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Let G be a locally compact group, M(G) denote its measure algebra and L^1(G)denote its group algebra. Also, let pi:G->U(H) be a strongly continuous unitaryrepresentation, and let CB^{sigma}(B(H)) be the space of normal completelybounded maps on B(H). We study the range of the mapGamma_pi:M(G)->CB^sigma(B(H)), Gamma_pi(mu)= int_G pi(s)\otimes pi(s)^*dmu(s)where we identify CB^sigma(B(H)) with the extended Haagerup tensor productB(H)\otimes^{eh}B(H)$. We use the fact that the C*-algebra generated byintegrating pi to L^1(G) is unital exactly when pi is norm continuous to showthat Gamma_pi(L^1(G))\subset B(H)\otimes^{eh}B(H) exactly when pi is normcontinuous. For the case that G is abelian, we study Gamma_pi(M(G)) as a subsetof the Varopoulos algebra. We also characterise positive definite elements ofthe Varopoulos algebra in terms of completely positive operators.

Representations of group algebras in spaces of completely bounded maps