier algebra. If these spaces are given their canonical operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the noncommutative ${{L}^{p}}$ spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca–Herz algebra built out of these non-commutative ${{L}^{p}}$ spaces, say ${{A}_{p}}(\hat{G})$. It is shown that ${{A}_{2}}(\hat{G})$ is isometric to ${{L}^{1}}(G)$, generalising the abelian situation.

A recent survey article by J. Fournier and J. Stewart (Bull.AMS 13 (1985), 1-21) explains how amalgams of Lp with lq (as function spaces over any locally compact abelian group G) can be used as an effective tool for the treatment of various problems in harmonic analysis. The present article may be seen as a complement to this survey, indicating further advantages that arise if one works with generalized amalgams (introduced in 1980 under the name of Wiener-type spaces by the author [10]). The main difference between amalgams and these more general spaces is the fact that they allow a more precise description of the local behavior of functions (or distributions) by rather arbitrary norms and that the conditions on the global behavior (of the quantity obtained using that chosen local norm) is described in a way that includes both growth and integrability conditions (not only lq-summability).