Representations of $p$-convolution algebras on $L^q$-spaces

  title={Representations of \$p\$-convolution algebras on \$L^q\$-spaces},
  author={Eusebio Gardella and Hannes Thiel},
  journal={Transactions of the American Mathematical Society},
For a nontrivial locally compact group $G$, and $p\in [1,\infty)$, consider the Banach algebras of $p$-pseudofunctions, $p$-pseudomeasures, $p$-convolvers, and the full group $L^p$-operator algebra. We show that these Banach algebras are operator algebras if and only if $p=2$. More generally, we show that for $q\in [1,\infty)$, these Banach algebras can be represented on an $L^q$-space if and only if one of the following holds: (a) $p=2$ and $G$ is abelian; or (b) $|\frac 1p - \frac 12|=|\frac… 
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