Module homomorphism

Question

Let $A$ be a Banach algebra with norm $\|.\|_A$ and $X$ be a Banach space with norm $\|.\|_X$. If there exists a operation $.:A\times X\to X$ such that for any $a,b\in A$ and $x,y\in X$ we have

1. $(a+b).x=a.x+b.x$,
2. $a.(x+y)=a.x+a.y$,
3. $(ab).x=a.(b.x)$,
4. $\|a.x\|_X\le\|a\|_A\|x\|_X$

then $X$ is called left Banach $A$-module. For example $A$ itself is a left Banach $A$-module by its algebraic product.

If $X,Y$ are two left Banach $A$-module the bounded linear operator $\phi:X\to Y$ is called left $A$-module homomorphism if $\phi(a.x)=a.\phi(x)$ for any $a\in A$ and $x\in X$.

Let $G$ be a locally compact group, consider the convolution group algebra $L^1(G)$, and left $L^1(G)$-modules $L^P(G)$ for $1< p\le\infty$. Is there any left $L^1(G)$-module homomorphism from $L^p(G)$ to $L^1(G)$?

Answer 1

When $G$ is not compact, there is no homomorphism. See MR0244764 (39 #6078) Reviewed Rieffel, Marc A. Multipliers and tensor products of Lp-spaces of locally compact groups. Studia Math. 33 1969 7182.