Representations of group algebras in spaces of completely bounded maps R. SmithNico Spronk 46L0722D2022D1022D2522B05group algebracompletely bounded mapextended Haagerup tensor product Let $G$ be a locally compact group, $\pi:G\to\mathcal{U}(\mathcal{H})$ be a strongly continuous unitary representation, and $\mathcal{C}\mathcal{B}^{\sigma}(\mathcal{B}(\mathcal{H}))$ the space of normal completely bounded maps on $\mathcal{B}(\mathcal{H})$. We study the range of the map $$\Gamma_{\pi}:\mathrm{M}(G)\to\mathcal{C}\mathcal{B}^{\sigma}(\mathcal{B}(\mathcal{H})),\quad \Gamma_{\pi}(\mu)=\int_{G}\pi(s)\otimes\pi(s)^{*}\,d\mu(s),$$ where we identify $\mathcal{C}\mathcal{B}^{\sigma}(\mathcal{B}(\mathcal{H}))$ with the extended Haa\-ger\-up tensor product $\mathcal{B}(\mathcal{H})\otimes^{eh}\mathcal{B}(\mathcal{H})$. We use the fact that the $\mathrm{C}^{*}$-algebra generated by integrating $\pi$ to $\mathrm{L}^{1}(G)$ is unital exactly when $\pi$ is norm continuous, to show that $\Gamma_{\pi}(\mathrm{L}^{1}(G))\subset\mathcal{B}(\mathcal{H})\otimes^{h}\mathcal{B}(\mathcal{H})$ exactly when $\pi$ is norm continuous. For the case that $G$ is abelian, we study $\Gamma_{\pi}(\mathrm{M}(G))$ as a subset of the Varopoulos algebra. We also characterize positive definite elements of the Varopoulos algebra in terms of completely positive operators. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2551 10.1512/iumj.2005.54.2551 en Indiana Univ. Math. J. 54 (2005) 873 - 896 state-of-the-art mathematics http://iumj.org/access/