We study a method of approximating representations of the group $M\left(n\right)$ by those of the group $SO(n+1)$. As a consequence we establish a version of a theorem of DeLeeuw for Fourier multipliers of $L^p$ that applies to the “restrictions” of a function on the dual of $M\left(n\right)$ to the dual of $SO(n+1)$.

Let $G$ be a locally compact group, for $p\in (1,\infty )$ let $P{f}_p\left(G\right)$ denote the closure of $L^1\left(G\right)$ in the convolution operators on $L^p\left(G\right)$. Denote $W_p\left(G\right)$ the dual of $P{f}_p\left(G\right)$ which is contained in the space of pointwise multipliers of the Figa-Talamanca Herz space $A_p\left(G\right)$. It is shown that on the unit sphere of $W_p\left(G\right)$ the $\sigma (W_p,P{f}_p)$ topology and the strong $A_p$-multiplier topology coincide.

We study the problem of $L^p$-boundedness ($1\<p\<\infty$) of operators of the form $m(L_1,\cdots ,L_n)$ for a commuting system of self-adjoint left-invariant differential operators $L_1,\cdots ,L_n$ on a Lie group $G$ of polynomial growth, which generate an algebra containing a weighted subcoercive operator. In particular, when $G$ is a homogeneous group and $L_1,\cdots ,L_n$ are homogeneous, we prove analogues of the Mihlin-Hörmander and Marcinkiewicz multiplier theorems.

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.

Soient $G_1$ et $G_2$ deux groupes abéliens localement compacts de dual ${\Gamma }_1$ et ${\Gamma }_2$. Soit $h:{\Gamma }_1\to {\Gamma }_2$ un homomorphisme continu d’image dense de ${\Gamma }_1$ dans ${\Gamma }_2$. Soit $1\le p\le \infty$ ; on prouve un théorème d’approximation des multiplicateurs de $\mathbf{F}{L}^p(G_2)$ et on utilise ce résultat pour démontrer le suivant : soit $m:{\Gamma }_2\to \mathbf{C}$ une fonction continue ; $m$ est un multiplicateur de $\mathbf{F}{L}^p(G_2)$ si, et seulement si, $m\circ h$ est un multiplicateur de $\mathbf{F}{L}^p(G_1)$.

We find representations for the automorphisms, derivations and multipliers of the Fréchet algebra $L{¹}_{loc}$ of locally integrable functions on the half-line ${ℝ}^{+}$. We show, among other things, that every automorphism θ of $L{¹}_{loc}$ is of the form $\theta ={\phi }_a{e}^{\lambda X}{e}^D$, where D is a derivation, X is the operator of multiplication by coordinate, λ is a complex number, a > 0, and ${\phi }_a$ is the dilation operator $\left({\phi }_{a}f\right)\left(x\right)=af\left(ax\right)$ ($f\in L{¹}_{loc}$, $x\in {ℝ}^{+}$). It is also shown that the automorphism group is a topological group with the topology of uniform convergence on bounded...