I am reading an appendix on Group algebras which contains the following Proposition which I am trying to prove:

Proposition: Let $G$ be a locally compact group, with $\zeta\in L^{p}(G)$ fixed. Define $L_{x}\in L^{p}(G)$ by $L_{x}(\zeta)[\lambda] = \zeta(x\lambda)$. The map $$G\mapsto L^{p}(G), x\mapsto L_{x}(\zeta)$$ is continuous.

My Proof Attempt:

It was not obvious to me that this map is well-defined; but I think I have verified it below. Can someone check my logic and confirm I am using the notation correctly? My measure theory background always leaves me feeling uneasy when abstract measure space show up.

Let $\mu$ denote the left Haar measure for $G$.

$\begin{align*} \int_{G}|L_{x}(\zeta)|^{p}d\mu &= \int_{G}|\zeta(xy)|^{p}\mu(dy)\\ &= \int_{G}|\zeta(y)|^{p}\mu(x\cdot dy)\text{, (transformation property)}\\ &= \int_{G}|\zeta(y)|^{p}\mu(dy)\text{, (by Left }G\text{-invariance)}\\ &= \|\zeta\|_{p}^{p} <\infty\text{, because }\zeta\in L^{p}(G) \end{align*}$

Therefore $L_{x}(\zeta)\in L^{p}(G)$, so the map is well-defined.

Now to show the map is continuous, it suffices by $G$-invariance of $\mu$ to show that whenever $x_{\alpha}\to 1$, we have $\|L_{x_{\alpha}}(\zeta) - \zeta\|_{p}\to 0$.

I'm stuck here though, as I have no idea how to estimate $\int_{G}|\zeta(x_{\alpha}y) - \zeta(y)|^{p}d\mu$. What is the trick?

  • 3
    $\begingroup$ Can you do it for the characteristic function of an open set? Notice that is enough, for they span $L^p$. $\endgroup$ – Mariano Suárez-Álvarez May 12 '13 at 2:57
  • $\begingroup$ Yes I think I can, thanks for the hint. $\endgroup$ – roo May 12 '13 at 3:24
  • $\begingroup$ Alternatively, show that the action of $G$ on $C_c(G)$ is continuous with respect to the sup-norm since functions in $C_c(G)$ are uniformly continuous. Then use that $C_c(G)$ is dense in $L^p$ for $p \lt \infty$. A more detailed outline is here. $\endgroup$ – Martin May 14 '13 at 1:06
  • $\begingroup$ Thank you very much for the reference. I'll try that approach. $\endgroup$ – roo May 14 '13 at 18:16

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